\documentclass[JEP,XML,SOM,Unicode]{cedram} \datereceived{2025-05-07} \dateaccepted{2026-10-02} \dateepreuves{2026-10-13} \usepackage[scr=rsfs,cal=euler]{mathalfa} \multlinegap0pt \graphicspath{{hebert-philippe_illustrations}} \hyphenation{assu-me coho-mo-lo-gy data datum defined dia-gram dia-grams exact-ly impor-tant inde-pen-dently remark remarks sequences variety varieties} \makeatletter \def\@settitle{% \vspace*{-8mm} \raggedleft\includegraphics[scale=.5]{titre-jep} \vtop to 50 mm{% \parindent=0pt {\abstractfont\article@logo\par} \medskip \hrule \vfil \begin{center} \def\baselinestretch{1.2}\large\vfil {\didottitraille\MakeUppercase\@title\par} \vfil\vfil \begin{minipage}{.8\textwidth}\centering \ifx\@empty\smfbyname\else {\smf@byfont\smfbyname\ifsmf@byauthor\enspace\else\ \fi}% \fi {\smf@authorfont \edef\smfandname{{\noexpand\smf@andfont \smfandname}} \andify\authors\authors\par} \end{minipage} \vfil \vrule height .4pt width .3\textwidth \vfil \end{center}}% \par\enlargethispage{.5\baselineskip}% } \def\@setthanks{\def\thanks##1{\par##1\@addpunct{{\upshape.}}}\vspace*{-5pt}\thankses} \makeatother \newenvironment{enumeratei} {\bgroup\def\theenumi{\roman{enumi}}\def\theenumii{\arabic{enumii}}\begin{enumerate}} {\end{enumerate}\egroup} \let\oldbigsqcup\bigsqcup \renewcommand{\bigsqcup}{\mathop{\textstyle\oldbigsqcup}\displaylimits} \newcommand{\quand}{\quad\text{and}\quad} \newcommand{\asr}{\mathrm{asr}} \newcommand{\cov}{\mathrm{cov}} \newcommand{\cocov}{\mathrm{cocov}} \usepackage{stmaryrd} \usepackage[nonewpage]{imakeidx} \makeindex[columns=4,columnsep=20pt] \usepackage[ ordering=Kac, edge/.style=blue, indefinite-edge={draw=green,fill=white,densely dashed}, indefinite-edge-ratio=5, mark=o, root-radius=.06cm] {dynkin-diagrams} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \newtheorem{Definition}[Theorem]{Definition} \newtheorem{Lemma}[Theorem]{Lemma} \theoremstyle{definition} \newtheorem{Definitions}[Theorem]{Definitions} \newtheorem{Example}[Theorem]{Example} \newtheorem{Remark}[Theorem]{Remark} \let\qp\varpi \newcommand{\A}{\mathbb{A}} \newcommand{\G}{\mathbb{G}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\htt}{\mathrm{ht}} \newcommand{\Inv}{\mathrm{Inv}} \newcommand{\pr}{\mathrm{proj}} \newcommand{\reg}{\mathrm{reg}} \newcommand{\sph}{\mathrm{sph}} \newcommand{\supp}{\mathrm{Supp}} \newcommand{\vv}{\mathrm{v}} \newcommand{\cC}{\mathcal{C}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cQ}{\mathcal{Q}} \newcommand{\sT}{\mathscr{T}} \newcommand{\bt}{\boldsymbol{t}} \newcommand{\bx}{\boldsymbol{x}} \newcommand{\by}{\boldsymbol{y}} \newcommand{\bz}{\boldsymbol{z}} \DeclareMathOperator{\sgn}{sgn} \begin{document} \frontmatter \title[Quantum roots for Kac--Moody root systems]{Quantum roots for Kac--Moody root systems and finiteness properties of the\nobreakspace Kac--Moody\nobreakspace affinized Bruhat order} \author[\initial{A.} \lastname{Hébert}]{\firstname{Auguste} \lastname{Hébert}} \address{\\Université de Lorraine, Institut Élie Cartan de Lorraine, CNRS, UMR 7502,\\ F-54000 Nancy, France} \email{auguste.hebert@univ-lorraine.fr} \author[\initial{P.} \lastname{Philippe}]{\firstname{Paul} \lastname{Philippe}} \address{Universität Münster,\\ Einsteinstr. 62, 48149 Münster, Germany} \email{paul.philippe@uni-muenster.de} \begin{abstract} The notion of quantum root comes up in the study of the quantum cohomology of generalized flag varieties, quantum Bruhat graphs or in the study of the affinized Bruhat order on an affinized Weyl semi-group. In this paper, we~provide a complete classification of quantum roots associated to general Kac–Moody root systems. In particular we prove that the set of quantum roots of a given root system is always finite. We then deduce several finiteness properties of the affinized Bruhat order, generalizing results of A.\,Welch. In particular, we~prove that its intervals are finite. This affinized Bruhat order was introduced by A.\,Braverman, D.\,Kazhdan and M.\,Patnaik in their study of Kac-Moody groups over discretely valued fields, and of the associated Iwahori-Hecke algebras. \end{abstract} \subjclass{20G44, 17B22, 20C08} \keywords{Kac--Moody groups, root systems, Bruhat order, Hecke algebras} \altkeywords{Groupes de Kac-Moody, systèmes de racines, ordre de Bruhat, algèbres de Hecke} \alttitle{Racines quantiques pour les systèmes de racines de Kac-Moody et propriétés de finitude de l’ordre de Bruhat affine de Kac-Moody} \begin{altabstract} Au sein d’un système de racines, les racines quantiques jouent un rôle particulier. Dans le cadre réductif elles aparaissent notamment dans l’étude de la cohomologie quantique des variétés de drapeaux, et la structure combinatoire des groupes de Weyl affines. Nous classifions les racines quantiques des systèmes de racines de Kac-Moody de type général, nous montrons en particulier que chaque système de racines n’en contient qu’un nombre fini. Nous en déduisons plusieurs propriétés de finitude pour l’ordre de Bruhat affinisé sur le semi-groupe de Weyl affinisé, généralisant des résultats de A.\,Welch. Nous montrons notamment que les intervalles sont finis. Cet ordre de Bruhat affinisé avait été introduit par A.\,Braverman, D.\,Kazhdan et M.\,Patnaik pour étudier les groupes de Kac-Moody sur les corps discrètement valués, et les algèbres de Iwahori-Hecke associées. \end{altabstract} \maketitle \vspace*{-\baselineskip} \tableofcontents \mainmatter \vspace*{-3.5\baselineskip}\mbox{} \section{Introduction} Kac--Moody root systems are particular arrangements of vectors in a free $\Z$-lattice, whose structure is encoded in a generalized Cartan matrix; they appear in various contexts in combinatorics and algebra. For instance, finite Kac--Moody root systems arise from the theory of reductive Lie algebras and reductive groups (such as $\mathfrak{sl}_n$ and~$\operatorname{SL}_n$), and the infinite Kac--Moody root systems allow one to define their infinite-dimensional counterparts, namely Kac--Moody Lie algebras and Kac--Moody groups. Certain roots within a given root system, called quantum roots, are of particular \hbox{interest}. Without being named as such, quantum roots of reductive root systems first came up in the computation of the quantum cohomology of flag manifolds and related varieties (\cite{brenti1999mixedbruhat,braverman2011quantum}). They are intrinsically connected to the combinatorics of affine Weyl groups (this is already apparent in Peterson's lectures on quantum cohomology \cite[Lect.\,12]{peterson1997lectures}) and they are essential to the definition of quantum Bruhat graphs (introduced by Postnikov \cite{postnikov2004quantum}). The term ``quantum root'' seems to first appear in the work of Lenart et al. \cite{lenart2012uniform}, in~which they apply the quantum Bruhat graph to the analysis of certain Kirillov--Reshetikhin crystals. More recently, in~his doctoral thesis, Schremmer \cite{schremmer2022thesis} established a wide range of further applications of quantum Bruhat graphs associated to reductive root systems. The notion of quantum root is also related to the notion of minuscule element defined by D.\,Peterson (\cite{stembridge2001minuscule}); see Proposition~\ref{p_minuscule}. The latter concept has applications in various areas, for example in the theory of Kac--Moody homogeneous spaces (see \cite{chaput2012littlewood}) or in the study of Schubert varieties (see \cite[Ch.\,9]{billey2000singular}), to cite but a~few. In this paper, we~study quantum roots associated with (infinite) Kac--Moody (\hbox{infinite}) root systems. We~provide a complete classification of these roots, and, in~particular, we~prove that the set of quantum roots is always finite. As observed by the second author in a previous work \cite[Prop.\,3.19]{philippe2023grading}, describing these roots is important for understanding the affinized Bruhat order on the Kac--Moody affine Weyl semi-group. This is a combinatorial object which naturally replaces the extended affine Weyl group when looking at non-reductive Kac--Moody groups over discretely valued fields. In~the second part of this paper, we~deduce several finiteness properties of the affinized Weyl semi-group, which we use in a follow-up work to define and compute affine Kazhdan--Lusztig $R$-polynomials for Kac--Moody groups \cite{hebert2024affine}. Independently, Dean has considered quantum Bruhat graphs associated to Kac--Moody root systems \cite{dean2025quantum}, mainly motivated by the definition of a Demazure product on affinized Weyl semi-groups. This relies on properties of the quantum Bruhat graph which remain conjectural in the non-reductive setting, but have been proven by Dean in affine type~$A_1$ affine. This type can be worked out very explicitly, especially because only the two simple roots are quantum (this is a particular case of Lemma~\ref{l_no_-1_coefficient}). We~expect a good understanding of quantum roots to be necessary to extend these results to the general Kac--Moody setting. Let us now introduce more precisely the main objects of study and the results presented in this paper. \subsubsection*{Quantum roots} Let $\Phi,\Phi^\vee$ be a (free and cofree) Kac--Moody real root system and its dual. Let $(\alpha_i)_{i\in I}$ and $(\alpha_i^\vee)_{i\in I}$ be its simple roots and coroots, where $I$ is a finite index set. There is a natural bijection $\Phi\rightarrow \Phi^\vee: \alpha\mapsto \alpha^\vee$ which sends $\alpha_i$ to $\alpha_i^\vee$. Each coroot $\alpha^\vee$ decomposes uniquely as $\alpha^\vee=\varepsilon\sum_{i\in I} n_i \alpha_i^\vee$ with $\varepsilon\in \{1,-1\}$ and~$(n_i)_{i\in I}\in \mathbb N^I$. The height of $\alpha^\vee$ is then defined as $\htt(\alpha^\vee):=\varepsilon\sum_{i\in I}n_i$. Intuitively, the height measures the size of a given (co)root; for instance, a root is simple if and only if its coroot has height $1$. Another quantity associated with such a system can be defined via its vectorial Weyl group $W^v$. This object is a Coxeter group whose set of simple reflections is in bijection with $(\alpha_i)_{i\in I}$. It acts on $\Phi$ (and dually on $\Phi^\vee$) and we have $\Phi=W^v\cdot\{\alpha_i\}_{i\in I}$. To any root $\alpha$ is associated a unique reflection $s_\alpha$, whose length $\ell(s_\alpha)$ also measures the size of $\alpha$. In general, we~have that \begin{equation}2\htt(\alpha^\vee)\geq \ell(s_\alpha)+1\end{equation} by Lemma~\ref{l_inequality_length}. We~say that a root is quantum when equality holds. While there is a concise characterization of quantum roots in reductive root systems (see \cite[Lem.\,4.2]{lenart2012uniform}), the situation seems very different in the non-reductive case. For example, every positive root is quantum in a reductive simply laced root system, but this property fails even for the simplest infinite root systems. Nonetheless, we~provide a complete classification of quantum roots using sequences of Dynkin diagrams. As a corollary, we~obtain: \begin{Theorem}[see Theorem~\ref{t_finiteness_almost_simple_roots}]\label{Theorem: finiteness quantum roots intro} Any Kac--Moody root system admits finitely many quantum roots. \end{Theorem} From Theorem~\ref{Theorem: finiteness quantum roots intro} we deduce finiteness properties for the affinized Bruhat order. In~the following paragraphs we introduce this affinized Bruhat order and explain its connection with quantum roots. \subsubsection*{Affine Weyl semi-group $W^+$} In their study of Kac--Moody groups over non-Archimedean local fields, A.\,Braverman, D.\,Kazhdan and M.\,Patnaik consider a semi-group $W^+$, called the affinized Weyl semi-group. It is defined as follows. When~$\Phi$ is the root system of a Kac--Moody group $\mathbb G$ (or of a Kac--Moody Lie algebra~$\mathfrak g$), $\Phi^\vee$ lies in the cocharacter lattice $Y$ and we have $\Phi\subset \operatorname{Hom}_\mathbb Z(Y,\Z)$. The action of $W^v$ extends to $Y$. The cocharacter lattice admits a $W^v$-stable subcone $Y^+$, called its integral Tits cone, which plays a role in the representation theory of the associated Kac--Moody Lie algebra. The affinized Weyl semi-group is then defined as $W^+:=Y^+\rtimes W^v$. When $\Phi$ is associated to a reductive group (and hence is finite), then $Y^+=Y$ $W^+$ is the extended affine Weyl group associated to $\mathbb G$. It is a finite extension of a Coxeter group, which is very useful for studying $\mathbb G$ over non-Archimedean local fields. In~the general Kac--Moody case, we~expect $W^+$ to play an analogous role but its structure is less well understood. It is not a group, so~it does not admit any Coxeter structure, but a few key notions of Coxeter theory have been extended to $W^+$. One of these notions is the Bruhat-like partial pre-order $<$ defined by Braverman, Kazhdan and Patnaik in \cite[App.\,B2]{braverman2016iwahori}. In~\cite{muthiah2018iwahori}, D.\,Muthiah proves that this relation is a partial order, and in \cite{muthiah2019bruhat} Muthiah and D.\,Orr define a strictly compatible $\mathbb Q$-valued length function $\ell^a:W^+\rightarrow \Q$. We~call this partial order the affinized Bruhat order. \subsubsection*{Covers for the affinized Bruhat order and quantum roots} The affinized Bruhat length gives a characterization of covers: Muthiah and Orr (in the affine ADE case) and the second-named author (in the general case) have proved that $\by$ covers $\bx$ (see Definition~\ref{Definition: covers}) if and only if $\by>\bx$ and $\ell^a(\by)=\ell^a(\bx)+1$ (\cite{muthiah2019bruhat}, \cite{philippe2023grading}). The notion of quantum roots naturally comes up in a more explicit description of covers for the affinized Bruhat order. This is sketched in Lemma~\ref{Lemma: form of covers} (see \cite[Prop.\,3.20]{philippe2023grading} for a complete statement). Therefore our classification of quantum roots also provides insight into the covers for the affinized Bruhat order. In~particular, using Theorem~\ref{Theorem: finiteness quantum roots intro}, we~prove: \begin{Theorem}[see Theorem~\ref{Theorem: finiteness of covers} and Theorem~\ref{thm_finiteness_co-covers}]\label{thm_finiteness_intro} Let $\bx\in W^+$. Then $\bx$ admits finitely many covers. If~furthermore $\bx$ is spherical, i.e., the fixator in $W^v$ of the associated cocharacter is finite, then $\bx$ also admits finitely many co-covers. \end{Theorem} \begin{Corollary}\label{Corollary: finiteness intervals introduction} Let $\bx,\by\in W^+$. Then the interval $[\bx,\by]$ is finite. \end{Corollary} In the reductive case, these theorems are easily obtained by considering reduced decompositions. In~the Kac--Moody framework, these theorems were first proved by A.\,Welch in the untwisted affine ADE case, using different methods in \cite{welch2022classification}. When $W^v$ admits an infinite proper parabolic subgroup (this implies that $\G$ is not affine), we~also prove the existence of elements of $W^+$ admitting infinitely many co-covers, see Lemma~\ref{l_reciprocal_thm_cocovers}. Our main motivation for the study of the combinatorics of $W^+$ is the construction of an affine Kazhdan--Lusztig theory for Kac--Moody groups. This theory was initiated by Muthiah in \cite{muthiah2019double}, where he uses Corollary~\ref{Corollary: finiteness intervals introduction} to give a conjectural definition of spherical $R$-polynomials in the affine ADE case. We~use our results to define an Iwahori version of Muthiah's polynomials in the general Kac--Moody framework (see \cite{hebert2024affine}). \subsubsection*{Organization of the paper} In Section~\ref{s_almost_simple_roots}, we~define the quantum roots and we prove that the set of quantum roots of a Kac--Moody datum is finite (see Theorem~\ref{t_finiteness_almost_simple_roots}). We~also obtain a general classification of quantum roots (see Theorem~\ref{Theorem: Classification quantum roots}). We~then describe the set of quantum roots in particular cases. In Section~\ref{s_almost_simple_roots_covers}, we~explain the link between covers for the affinized Bruhat order and quantum roots. We~then prove Theorem~\ref{thm_finiteness_intro}. We~also show that the spherical condition in Theorem~\ref{thm_finiteness_intro} is necessary by constructing non-spherical elements with infinitely many co-covers (see Proposition~\ref{Proposition: nonspherical infinite cocovers}). We~end the section with an explicit construction of covers associated to any given quantum root, thus showing that the notion of quantum root is well-suited. Note that this section uses the finiteness of the set of quantum roots (Theorem~\ref{Theorem: finiteness quantum roots intro}) but is largely independent of Section~\ref{s_almost_simple_roots}: for the reader willing to admit this theorem, the only prerequisites for this section are contained in Section~\ref{ss_gen_def_not}. \subsubsection*{Acknowledgements} We thank Dinakar Muthiah and Stéphane Gaussent for helpful conversations on the subject. We~thank Pierre-Emmanuel Chaput for pointing out to us the notion of minuscule elements and for discussions on this topic. We~thank the referees for their careful reading, valuable comments and suggestions. \tableofcontents \section{Quantum roots}\label{s_almost_simple_roots} \subsection{General definitions and notations}\label{ss_gen_def_not} We let $\N\subset \mathbb \Z$ denote the set of non-negative integers. For $a,b\in \Z$, we~denote by $\llbracket a,b\rrbracket$ the integer interval $[a,b]\cap \Z$. \Subsubsection*{Generalized Cartan matrices and Dynkin diagrams} \begin{Definition} Let $I$ be a finite set. A \emph{generalized Cartan matrix} over $I$ is a matrix $A=(a_{i,j})_{(i,j)\in I\times I}\in M_I(\mathbb Z)$ with integral coefficients satisfying the following properties: \begin{enumerate} \item $\forall i\in I \; a_{i,i}=2$ \item $\forall (i,j)\in I^2,\; i\neq j \implies a_{i,j}\leq 0$ \item $\forall (i,j)\in I^2,\; a_{i,j}=0 \implies a_{j,i}= 0$. \end{enumerate} \end{Definition} \begin{Definition} A \emph{Dynkin diagram} is a tuple $(I,E,w)$ where $I$ is a finite set (its vertices), $E$ is a symmetric subset of $I\times I \setminus \{(i,i)\mid i \in I\}$ (its edges) and $w$ is a map $E \rightarrow \mathbb Z_{>0}$ (its weight function). In~other words, it is a positively weighted directed graph such that $(i,j) \in E \iff (j,i) \in E$. Let $\Gamma=(I,E,w)$ be a Dynkin diagram. Then any subset $J\subset I$ admits an induced Dynkin diagram structure: $(J,E\cap (J\times J), w|_J)$, where $w|_J$ is the restriction of $w$ to $E\cap (J\times J)$. By~a subdiagram (or subgraph) of $\Gamma$, we~mean any Dynkin diagram of this form. A \emph{leaf} in a Dynkin diagram is any vertex which lies in at most one unoriented edge. That is to say, it is any $i\in I$ for which there is at most one $j\in I$ such that $(i,j)\in E$. \end{Definition} Dynkin diagrams are graphical representations of generalized Cartan matrices: suppose that $A=(a_{i,j})_{(i,j)\in I\times I}$ is a generalized Cartan matrix indexed by $I$. Then, setting $E=\{(i,j)\in I^2 \mid a_{i,j}<0\}$ and $w: (i,j)\mapsto -a_{i,j}$, we~obtain a Dynkin diagram $(I,E,w)$. Conversely, to any Dynkin diagram $(I,E,w)$ we associate a generalized Cartan matrix $(a_{i,j})_{(i,j)\in I\times I }$ indexed by $I$, setting \[ a_{i,j}=\begin{cases}2 &\text{ if } i=j, \\ -w(i,j) &\text{ if } (i,j)\in E, \\ 0 &\text{ otherwise.} \end{cases} \] \begin{Definition}[Connectivity and $1$-star-convexity] A Dynkin diagram $(I,E,w)$ is said to be \emph{connected} if it is connected as a graph. That is to say, if, for any $i,j\in I$, there is a sequence $i=i_0,i_1,\dots,i_n=j$ such that $(i_k,i_{k+1})\in E$ for all $k\in \llbracket0,n-1\rrbracket$. For a general Dynkin diagram $\Gamma$, a \emph{connected component} of $\Gamma$ is any connected subdiagram which is maximal for the inclusion. Therefore a Dynkin diagram is connected if and only if it has exactly one connected component. A connected Dynkin diagram is a \emph{Dynkin tree} if it does not contain any closed circuit: for any sequence $i_0,\dots, i_n$ such that $(i_k,i_{k+1})\in E$ for all $k\in \llbracket0,n-1\rrbracket$ and $i_{k-1}\neq i_{k+1}$ for all $k \in \llbracket1,n-1\rrbracket$, we~have $i_0\neq i_n$. A Dynkin tree is a \emph{Dynkin segment} if there are at most two edges going out of any given vertex. We introduce a refinement of the notion of connectivity: we say that a Dynkin diagram $\Gamma=(I,E,w)$ is \emph{$1$-star-convex} if there exists a vertex $i_0\in I$ such that, for any $j\in I$, there is a sequence $(i_1,\dots,i_n)$ with $i_n=j$ such that, for all $k\in \llbracket0,n-1\rrbracket$, \begin{equation} (i_k,i_{k+1})\in E \quand w(i_k,i_{k+1})=1. \end{equation} In this case, we~say that $\Gamma$ is $1$-star-convex at $i_0$. \end{Definition} By symmetry of $E$, a $1$-star-convex Dynkin diagram is always connected, but the converse is not true. \subsubsection*{Kac--Moody root systems} Let $\mathcal D = (A,X,Y,(\alpha_i)_{i \in I},(\alpha_i^\vee)_{i \in I})$ be a Kac--Moody root datum in the sense of \cite[\S 8]{remy2002groupes}. It is a quintuple such that: \begin{itemize} \item $I$ is a finite indexing set and $A=(a_{ij})_{(i,j)\in I\times I}$ is a generalized Cartan matrix. \item $X$ and $Y$ are two dual free $\mathbb Z$-modules of finite rank, we~write $\langle , \rangle$ for the duality bracket. \item $(\alpha_i)_{i\in I}$ (\resp $(\alpha_i^\vee)_{i \in I}$) is a family of linearly independent elements of $X$ (\resp $Y$), the simple roots (\resp simple coroots). \item For all $(i,j) \in I^2$ we have $\langle \alpha_i^\vee,\alpha_j \rangle = a_{ij}$. \end{itemize} Let $(I,E,w)$ be the Dynkin diagram corresponding to $A$, with vertices $I$, which is fixed once and for all. In~this article, every Dynkin diagram we will consider will be a subdiagram of $(I,E,w)$. Since they are identified with subsets of $I$, any $J\subseteq I$ now denotes either the subset $J$ of $I$, or the corresponding subdiagram of $(I,E,w)$. \subsubsection*{Height function} We define a height function on $Y$ as follows: let $\rho\in X\otimes \Q$ be any element such that $\langle \alpha_i^\vee,\rho\rangle =1$ for all $i\in I$, then, for any $\lambda\in Y$, the \emph{height} of~$\lambda$~is: \index{h@$\htt$} $\htt(\lambda)=\langle \lambda,\rho\rangle$. This definition depends on the choice of $\rho$, but its restriction to $Q^\vee=\bigoplus_{i\in I} \mathbb Z\alpha_i^\vee$ does not: $\htt(\sum_{i\in I} n_i \alpha_i^\vee)=\sum_{i\in I} n_i$. \subsubsection*{Vectorial Weyl group} For every $i \in I$, set $r_i \in \operatorname{Aut}_\mathbb Z(X): x \mapsto x- \langle \alpha_i^\vee, x \rangle \alpha_i$. The generated group \index{w@$W^v$} $W^v=\langle r_i \mid i \in I \rangle$ is the \emph{vectorial Weyl group} of the Kac--Moody root datum. The duality bracket $\langle Y,X\rangle$ induces a contragredient action of~$W^v$ on~$Y$, explicitly $r_i(y)=y-\langle y,\alpha_i\rangle \alpha_i^\vee$. By~construction the duality bracket is then $W^v$\nobreakdash-invariant. The group $W^v$ is a Coxeter group; in particular, it has a Bruhat order $<$ and a length function $\ell$ compatible with the Bruhat order. \subsubsection*{Real roots} Let $\Phi=W^v\cdot\{\alpha_i \mid i \in I \}$ be the set of real roots of $\mathcal D$; it is a, possibly infinite, root system (see \cite[1.2.2\,Def.]{kumar2002kac}). In particular, let $\Phi_+=\Phi\cap \oplus_{i\in I} \mathbb N \alpha_i$ be the set of positive real roots, then $\Phi=\Phi_+ \sqcup -\Phi_+$. We~write $\Phi_-=-\Phi_+$ for the set of negative roots. The set $\Phi^\vee=W^v\cdot\{\alpha_i^\vee \mid i \in I\}$ is the set of \emph{coroots}, and its subset $\Phi^\vee_+=\Phi^\vee \cap \oplus_{i\in I} \mathbb N \alpha_i^\vee$ is the set of \emph{positive coroots}. To each root $\beta$ corresponds a unique coroot $\beta^\vee$: if $\beta=w(\alpha_i)$ then $\beta^\vee=w(\alpha_i^\vee)$. This map $\beta\mapsto\beta^\vee$ is well defined, is a bijection between $\Phi$ and $\Phi^\vee$, and sends positive roots to positive coroots. Note that $\langle \beta^\vee,\beta\rangle = 2$ for all $\beta \in \Phi$. Moreover, to each root $\beta$ one associates a reflection $s_\beta \in W^v$: if $\beta=w(\pm\alpha_i)$ then $s_\beta:= wr_iw^{-1}$\index{s@$s_\beta$}. It is well-defined, independently of the choices of $w$ and $i$. Explicitly, it is the map $x \mapsto x-\langle \beta^\vee , x \rangle \beta$. We~have $s_\beta=s_{-\beta}$ and the map $\beta\mapsto s_\beta$ is a bijection between the set of positive roots and the set $\{wr_iw^{-1}\mid (w,i)\in W^v\times I\}$ of reflections of $W^v$. \subsubsection*{Dynkin sequence associated to a positive root} Let $\beta\in \Phi_+$. Since $(\alpha^\vee_i)_{i\in I}$ are linearly independent, there is a unique decomposition $\beta^\vee= \sum_{i\in I} N_i(\beta)\alpha_i^\vee$, and for all $i \in I$, $N_i(\beta)\in \mathbb Z_{\geq0}$. For any $n \in \Z_{\geq 0}$, let us define \begin{equation}\label{eq: I_n definition} I_n(\beta)=\{i\in I \mid N_i(\beta) \geq n \}. \end{equation} We consider $I_n(\beta)$ as a subdiagram of $\Gamma$. We~call $(I_n(\beta))_{n\geq 1}$ the \emph{Dynkin sequence} of~$\beta$. The Dynkin sequence associated to a root is, by~definition, non-increasing (for the inclusion) and finitely supported (in the sense that $I_n=\emptyset$ for $n$ large enough). Note that the Dynkin sequence of a simple root $\alpha_i$ is given by $I_1(\alpha_i)=\{i\}$ and $I_n(\alpha_i)=\emptyset$ for $n\geq 2$. We will provide a classification of quantum roots using Dynkin sequences. Note that the Dynkin sequence $(I_n(\beta))_{n\geq 1}$ fully determines $\beta$. Indeed, we~have $\beta^\vee=\sum_{n=1}^{+\infty} \sum_{i\in I_n(\beta)}\alpha_i^\vee$, which proves that $\beta^\vee$, and thus $\beta$, is determined by its Dynkin sequence. \subsubsection*{Inversion sets} For any $w \in W^v$, let\index{i@$\Inv(w)$} \[ \Inv(w)=\Phi_+ \cap w^{-1}\cdot\Phi_-= \{ \alpha \in \Phi_+ \mid w(\alpha) \in \Phi_-\}. \] These sets are closely related to the Bruhat order since, by \cite[1.3.13]{kumar2002kac}, $$\forall \alpha\in\Phi_+, \; \alpha \in \Inv(w) \iff ws_\alpha < w \iff s_\alpha w^{-1} < w^{-1}.$$ Moreover, they are related to the length $\ell$ on $W$: $\ell(w)=|\Inv(w)|$ (\cite[1.3.14]{kumar2002kac}). \Subsection{Quantum roots: definition, motivation and notations}\label{ss_Quantum_roots} \begin{Definition}\label{d_almost_simple_root} Let $\beta\!\in\mkern-4mu \Phi_+$. We~say that $\beta$ is \emph{quantum} if for every \hbox{$\gamma\!\in\mkern-4mu \Inv(s_\beta)\!\setminus\! \{\beta\}$}, we~have $\langle \beta^\vee,\gamma\rangle=1$. We~denote by $\cQ(\Phi_+)$\index{q@$\cQ(\Phi_+)$} the set of quantum roots of $\Phi_+$. \ \end{Definition} Note that $\Inv(r_i)=\{\alpha_i\}$ for every $i\in I$, and thus every simple root is quantum. \begin{Remark} This definition is not the one we presented in the introduction. By~Lemma~\ref{l_characterization_almost_simpleness} below, the two definitions are equivalent. \end{Remark} Note that the notion of quantum root is related to the notion of minuscule element attributed to Peterson (see \cite[2]{stembridge2001minuscule}). J.\,Stembridge assumes the generalized Cartan matrix to be symmetrizable, but this assumption is not needed for this definition to make sense, so~we drop it here. \skpt \begin{Proposition}\label{p_minuscule} \begin{enumerate} \item\label{p_minuscule1} Let $\beta\in \Phi_+$. Write $\beta=w(\alpha_i)$, with $i\in I$ and $w\in W^{\vv}$ of minimal possible length. Then $\{\langle \beta^\vee,\gamma\rangle\mid \gamma\in \Inv(s_\beta)\setminus \{\beta\}\}=\{-\langle \alpha_i^\vee,\gamma\rangle\mid \gamma\in \Inv(w)\}$. \item\label{p_minuscule2} Let $\beta\in \Phi_+$. Write $\beta=w(\alpha_i)$, with $i\in I$ and $w\in W^{\vv}$ of minimal possible length. Then $\beta\in \cQ(\Phi_+)$ if and only if $w$ is $\alpha_i^\vee$-minuscule in the sense of \cite[(2.1)]{stembridge2001minuscule}. \end{enumerate} \end{Proposition} \skpt \begin{proof} \begin{enumerate} \item Write $w=r_{i_n}\ldots r_{i_1}$, with $n=\ell(w)$ and $i_1,\ldots,i_n\in I$. Then by \cite[1.3.14\,Lem.]{kumar2002kac}, we~have: \begin{multline*} \Inv(s_\beta)=\bigl\{\alpha_{i_n},r_{i_n}(\alpha_{i_{n-1}}),\ldots, r_{i_n}\ldots r_{i_2}(\alpha_{i_1}),\\[-5pt] w(\alpha_i), wr_i(\alpha_{i_1}),\ldots, w r_{i}r_{i_1}\ldots r_{i_{n-1}}(\alpha_i)\bigr\}. \end{multline*} Set \begin{align*} E_1&=\{r_{i_n}\ldots r_{k}(\alpha_{i_{k-1}})\mid k\in \llbracket 2,n+1\rrbracket\}\\ \tag*{and} E_2&=\{w r_ir_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})\mid k\in \llbracket 0,n-1\rrbracket\}. \end{align*} Then $\Inv(s_\beta)=E_1\sqcup \{\beta\}\sqcup E_2.$ Let $\gamma \in E_1$. Write $\gamma=r_{i_n}\ldots r_{i_k}(\alpha_{i_{k-1}})$, where $k\in \llbracket 2,n+1\rrbracket$. Then \begin{align*} \langle \beta^\vee,\gamma\rangle&=\langle w(\alpha_i^\vee),\gamma\rangle\\ &=\langle \alpha_i^\vee,w^{-1} r_{i_n}\ldots r_{i_{k}}(\alpha_{i_{k-1}})\rangle\\ &=\langle \alpha_i^\vee, r_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})\rangle\\ &=-\langle \alpha_i^\vee,r_{i_1}\ldots r_{i_{k-2}}(\alpha_{i_{k-1}})\rangle. \end{align*} Therefore $\{\langle \beta^\vee,\gamma\rangle \mid \gamma\in E_1\}=\{-\langle \alpha_i^\vee,\gamma\rangle \mid \gamma\in \Inv(w)\}$. Let $\gamma$ be in $E_2$. Write $\gamma=wr_i r_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})$, with $k\in \llbracket 0,n-1\rrbracket$. Then \[ \langle \beta^\vee,\gamma\rangle=\langle \alpha_i^\vee,r_{i} r_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})\rangle=-\langle \alpha_i^\vee,r_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})\rangle. \] Therefore \[ \{\langle \beta^\vee,\gamma\rangle \mid \gamma\in E_2\}=\{-\langle \alpha_i^\vee,\gamma\rangle \mid \gamma\in \Inv(w)\}. \] This proves \eqref{p_minuscule1}. \item If $k\in \llbracket 0,n-1\rrbracket$, then $\langle \alpha_i^\vee,r_{i_1}\ldots r_{i_{k-1}}(\alpha_{i_{k-1}})\rangle=\langle r_{i_k}\ldots r_{i_1}(\alpha_i^\vee),\alpha_{i_{k+1}}\rangle$ and thus Definition~\ref{d_almost_simple_root} for $w(\alpha_i)$ is equivalent to \cite[(2.1)]{stembridge2001minuscule} for $w$ and $\alpha_i^\vee$.\qedhere \end{enumerate} \end{proof} \begin{Remark} In \cite{proctor1999minuscule} and \cite{proctor1999dynkin}, R.\,Proctor classifies $\lambda$-minuscule elements, for $\lambda\in Y$ dominant (i.e., $\langle\lambda,\alpha_i\rangle\geq 0$, for all $i\in I$), in~the simply laced case. The case we study here is somewhat different: in general simple coroots are not dominant, and furthermore, when the generalized Cartan matrix is indecomposable and not of finite type, they do not lie in the Tits cone. It would be interesting to have a theory unifying these two situations. \end{Remark} The aim of this section is to study quantum roots and in particular to prove that for every Kac--Moody datum, the number of quantum roots is finite (see Theorem~\ref{t_finiteness_almost_simple_roots}). This result is a key step in proving our finiteness results on covers and co-covers (see Theorems~\ref{Theorem: finiteness of covers} and \ref{thm_finiteness_co-covers}). We~then classify the quantum roots. We~end up this section with examples: we describe the set of quantum roots for particular choices of generalized Cartan matrices. Let us describe the main steps of the proofs of Theorem~\ref{t_finiteness_almost_simple_roots} and Theorem~\ref{Theorem: Classification quantum roots}. Let $\beta\in \cQ(\Phi_+)$. We~write\vspace*{-3pt} \begin{equation}\label{e_reduced_writting_beta} \beta=r_{i_L}\ldots r_{i_2}(\alpha_{i_1}), \end{equation} with $i_1, \ldots,i_L\in I$ and $L$ as small as possible. \begin{enumerate} \item\label{enum:1} We begin by proving that, up to renumbering, we~can assume that, if $L_1=|I_1(\beta)|$, then $\{i_1,\ldots,i_{L_1}\}=I_1(\beta)$ (see Lemma~\ref{l_I1_placed_beggining}). \item\label{enum:2} We prove that $I_1(\beta)$ is a $1$-star-convex tree (see Proposition~\ref{p_characterization_I1}). \item\label{enum:3} We prove that the ``orders of appearance'' in \eqref{e_reduced_writting_beta} are highly constrained. If~$i,j\in I_2(\beta)$ belong to the same connected component and if the second appearance of $i$ (when we read the right-hand side of \eqref{e_reduced_writting_beta} from right to left) precedes the second appearance of $j$, then for $n\in \Z_{\geq 3}$,\vspace*{-3pt} \[ j\in I_n(\beta)\implies i\in I_n(\beta), \] and the $n$-th appearance of $i$ precedes the $n$-th appearance of $j$ (see Lemma~\ref{l_description_chains_In}). \item\label{enum:4} We prove that the connected components of $I_2(\beta)$ are as follows. Let $\cC$ be a connected component of $I_2(\beta)$ and $j$ be the element of $\cC$ whose second appearance in \eqref{e_reduced_writting_beta} is the earliest. Then $\cC$ is the union of at most three segments based at $j$ (see Lemma~\ref{l_description_chains_I2} and Proposition~\ref{p_connected_components_I2}). \item\label{enum:5} Using the results described above, we~obtain restrictions on the form of the Dynkin sequence $(I_n(\beta))$. We~deduce that if $k=\max \{n\in \Z\mid I_n(\beta)\neq \emptyset\}$, then $k\leq \max(6,k'+1)$, where $k'$ is the maximal length of a segment appearing in point \eqref{enum:4} (see Proposition~\ref{p_majoration_coefficients_AS_roots}). As $k'$ is bounded by the size of $I$, we~deduce an upper bound on the height of $\beta^\vee$ and hence that $\cQ(\Phi_+)$ is finite. \item\label{enum:6} Refining the results of \eqref{enum:5}, we~describe all the possible forms of the Dynkin sequences. We~then construct, for each of these Dynkin sequences, a quantum root to which it is associated (see Theorem~\ref{Theorem: Classification quantum roots}). \end{enumerate} Let $\beta\in \Phi_+$. We~write $\beta=w(\alpha_{i_1})$, with $i_1\in I$ and $w\in W^v$ and we assume that $L:=\ell(w)+1$\index{l@$L$} is as small as possible. We~fix a reduced decomposition $\underline{w}=(r_{i_L},\ldots ,r_{i_2})$ of $w$, with $i_L,\ldots,i_2\in I$. By \cite[1.3.14\,Lem.]{kumar2002kac} we have\vspace*{-5pt} \begin{multline}\label{e_Inversion_set} \Inv(s_\beta)=\bigl\{\alpha_{i_L},r_{i_{L}}(\alpha_{i_{L-1}}),\ldots,r_{i_L}\ldots r_{i_{2}}(\alpha_{i_1}),r_{i_L}\ldots r_{i_1}(\alpha_{i_2}),\ldots,\\[-5pt] r_{i_L}\ldots r_{i_1} \ldots r_{i_{L-1}}(\alpha_{i_L})\bigr\}. \end{multline} Set \[ \gamma^\vee_1=\alpha_{i_1}^\vee,\ \gamma^\vee_2=r_{i_2}(\alpha_{i_1}^\vee),\ \ldots,\ \gamma^\vee_L=r_{i_L}\ldots r_{i_2}(\alpha_{i_1}^\vee)=\beta^\vee. \] \begin{Lemma}\label{l_inequality_length} Let $\beta\in \Phi_+$. Then $\ell(s_\beta)\leq 2\htt(\beta^\vee)-1$. \end{Lemma} \begin{proof} By \cite[Cor.\,1.10]{philippe2023grading}, we~have \[ 2\htt(\beta^\vee)=2+\sum_{\gamma\in \Inv(s_\beta)\setminus\{\beta\}} \langle \beta^\vee,\gamma\rangle \] and all the terms on the right-hand side are positive integers. Therefore $2\htt(\beta^\vee)-2\geq |\Inv(s_\beta)|-1=\ell(s_\beta)-1$. \end{proof} \begin{Lemma}\label{l_characterization_almost_simpleness} Let $\beta\in \Phi_+$. Using the same notation as above, set $\beta_1=r_{i_L}\ldots r_{i_3}(\alpha_{i_2})$, $\beta_2=r_{i_L}\ldots r_{i_4}(\alpha_{i_3}),\ldots, \beta_{L-1}=\alpha_{i_L}$. Then the following conditions are equivalent: \begin{enumerate} \item\label{l_characterization_almost_simpleness1} $\beta$ is quantum, \item\label{l_characterization_almost_simpleness2} for every $t\in \llbracket 1,L-1\rrbracket$, $\langle \beta^\vee,\beta_t\rangle=1$, \item\label{l_characterization_almost_simpleness3} $%\label{e_roots} \forall t\in \llbracket 1,L\rrbracket, \quad \gamma^\vee_t=\sum_{k=1}^t \alpha_{i_k}^\vee \quand\langle \gamma_{t-1}^\vee,\alpha_{i_t}\rangle=-1,\text{ if }t\neq 1 $, \item\label{l_characterization_almost_simpleness4} $\ell(s_\beta)=2\htt(\beta^\vee)-1$. \end{enumerate} \end{Lemma} \begin{proof} Let $t\in \llbracket 2,L\rrbracket$. We~have: \begin{align*} \langle \beta^\vee,\beta_{t-1}\rangle&=\langle r_{i_L}\ldots r_{i_2}(\alpha_{i_1}^\vee),r_{i_L}\ldots r_{i_{t+1}}(\alpha_{i_{t}})\rangle\\ &=\langle r_{i_t}\ldots r_{i_2}(\alpha_{i_1}^\vee),\alpha_{i_t}\rangle=-\langle r_{i_{t-1}}\ldots r_{i_2}(\alpha_{i_1}^\vee),\alpha_{i_t}\rangle. \end{align*} We thus have: \[ \begin{aligned}\langle r_{i_L}\ldots r_{i_2}(\alpha_{i_1}^\vee),r_{i_L}\ldots r_{i_1} \ldots r_{i_{t-1}}(\alpha_{i_t})\rangle &=\langle \alpha_{i_1}^\vee,r_{i_1}\ldots r_{i_{t-1}}(\alpha_{i_t})\rangle\\ &=-\langle r_{i_{t-1}}\ldots r_{i_2}(\alpha_{i_1}^\vee),\alpha_{i_t}\rangle=\langle \beta^\vee,\beta_{t-1}\rangle. \end{aligned} \] Therefore, \[ \{\langle\beta^\vee,\gamma\rangle\mid \gamma\in \Inv(s_\beta)\setminus\{\beta\}\}=\{\langle\beta^\vee,\beta_t\rangle\mid t\in \llbracket 2,L \rrbracket\}, \] which proves the equivalence between \eqref{l_characterization_almost_simpleness1} and \eqref{l_characterization_almost_simpleness2}. Let $t\in \llbracket 1,L-1\rrbracket$. We~have: \begin{align*} \langle \beta^\vee,\beta_t\rangle&=\langle r_{i_L} \ldots r_{i_2}(\alpha_{i_1}^\vee),r_{i_L}\ldots r_{i_{t+2}}(\alpha_{i_{t+1}})\rangle\\ &=\langle r_{i_{t+1}}\ldots r_{i_2}(\alpha_{i_1}^\vee),\alpha_{i_{t+1}}\rangle=-\langle \gamma_t^\vee,\alpha_{i_{t+1}}\rangle. \end{align*} Therefore, \[ \gamma_{t+1}^\vee=r_{i_{t+1}}(\gamma_t^\vee)=\gamma_t^\vee-\langle \gamma_t^\vee,\alpha_{i_{t+1}}\rangle \alpha_{i_{t+1}}^\vee=\gamma_t^\vee+\langle \beta^\vee,\beta_t\rangle\alpha_{i_{t+1}}^\vee. \] The equivalence between \eqref{l_characterization_almost_simpleness1} and \eqref{l_characterization_almost_simpleness3} follows. Let $\beta\in \Phi_+$ satisfying \eqref{l_characterization_almost_simpleness3}. We~have $\gamma_L^\vee=\beta^\vee$ and $\htt(\gamma_L^\vee)=L$. As $\ell(s_\beta)=2L-1$, we~deduce that \eqref{l_characterization_almost_simpleness3} implies \eqref{l_characterization_almost_simpleness4}. Let $\beta^\vee\in \Phi^\vee_+$. Assume that $\beta^\vee$ satisfies \eqref{l_characterization_almost_simpleness4}. Then by \cite[Cor.\,1.10]{philippe2023grading}, we~have $2\htt(\beta^\vee)=\sum_{\gamma\in \Inv(s_\beta)} \langle \beta^\vee,\gamma\rangle=2+\sum_{\gamma\in \Inv(s_\beta)\setminus\{\beta^\vee\}} \langle \beta^\vee,\gamma\rangle=\ell(s_\beta)+1$ and all the terms of this sum are positive. Therefore $\langle \beta^\vee,\gamma\rangle=1$ for all $\gamma\in \Inv(s_\beta)\setminus \{\beta\}$, which proves that $\beta$ is quantum. \end{proof} \begin{Remark}\label{r_height_almost_simple_coroot} By Lemma~\ref{l_characterization_almost_simpleness}, for all $t\in \llbracket 1,L\rrbracket$, $\gamma_t=r_{i_t}\dots r_{i_2}(\alpha_{i_1})$ is quantum and $\htt(\gamma_t^\vee)=t$. \end{Remark} The following proposition shows that the set $\cQ(\Phi_+)$ can be constructed recursively. This will be used extensively in Section~\ref{subsection: classification} in order to prove Theorem~\ref{Theorem: Classification quantum roots}. \pagebreak[2] \skpt \begin{Proposition}\label{Proposition: simplereflectionquantumroot} \begin{enumerate} \item\label{Proposition: simplereflectionquantumroo1} If $\beta\in \cQ(\Phi_+)$ is a quantum root which is not simple, then there exists $i\in I$ such that $r_i(\beta)\in \cQ(\Phi_+)$ and $\htt(r_i(\beta^\vee))=\htt(\beta^\vee)-1$. \item\label{Proposition: simplereflectionquantumroot2} Let $\beta\in \cQ(\Phi_+)$ and $i\in I$. Then $r_i(\beta)\in \cQ(\Phi_+)$ if and only if $|\langle \beta^\vee,\alpha_i\rangle|\leq 1$. \end{enumerate} \end{Proposition} \skpt \begin{proof} \begin{enumerate} \item We simply take $i\!=\!i_L$ with the notation of Formula \eqref{e_Inversion_set}; then, by Remark~\ref{r_height_almost_simple_coroot}, $r_i(\beta)=\gamma_{L-1}$ is a quantum root. \item Let $\beta\in \cQ(\Phi_+)$ and $i\in I$. Let $\tilde{\beta}=r_i(\beta)$. We~have $s_{\tilde{\beta}}=r_i s_{\beta}r_i$ and thus: \begin{equation}\label{e_length} \ell(s_{\tilde{\beta}})\in \{\ell(s_\beta)-2,\ell(s_\beta),\ell(s_{\beta})+2\} . \end{equation} We have $\htt(\tilde{\beta^\vee})=\htt(\beta^\vee)-\langle \beta^\vee,\alpha_i\rangle$. If~$r_i(\beta)\in \cQ(\Phi_+)$, then by Lemma~\ref{l_characterization_almost_simpleness}, we~have $\ell(s_{\tilde{\beta}})=2\htt(\tilde{\beta}^\vee)-1=2\htt(\beta^\vee)-1-2\langle \beta^\vee,\alpha_i\rangle=\ell(s_\beta)-2\langle \beta^\vee,\alpha_i\rangle$. Using \eqref{e_length}, we~deduce that if $r_i(\beta)\in \cQ(\Phi_+)$, then $|\langle \beta^\vee,\alpha_i\rangle|\leq 1.$ Now let $\beta\in \cQ(\Phi_+)$ and $i\in I$. If~$\langle \beta^\vee,\alpha_i\rangle=0$, then $r_i(\beta)=\beta\in \cQ(\Phi_+)$. Assume now $\langle \beta^\vee,\alpha_i\rangle =1$. Set $\tilde{\beta}=r_i(\beta)$. We~have $\htt(\tilde{\beta}^\vee)=\htt(\beta^\vee)-1$ and by Lemma~\ref{l_inequality_length}, \[ \ell(s_{\tilde{\beta}})\leq 2\htt(\tilde{\beta})-1=2\htt(\beta^\vee)-3=\ell(s_{\beta})-2. \] By \eqref{e_length} we deduce $\ell(s_{\tilde{\beta}})=\ell(s_{\beta})-2=2\htt(\tilde{\beta})-1$, which proves that $\tilde{\beta}=r_i(\beta)$ is quantum by Lemma~\ref{l_characterization_almost_simpleness}. Finally, assume that $\langle \beta^\vee,\alpha_i\rangle = -1$. Let $\tilde \beta:=r_i(\beta)$. Note that the associated coroot is $\tilde \beta^\vee =r_i(\beta^\vee)=\beta^\vee+\alpha_i^\vee$. Therefore $\htt(\tilde \beta^\vee)=\htt(\beta^\vee)+1$ and, by~Lemma~\ref{l_characterization_almost_simpleness}, $\tilde \beta$ is quantum if and only if $$\ell(s_{\tilde \beta})=\ell(r_i s_\beta r_i)=\ell(s_\beta)+2.$$ Since $r_i$ is a simple reflection, $\{\ell(r_i s_\beta),\ell(s_\beta r_i)\}\subset \{\ell(s_\beta)+1,\ell(s_\beta)-1\}$. Moreover since $s_\beta$ is a reflection, $\ell(r_i s_\beta)=\ell((r_is_\beta)^{-1})=\ell(s_\beta r_i)$. We have $\langle \beta^\vee,\alpha_i\rangle <0$, so~$s_\beta(\alpha_i)\in \Phi_+$. Hence $\alpha_i\notin \Inv(s_\beta)$, $s_\beta r_i>s_\beta$ and thus $\ell(s_\beta r_i)=\ell(r_i s_\beta)=\ell(s_\beta)+1$. Note that $s_\beta(\alpha_i)=\alpha_i-\langle \beta^\vee,\alpha_i\rangle \beta=\beta+\alpha_i$. Moreover since $\langle \beta^\vee,\alpha_i\rangle <0$, by~\cite[Lem.\,1.1.10]{bardy1996systemes} we also have $\langle \alpha_i^\vee,\beta\rangle <0$, that is to say $\langle \alpha_i^\vee,\beta\rangle+1 \leq 0$ (because it is an integer). Therefore, $$r_i s_\beta(\alpha_i)=r_i(\beta+\alpha_i)=r_i(\beta)-\alpha_i=\beta-(\langle \alpha_i^\vee,\beta\rangle +1)\alpha_i \in \Phi_+.$$ So $\alpha_i\notin \Inv(r_i s_\beta)$ and $\ell(r_i s_\beta r_i)=\ell(r_i s_\beta)+1=\ell(s_\beta)+2$, which proves the proposition.\qedhere \end{enumerate} \end{proof} From now on we assume that $\beta$ is quantum. For $i\in I$ and $t \in \llbracket 1,L\rrbracket$, set $N_i(t)=|\{t'\in \llbracket 1,t\rrbracket \mid i_{t'}=i\}|$\index{n@$N_i(\cdot)$}. Then, by Lemma~\ref{l_characterization_almost_simpleness}\eqref{l_characterization_almost_simpleness3}, \begin{equation}\label{e_coordinate_root} \gamma^\vee_t=\sum_{i\in I_1} N_i(t) \alpha_i^\vee. \end{equation} Note that, by Lemma~\ref{l_characterization_almost_simpleness}\eqref{l_characterization_almost_simpleness3}, $N_i$ is non-decreasing for every $i\in I_1(\beta)$, and for every $t\in \llbracket 1,L-1\rrbracket$, $N_i(t)=N_i(t+1)$ for all but one element $k$ of $I_1(\beta)$, and we have $N_k(t+1)=N_k(t)+1$. Note that the Dynkin sequence of $\beta$ can be defined by \[ I_n(\beta):=\{i\in I\mid \max N_i\geq n\}, \] with the induced diagram structure from $I$. The set $I_n(\beta)$ is also the set of elements which appear at least $n$ times in $(\underline{w},r_{i_1})$. In this section, we~have fixed a quantum root $\beta$; therefore, to simplify notation, we~set \begin{equation}I_n:=I_n(\beta). \end{equation}\index{i@$I_1,I_2,I_n,I_n(\beta)$} For $j\in I$ and $n\in \Z_{\geq 0}$, we~set \begin{equation}\bt(j,n)=\min \{t\in \llbracket 1,L\rrbracket\mid N_j(t)=n\}-1\in \llbracket 0,L-1\rrbracket\cup \{\infty\}. \end{equation}\index{t@$\bt(\cdot,\cdot)$} We then have $N_j(\bt(j,n))=n-1$ and $N_j(\bt(j,n)+1)=n$, if~$\bt(j,n)\neq \infty$. \begin{Lemma}\label{l_I1_placed_beggining} Let $L_1=|I_1|$\index{l@$L_1$}. Then up to changing the reduced decomposition of $w$, we~may assume $I_1=\{i_1,\ldots,i_{L_1}\}$. \end{Lemma} \begin{proof} Assume $\{i_1,\ldots,i_{L_1}\}\neq I_1$. Then $(i_j)_{j\in \llbracket 1,L_1\rrbracket}$ is not injective. Let $k\in \llbracket 1,L_1\rrbracket$ be minimal such that $i_k\in \{i_1,\ldots, i_{k-1}\}$. Let $k'\in \llbracket k+1,L\rrbracket$ be minimal such that $i_{k'}\notin \{i_1,\ldots,i_{k'-1}\}$. Then for every $k''\in \llbracket k,k'-1\rrbracket$, $N_{i_{k''}}(k')=N_{i_{k''}}(k'-1)\geq 2$ and $N_{i_{k'}}(k'-1)=0=N_{i_{k'}}(k')-1$. By~assumption on $\beta$, we~have $\langle r_{i_{k'-1}}\ldots r_{i_2}(\alpha_{i_1}^\vee), \alpha_{i_{k'}}\rangle=-1$. Using \eqref{e_coordinate_root}, we~deduce $\sum_{i\in I_1} N_i(k'-1) a_{i,i_{k'}}=-1$. As $N_{i_{k'}}(k'-1)=0$, all the terms appearing on the left-hand side are non-positive integers. Therefore exactly one is nonzero, and it is equal to $-1$. Therefore $a_{i_{k''},i_{k'}}=0$ for every $k''\in \llbracket k, k'-1\rrbracket$. In~other words, $r_{i_{k''}}$ commutes with $r_{i_{k'}}$ for every $k''\in \llbracket k,k'-1\rrbracket$. Therefore we can replace $r_{i_{k'}}r_{i_{k'-1}}\ldots r_{i_k} r_{i_{k-1}}\ldots r_{i_2}$ by $r_{i_{k'-1}}\ldots r_{i_k} r_{i_{k'}} r_{i_{k-1}}\ldots r_{i_2}$ in the reduced decomposition of $w$. Repeating this process as many times as necessary, we~end up with a reduced decomposition $\underline{w}$ such that $(i_j)_{j\in \llbracket 1,L_1\rrbracket}$ is injective and then $\{i_1,\ldots,i_{L_1}\}=\nobreak I_1$. \end{proof} From now on, we~assume that the reduced decomposition $\underline{w}$ is chosen so that $\{i_1,\ldots,i_{L_1}\}=I_1$, which is possible by Lemma~\ref{l_I1_placed_beggining}. \subsection{General structure of $I_1(\beta)$} In this section, we~study the Dynkin diagram structure of $I_1(\beta)$ when $\beta$ is a quantum root. We~have fixed a quantum root $\beta$ and we have set $I_1=I_1(\beta)$. For $i\in I$, set \begin{equation}\label{e_definition_degree} \deg(i)=-\sum_{j\in I_1\setminus \{i\}} a_{j,i}\in \Z_{\geq 0}\end{equation}\index{d@$\deg$} and \[ \supp(i)=\{j\in I_1\mid a_{j,i}\neq 0\}. \]\index{s@$\supp$} \begin{Lemma}\label{l_I1_tree} The graph $I_1$ is a tree: it is connected and has no circuit. \end{Lemma} \begin{proof} By \cite[Lem.\,1.6]{kac1994infinite}, $I_1$ is connected ($I_1$ is the support of $\beta$ for the terminology of \cite{kac1994infinite}). Assume, for a contradiction, that $I_1$ contains a circuit. Let $n\in \Z_{>0}$ and $\tau:\llbracket 0,n\rrbracket\rightarrow I_1$ be such that $\tau(0)=\tau(n)$ and $\{\tau(i) ,\tau(i+1)\}$ is an edge of $I_1$ for all $i\in \llbracket 0,n-1\rrbracket$. We~assume moreover that $\tau(i)\neq \tau(0)$ for $i\in \llbracket 1,n-1\rrbracket$. Then every element of $\tau(\llbracket 0,n\rrbracket)$ has at least two neighbors in $I_1$. Let $i\in \llbracket 0,n-1\rrbracket$ be such that $\bt(\tau(i),1)=t:=\max\{\bt(\tau(j),1)\mid j\in \llbracket 0,n-1\rrbracket\}$. Then there exist $i_1,i_2\in I_1\setminus \{\tau(i)\}$ such that $a_{i_1,\tau(i)}, a_{i_2,\tau(i)}\neq 0$ and $\gamma_t^\vee=\alpha_{i_1}^\vee+\alpha_{i_2}^\vee+x$, for some $x\in \bigoplus_{j\in I_1\setminus \{\tau(i)\}}\N \alpha_j^\vee$. Then $\alpha_{\tau(i)}(\gamma^\vee_t)\leq \alpha_{\tau(i)}(\alpha_{i_1}^\vee+\alpha_{i_2}^\vee)\leq -2<-1$: a contradiction. The lemma follows. \end{proof} A first step towards determining $\cQ(\Phi_+)$ is to understand what the support of quantum roots can be. That is to say, to determine $\{I_1(\beta) \mid \beta \in \cQ(\Phi_+)\}$. Proposition~\ref{p_characterization_I1} below characterizes this set. \begin{Proposition}\label{p_characterization_I1} Let $T$ be a subset of $I$. Then the following properties are equivalent: \begin{enumerate} \item\label{p_characterization_I11} As a subdiagram of $I$, $T$ is a $1$-star-convex Dynkin tree. \item\label{p_characterization_I12} There exists $\beta \in \cQ(\Phi_+)$ such that $T=I_1(\beta)$. \item\label{p_characterization_I13} There exists $\beta \in \cQ(\Phi_+)$ such that $T=I_1(\beta)$ and $I_2(\beta)=\emptyset$. More precisely, $\beta$~is the unique element of $\Phi$ such that $\beta^\vee=\sum_{i\in T} \alpha_i^\vee$. \end{enumerate} \end{Proposition} \skpt \begin{proof} $\eqref{p_characterization_I12}\Rightarrow \eqref{p_characterization_I11}$ Let $\beta\in \cQ(\Phi_+)$ and $T=I_1(\beta)$. By~Lemma~\ref{l_I1_placed_beggining} and Remark~\ref{r_height_almost_simple_coroot}, $\beta'^\vee:=\sum_{i\in T}\alpha_i^\vee$ is a quantum root. Therefore, up to replacing $\beta$ by~$\beta'$, we~can assume $\beta^\vee=\sum_{i\in T}\alpha_i^\vee$. Write $\beta=r_{i_L}\ldots r_{i_2}(\alpha_{i_1})$, where $L=|T|$ and $\{i_L,\ldots,i_1\}=T$. Let us prove that $T$ is $1$-star-convex at $i_1$. Let $t\in\llbracket 1,L-1\rrbracket$ and assume that, as a subdiagram of $T$, $\{i_1,\ldots,i_t\}$ is $1$-star-convex at $i_1$. Then we have $r_{i_t}\ldots r_{i_2}(\alpha_{i_1}^\vee)=\sum_{s=1}^{t} \alpha_{i_{s}}^\vee$ and $r_{i_{t+1}}\cdot(r_{i_t}\ldots r_{i_2}(\alpha_{i_1}^\vee))= (r_{i_t}\ldots r_{i_2}(\alpha_{i_1}^\vee))+\alpha_{i_{t+1}}^\vee$, by~Lemma~\ref{l_characterization_almost_simpleness}. Therefore $\langle \sum_{s=1}^{t} \alpha_{i_{s}}^\vee,\alpha_{i_{t+1}}\rangle=\sum_{s=1}^{t}\langle \alpha_{i_{s}}^\vee,\alpha_{i_{t+1}}\rangle=-1$. As this is a sum of non-positive integers, exactly one term is $-1$ and the others are $0$. Let $t_1\in \llbracket 1,t\rrbracket$ be such that $a_{i_{t_1},i_{t+1}}=-1$. By~assumption, there exist $t'\in \llbracket 1,t\rrbracket$ and a sequence $(u_i)_{i\in \llbracket 1,t'\rrbracket}\in \{i_1,\ldots,i_t\}^{\llbracket 1,t'\rrbracket}$ such that $a_{u_i,u_{i+1}}=-1$ for all $i\in \llbracket 1,t'-1 \rrbracket$, $u_1=i_1$ and $u_{t'}=i_{t_1}$. Then $((u_i)_{i\in \llbracket 1,t'\rrbracket},i_{t+1})$ is a path in $I_1$ relating $i_1$ and $i_{t+1}$, which proves the $1$-star-convexity of $\{i_1,\ldots,i_{t+1}\}$ at $i_1$. By~induction, we~deduce that $T$ is $1$-star-convex. Using Lemma~\ref{l_I1_tree}, we~deduce that $I_1$ is a $1$-star-convex tree. $\eqref{p_characterization_I11}\Rightarrow\eqref{p_characterization_I13}$ Let $T$ be a subdiagram of $I$ which is $1$-star-convex and which is a tree. We~assume that $|T|\geq 2$ and that for all $T'\subset I$ such that $|T'|<|T|$ and for which $T'$ is $1$-star-convex, there exists $\beta'\in \cQ(\Phi_+)$ such that $T'=I_1(\beta')$. Let $j$ be a leaf of $T$. Since a finite tree with at least two vertices has at least two leaves, after possibly changing $j$, we~can assume that $T$ is $1$-star-convex at a node~$j_0$ distinct from $j$. Let $\{j,k\}$ be the only edge containing $j$, then any path from~$j_0$ to~$j$ contains $\{k,j\}$ and so $a_{k,j}=-1$ by $1$-star-convexity of $T$ at $j_0$. Moreover, $T\setminus\nobreak \{j\}$ is $1$-star-convex (at $j_0$). Let $\beta_1\in \cQ(\Phi_+)$ be such that $(\beta_1)^\vee=\sum_{i\in T\setminus\nobreak \{j\}} \alpha_i^\vee$. We~have $\langle (\beta_1)^\vee,\alpha_j\rangle=-1$, thus $\beta^\vee=r_j(\beta_1^\vee)$ and thus by Proposition~\ref{Proposition: simplereflectionquantumroot}, $\beta=r_j(\beta_1)$ is quantum. Therefore \eqref{p_characterization_I11} implies \eqref{p_characterization_I13}. As \eqref{p_characterization_I13} obviously implies \eqref{p_characterization_I12}, we~deduce the proposition. \end{proof} \subsection{Order of appearance and general structure of $I_2(\beta)$} Let $n\!\in\! \N$. For \hbox{$i,j\in I_n$}, we~write $i \lhd_n j$\index{z@$\lhd_n$, $\lhd_2$} if $(i,j)$ is an edge of $I_n$, and if $\bt(i,n)<\bt(j,n)$. The first condition means that $a_{i,j}\neq 0$ and the second condition simply means that, if we read the decomposition of $w$ from right to left, we~meet $i$ $n$ times before we meet $j$ for the $n$-th time. We~denote by $\leq_n$\index{z@$\leq_n$} the preorder generated by $\lhd_n$. Note that for $i,j\in I_n$, if~$i\leq_n j$ and $j\leq_n i$, then $i$ and $j$ appear at the same time for the $n$-th time when we read $(\underline{w},i_1)$ from right to left, and thus $i=j$. Therefore $\leq_n$ is an order on $I_n$. Note that, although this is not obvious from the definition, this order is independent of the chosen decomposition of $w$, as we will see in Remark~\ref{r_connected_components_In}. As $I_n$ is finite, every chain is contained in a maximal one. \medbreak In this section, we~use $\leq_2$ to characterize the diagram $I_2$. In~Lemma~\ref{l_description_chains_In} we then describe chains for $\leq_n$. This will allow us to characterize the diagrams $I_n$ in Section~\ref{subsection: Dynkin sequences quantum roots}. \begin{Lemma}\label{l_description_chains_I2} Assume $I_2\neq \emptyset$. Let $j_1,\ldots,j_k\in I_2$\index{j@$j_1,\ldots,j_k$}\index{k@$k$} be a maximal chain for $\leq_2$, i.e.: \begin{enumerate} \item\label{l_description_chains_I21} $j_1$ is minimal in $I_2$ for $\leq_2$, \item\label{l_description_chains_I22} $j_k$ is maximal in $I_2$ for $\leq_2$, \item\label{l_description_chains_I23} $j_1\lhd_2 j_2\lhd_2 \ldots \lhd_2 j_k$. \end{enumerate} Then:\vspace*{-3pt} \begin{enumerate} \item $\deg(j_1)=3$, $|\supp(j_1)|\leq 4$. \item for every $t\in \llbracket 2,k-1\rrbracket$, $\supp(j_t)=\{j_{t-1},j_t,j_{t+1}\}$, $\deg(j_t)=2$ and $a_{j_{t-1},j_t}=a_{j_{t+1},j_t}=-1$. \item We have $\deg(j_k)\leq 2$, $|\supp(j_k)|\leq 3$ and $\supp(j_k)\supset \{j_{k-1},j_k\}$. \begin{enumerate} \item If $|\supp(j_k)|=3$, then $\supp(j_k)=\{j_{k-1},j_k,j_{k+1}\}$ for some $j_{k+1}\in I_1\setminus I_2$, and we have $a_{j_{k-1},j_k}=a_{j_{k+1},j_k}=-1$. \item If $|\supp(j_{k})|=2$, then $\supp(j_{k})=\{j_{k-1},j_k\}$, then\vspace*{-3pt} \begin{equation}\label{e_passage_3_before_2} a_{j_{k-1},j_{k}}=-1\quand\bt(j_{k-1},3)<\bt(j_k,2)<\bt(j_{k-1},4). \end{equation}\end{enumerate} \end{enumerate} In particular, if~$j\in I_2$ is such that $|\supp(j)|=2$, then $j$ is either minimal or maximal for $\leq_2$. \end{Lemma} \begin{proof} Let $t_1=\bt(j_1,2)$. By~\eqref{e_coordinate_root}, we~have: $\gamma^\vee_{t_1}=\sum_{i\in I_1} N_i(t_1)\alpha_i^\vee$ and $N_i(t_1)\geq 1$ for every $i\in I_1$ by choice of $\underline{w}$. Then $\alpha_{i_{t_1}}(\gamma^\vee_{t_1})=-1$. Let $i\in I_1$ and assume that $N_i(t_1)\geq 2$. Then as $i{\not\!\! \lhd}_2 j_1$, we~have $a_{i,j_1}=0$. Therefore\vspace*{-3pt} \[ \begin{aligned} \alpha_{j_1}(\gamma^\vee_{t_1})&=-1=\alpha_{j_1}\Bigl(\sum_{i\in \supp(j_1)} N_i(t_1) \alpha_i^\vee\Bigr)=\alpha_{j_1}\Bigl(\sum_{i\in \supp(j_1)} \alpha_{i}^\vee\Bigr)\\ &=2+\sum_{i\in \supp(j_1)\setminus\{j_1\}} a_{i,j_1}=2-\deg(j_1),\end{aligned} \] and thus $\deg(j_1)=3$. Now let $x\in \llbracket 2,k\rrbracket$. Let $t_x=\bt(j_x,2)$. We~have: \[ \alpha_{j_x}(\gamma^\vee_{t_x})=-1=\alpha_{j_x}\Bigl(\sum_{i\in \supp(j_x)} N_i(t_x) \alpha_i^\vee\Bigr)=2+\sum_{i\in \supp(j_x)\setminus \{j_x\}}N_i(t_x)a_{i,j_x}. \] Therefore $\sum_{i\in \supp(j_x)\setminus \{j_x\}}N_i(t_x)a_{i,j_x}=-3$, and this is a sum of non-positive integers. Moreover, as $N_{j_{x-1}}$ is non-decreasing and $t_x>t_{x-1}$, we~have $N_{j_{x-1}}(t_x)\geq 2$. Consequently, $|\supp(j_i)|\leq 3$. Assume $x\leq k-1$. As $j_{x-1}\lhd_2 j_x\lhd_2 j_{x+1}$, we~have $\supp(j_x)=\{j_{x-1},j_x,j_{x+1}\}$ and $a_{j_{x+1},j_x}=a_{j_{x-1},j_x}=-1$. In~particular, $\deg(j_x)=2$. Assume now $x=k$. Then if $|\supp(j_k)|=3$, we~have $a_{j_{k+1},j_k}=-1$, for some $j_{k+1}\in I_1$. If~$j_{k+1}\in I_2$, then we would have $j_k\lhd_2 j_{k+1}$, which would contradict our maximality assumption. Therefore, $j_{k+1}\in I_1\setminus I_2$. Assume $|\supp(j_k)|=2$. Then $\supp(j_k)=\{j_{k-1},j_k\}$. Let $t=\bt(j_k,2)$. Then \[ \alpha_{j_k}(\gamma^\vee_t)=-1=\alpha_{j_k}(N_{j_{k-1}}(t)\alpha_{j_{k-1}}^\vee+\alpha_{j_k}^\vee)=2-3, \] which proves \eqref{e_passage_3_before_2}. \end{proof} In terms of Dynkin diagrams, Lemma~\ref{l_description_chains_I2} implies that the form of the connected components of $I_2$ have a very constrained form, see Proposition~\ref{p_connected_components_I2} and Figure~\ref{f_conneceted_components_I2}. \begin{figure}[htb] \centering \includegraphics[scale=0.3]{Connected_component_I2.png} \caption{Generic form of a connected component of $I_2$, when $m=\nobreak3$ . Actually, as we shall see in Lemma~\ref{Lemma: Class 4S}, if~$j_1\in I_3$, then we have $\min\{k(1),k(2),k(3)\}\leq 2$. }\label{f_conneceted_components_I2} \end{figure} \begin{Proposition}\label{p_connected_components_I2} Let $\cC$ be a connected component of $I_2$. Then: \begin{enumerate} \item $\cC$ admits a minimum $j_1$ and $m:=|\supp(j_1)|-1\leq 3$. \item Write $\supp(j_1)\setminus \{j_1\}=\{j_2^{(i)}\mid i\in \llbracket 1,m\rrbracket\}$. For $i\in \llbracket 1,m\rrbracket$, $\{x\in I_2\mid x\geq_2 j_2^{(i)}\}$ is a segment: we can write it as $\{j_2^{(i)},j_3^{(i)},\ldots, j_{k(i)}^{(i)}\}$, for some $j_2^{(i)},\ldots,j_{k(i)}^{(i)}\in I_2$ and the edges are the $(j_t^{(i)},j_{t+1}^{(i)})$ for $t\in \llbracket 1,k(i)-1\rrbracket$. \item We have $\cC=\{j_1\}\sqcup \bigsqcup_{i=1}^m \{j_2^{(i)},j_{3}^{(i)},\ldots, ,j_{k(i)}^{(i)}\}$. \item If $j\in \cC\setminus\{j_1\}$ then $|\supp(j)|\leq 3$ and $a_{j',j}=-1$ for all $j'\in \supp(j)$. \end{enumerate} \end{Proposition} \begin{proof} Let $j_1\!\in\! \cC$ be the minimum for $\leq_2$. By~Lemma~\ref{l_description_chains_I2}, \hbox{$m\!:=\!|\supp(j_1)|\!-\!1\!\leq\! 3$}. Write $\supp(j_1)=\{j_1\}\cup \{j_2^{(i)}\mid i\in \llbracket 1,m\rrbracket\}$. Let $i\in \llbracket 1,m\rrbracket$. By~Lemma~\ref{l_description_chains_I2}, the set $E_i$ of the elements of $I_2$ which are greater than $j_2^{(i)}$ for $\leq_2$ is of the form $\{j_2^{(i)},j_3^{(i)},\ldots, j_{k(i)}^{(i)}\}$ for some $k(i)\in \Z_{\geq 2}$, and as a subgraph of $I_2$, $E_i$ is the line segment $[j_2^{(i)},j_{k(i)}^{(i)}]$. In~particular, if~$\cC'=\{j_1\}\cup \bigcup_{i=1}^m E_i$, then for each $x\in \cC'$, we~have $\supp(x)\cap I_2\subset \cC'$. Therefore $\cC'$ is connected and $\cC=\cC'$. Moreover $\cC$ is a connected subgraph of the tree~$I_1$ (by Proposition~\ref{p_characterization_I1}) and thus it is a tree. Therefore the sets $E_i$ are disjoint. Finally, by~Lemma~\ref{l_description_chains_I2}\,\eqref{l_description_chains_I22}\,\&\,\eqref{l_description_chains_I23}, for all $j\in \cC'$ we have $|\supp(j)|\leq 3$ and, for all $j'\in \supp(j)$, $a_{j',j}=-1$. The lemma follows. \end{proof} \begin{Remark}\label{r_independence_leq_2} The order $\leq_2$ does not depend on the choice of the reduced decomposition made at the beginning of Section~\ref{ss_Quantum_roots}, provided that it satisfies Lemma~\ref{l_I1_placed_beggining}. Indeed, if~two elements of $I_2$ are comparable for $\leq_2$, they belong to the same connected component of $I_2$. Thus it suffices to prove that for each connected component $\cC$ of $I_2$ the restriction of $\leq_2$ to $\cC$ is independent of the choice of the reduced decomposition. By~Lemma~\ref{l_description_chains_I2} and Proposition~\ref{p_connected_components_I2}, the minimum $j_1$ for $\leq_2$ is the unique element of $\cC$ whose degree is $3$. Then the elements $j_2^{(1)}, \ldots, j_2^{(m)}$ (where $m=|\supp(j_1)-1|$) that cover $j_1$ are the elements of $\supp(j_1)\setminus \{j_1\}$. If~$i\in \llbracket 1,m\rrbracket$, the only element (if it exists) covering $j_2^{(i)}$ is the element of $\supp(j_2^{(i)})\setminus \{j_1,j_2^{(i)}\}$ etc. All these elements are defined uniquely from the graph $I_2$, which is independent of the choice of the reduced decomposition. \end{Remark} With the above notation, we~set: \begin{equation}\label{e_notation_k_kprime} k'=\begin{cases} k,&\text{if }|\supp(j_k)|=2,\\ k+1&\text{if }\supp(j_k)=\{j_{k-1},j_k,j_{k+1}\}. \end{cases} \end{equation}\index{k@$k'$} The following lemma implies that \begin{equation}\label{e_inequalities_N} N_{j_1}(t)\geq N_{j_2}(t)\geq \ldots \geq N_{j_{k'}}(t),\quad \forall t\in \llbracket L_1+1,L\rrbracket. \end{equation} \begin{Lemma}\label{l_description_chains_In} Let $j_1,\ldots,j_k\in I_2$ be a maximal chain for $\leq_2$. Let $n\in \Z_{\geq 3}$. Then: \begin{enumerate} \item\label{l_description_chains_In1} there exists $k_n\in \llbracket 0,k\rrbracket$ such that $\{i\in \llbracket 1,k\rrbracket\mid j_i\in I_n\}=\llbracket 1,k_n\rrbracket$, \item\label{l_description_chains_In2} we have $j_1\lhd_n j_2\lhd_n\ldots \lhd_n j_{k_n}$, \item\label{l_description_chains_In3} the sequence $(k_n)$ is non-increasing. \end{enumerate} Suppose moreover that for all $n\geq 2$, $\bt(j_1,n)\geq\bt(j_2,n-1)$. Then: \begin{enumerate} \setcounter{enumi}{3} \item\label{l_description_chains_In4} we have $\supp(j_k)=\{j_{k-1},j_k,j_{k+1}\}$, \item\label{l_description_chains_In5} for all $i\in \llbracket 1,k\rrbracket$ and $n\geq 2$\begin{equation}\label{eq: k_n decreasing} \bt(j_i,n)\geq \bt(j_{i+1},n-1), \end{equation} \item\label{l_description_chains_In6} the sequence $(k_n)$ is decreasing until it reaches $0$. In~particular $\max N_{j_1} \leq k+1$. \end{enumerate} \end{Lemma} \begin{proof} We assume that $E:=\{x\in \llbracket 1,k\rrbracket\mid j_x\in I_n\}$ is non-empty. Let $y=|E|$. For $x\in \llbracket 1,y\rrbracket$, let $\tilde{t}_x\in \llbracket 2,L\rrbracket$ be such that $|\{x'\in \llbracket 1,k\rrbracket\mid N_{j_{x'}}(\tilde{t}_{x})\geq n\}|\geq x$ and such that~$\tilde{t}_x$ is minimal for this property. By~definition, $\tilde{t}_1<\tilde{t}_2<\ldots <\tilde{t}_y$. Let $x\in \llbracket 1,k\rrbracket$ be such that $N_{j_x}(\tilde{t}_1)=n$. We~then have $N_{j_x}(\tilde{t}_1-1)=n-1$ and $\alpha_{j_x}(\gamma^\vee_{\tilde{t}_1-1})=-1$. Assume $x\in \llbracket 2,k'-1\rrbracket$. Then by Lemma~\ref{l_description_chains_I2} we have \[ \supp(j_x)=\{j_{x-1},j_x,j_{x+1}\}\quand a_{j_{x-1},j_x}=-1=a_{j_{x+1},j_x}. \] We~have $N_{j_{x-1}}(\tilde{t}_1-1),N_{j_{x+1}}(\tilde{t}_1-1)\leq n-1$. Then: \begin{align*} \alpha_{j_x}(\gamma^\vee_{\tilde{t}_1-1})=&\alpha_{j_x}\left(N_{j_{x-1}}(\tilde{t}_1-1)\alpha_{j_{x-1}}^\vee+N_{j_{x}}(\tilde{t}_1-1)\alpha_{j_{x}}^\vee+N_{j_{x+1}}(\tilde{t}_1-1)\alpha_{j_{x+1}}^\vee\right)\\ &\geq 2(n-1)-2(n-1)=0: \end{align*} a contradiction. Therefore $x\notin \llbracket 2,k'-1\rrbracket$. Assume $k=k'$ and $x=k$. Then $N_{j_{x-1}}(\tilde{t}_1-\nobreak1)\leq n-1$. By~\eqref{e_passage_3_before_2}, $a_{j_{k-1},j_k}=-1$ and we reach a contradiction again. Therefore, $x=1$. Let $x\in \llbracket 2,y\rrbracket$ and $i\in \llbracket 1,k\rrbracket$ be such that $N_{j_i}(\tilde{t}_x-1)=n-1=N_{j_i}(\tilde{t}_x)-1$. We~assume that for all $x'\in \llbracket 1,x-1\rrbracket$, \[ \{z\in \llbracket 1,k\rrbracket \mid N_{j_z}(\tilde{t}_{x'})\geq n\}=\llbracket 1,x'\rrbracket \quand\{z\in \llbracket 1,k\rrbracket \mid N_{j_z}(\tilde{t}_{x'})\leq n-1\}=\llbracket x'+1,k\rrbracket. \] Then $i\geq x$. Assume $i\in \llbracket x+1,y\rrbracket$ and $i1$, which proves that $\max N_{j_1}\leq k+1.$ Finally, suppose that $\bt(j_2',2)=\infty$, that is to say $j_2'\notin I_2$. Let $n\geq 2$ and suppose that $t:=\bt(j_1,n)$ is finite. Then $-1=\alpha_{j_1}(\gamma^\vee_t)=2(n-1)-2N_{j_2}(t)-1$, so~$N_{j_2}(t)=n-1$. Hence $\bt(j_2,n-1)\leq \bt(j_1,n)$. The second inequality is given by Lemma~\ref{l_description_chains_In}\eqref{l_description_chains_In2}. By~Lemma~\ref{l_description_chains_In}\eqref{l_description_chains_In6}, we~deduce $\max N_{j_1} \leq k+1$. \end{proof} \Subsubsection{Case where $|\supp(j_1)|=4$}\label{subsection: supp = 3} \begin{Proposition}\label{Proposition: supp=4 kmax} Let $j_1 \in I_2$ be minimal for $\leq_2$ such that $|\supp(j_1)|=4$. Let $k$ denote the maximal length of a $\leq_2$-chain starting at $j_1$. Then $\max N_{j_1} \leq \max(6,k+1)$. \end{Proposition} \begin{proof} If $j_1 \notin I_3$ then $\max N_{j_1}\leq 2$ and the result is clear, so~we can assume $j_1 \in I_3$. If $|\supp(j_1)|=4$ and $\supp(j_1)\not\subset I_2$, then by Lemma~\ref{Lemma: class 4A} $\max N_{j_1} \leq k+1$. Otherwise, since we assume $j_1\!\in\! I_3$, either there is $j\!\in\! \supp(j_1)$ such that $\supp(j)\!\not\subset\! I_2$ and, by~Lemma~\ref{Lemma: class 4D}, $\max N_{j_1} \leq k+1$; or $\supp(j)\subset I_2$ for all $j\in \supp(j_1)$ and, by~Lemma~\ref{Lemma: class 4E}, we~have $\max N_{j_1}\leq 6$. \end{proof} \begin{Lemma}\label{Lemma: class 4A} Let $j_1\in I_2$ be minimal for $\leq_2$ such that $|\supp(j_1)|=4$ and suppose $\supp(j_1)\not\subset I_2$. Then, any maximal chain $j_1 \lhd_2 j_2 \dots \lhd_2 j_k$ with $k\geq 2$ satisfies properties of Lemma \ref{l_description_chains_In}\,\eqref{l_description_chains_In4},\,\eqref{l_description_chains_In5}\,\&\,\eqref{l_description_chains_In6} and in particular $\max N_{j_1}\leq k+1$. \end{Lemma} \begin{proof} Let us write $\supp(j_1)=\{j_1,j_2,j_2',j_2''\}$ with $j_2''\notin I_2$. By Lemma~\ref{l_description_chains_In}, to prove the result it suffices to prove that $\bt(j_1,n)\geq \bt(j_2,n-1)$ for all $n\geq 2$. If $\bt(j_1,n)=\infty$ then the inequality is clear, so~suppose $t=\bt(j_1,n)\neq \infty$. Since $N_{j_2''}(t)=1$ we have $$-1=\alpha_{j_1}(\gamma_t^\vee)=2N_{j_1}(t)-N_{j_2}(t)-N_{j_2'}(t)-N_{j_2''}(t)=2(n-1)-1-N_{j_2}(t)-N_{j_2'}(t),$$ so $N_{j_2}(t)+N_{j_2'}(t)=2(n-1)$. Moreover, $\max(N_{j_2}(t),N_{j_2'}(t))\leq n-1$, since, by~Lemma~\ref{l_description_chains_In}\eqref{l_description_chains_In2}, $j_1$ is also minimal for $\leq_n$. Therefore $N_{j_2}(t)+N_{j_2'}(t)=2(n-1)$ implies $N_{j_2}(t)=N_{j_2'}(t)=n-1$, which proves $\bt(j_1,n)=t>\bt(j_2,n-1)$. \end{proof} \begin{Lemma}\label{Lemma: Class 4S} Let $j_1\in I_2$ be minimal for $\leq_2$ such that $|\supp(j_1)|=4$, $\supp(j_1) \subset I_2$ and $j_1\in I_3$. Then there is a unique $j\in \supp(j_1)$ such that $|\supp(j)|=2$, and for any $j_2\in \supp(j_1)\setminus \{j_1,j\}$, we~have $|\supp(j_2)|=3$ and \begin{equation}\label{eq: Class 4S statement}\bt(j_2,2)<\bt(j_1,3)<\bt(j,2)\leq \bt(j_1,4). \end{equation} \end{Lemma} \begin{proof} Write $\supp(j_1)\setminus \{j_1\}=\{j_2^{(1)},j_2^{(2)},j_2^{(3)}\}$. Fix $i\in \llbracket 1,3\rrbracket$. Let $t=\bt(j_1,3)\neq \infty$. Since $j_1$ is minimal for $\leq_2$, it is also minimal for $\leq_3$. Then \[ N_{j_2^{(1)}}(t)+N_{j_2^{(2)}}(t)+N_{j_2^{(3)}}(t)=5\quand N_{j_2^{(i)}}(t)\leq 2\quad \text{for all $i$}, \] so, upon relabeling, we~can suppose $(N_{j_2^{(1)}}(t),N_{j_2^{(2)}}(t),N_{j_2^{(3)}}(t))=(2,2,1)$. For $i\in \{1,2\}$, $t_i:=\bt(j_2^{(i)},2)$ satisfies \[ t_i\bt(j_2',n+1)$. Let $t=\bt(j_1,2n+2)$ which we suppose finite, then $N_{j_1}(t)=2n+1$, $N_{j_2'}(t)\leq n+1$ (by Lemma~\ref{l_relation_Nk-1_Nk}) and $N_{j_2}(t)\leq 2n+1$. Therefore \[ -1=\alpha_{j_1}(\gamma_t^\vee)\geq 4n+2-(2n+1)-(n+1)-N_{j_2''}(t)=n-N_{j_2''}(t), \] so $N_{j_2''}(t)\geq n+1$ and $\bt(j_2'',n+1)\leq t$. This concludes the proof of Formula~\eqref{eq: type 4D}. Let us check that $\bt(j_1,n)\geq \bt(j_2,n-1)\geq\bt(j_1,n-1)$ for any $n\geq 3$. The inequality $\bt(j_2,n-1)\geq \bt(j_1,n-1)$ holds because $j_1 \lhd_2 j_2 \Rightarrow j_1 \lhd_{n-1} j_2$ by Lemma~\ref{l_description_chains_In}\eqref{l_description_chains_In2} (if~they belong to $I_{n-1}$). For the remaining inequality, suppose $t:=\bt(j_1,n)\neq \infty$: \begin{itemize} \item If $n$ is even, write $n=2p$. Then by Formula~\eqref{eq: type 4D}, $\max(N_{j_2'}(t),N_{j_2''}(t))\leq p$, so~$-1=\alpha_{j_1}(\gamma_t^\vee)\geq 2(2p-1)-2p-N_{j_2}(t)$ . We~deduce $N_{j_2}(t)\geq n-1$. \item Otherwise, write $n=2p+1$. Then similarly $N_{j_2''}(t)\leq p$ and $N_{j_2'}(t)\leq p+1$, so~$-1=\alpha_{j_1}(\gamma_t^\vee)\geq 2(2p)-2p-1-N_{j_2}(t)$, so~$N_{j_2}(t)\geq 2p=n-1$. \end{itemize} Either way, we~have proved $\bt(j_1,n)\geq\bt(j_2,n-1)$. Then \eqref{Lemma: class 4D2} follows from Lemma~\ref{l_description_chains_In}. \end{proof} \begin{Lemma}\label{Lemma: class 4E} Let $j_1\in I_2$ be minimal for $\leq_2$ with $|\supp(j_1)|=4$, $j_1\in I_3$ and, for all $j\in \supp(j_1)$, $\supp(j)\subset I_2$. Then $\max N_{j_1} \leq 6$. More precisely, using Lemma~\ref{Lemma: Class 4S}, we~write $\supp(j_1)=\{j_1,j_2,j_2',j_2''\}$ such that \[ |\supp(j_2'')|=2 \quand\bt(j_2,3)\leq \bt(j_2',3), \] and write $\supp(j_2')=\{j_1,j_2',j_3'\}$. Then: \begin{enumerate} \item\label{Lemma: class 4E1} If $\bt(j_2',3)\leq \bt(j_1,4)$, then $j_1\notin I_4$ and $j_2''\notin I_3$ \item\label{Lemma: class 4E2} If $\bt(j_1,4)\neq \infty$ and $\bt(j_2',3)=\infty$, then $j_1\notin I_5$ and $j_2''\notin I_3$ \item\label{Lemma: class 4E3} Otherwise, if $\bt(j_1,4)< \bt(j_2',3)<\infty$, then $\bt(j_2,n-1)\leq \bt(j_1,n)$ for all $n\in \llbracket2, 5\rrbracket$, $\supp(j_3')=\{j_2',j_3'\}$ and $j_1 \notin I_7$, $j_2'\notin I_5$, $j_3'\notin I_3$. \end{enumerate} \end{Lemma} \begin{proof} Suppose first that $\bt(j_2',3)\leq \bt(j_1,4)$ and assume, for a contradiction, that $t:=\bt(j_1,4)$ is finite. By~Lemma~\ref{Lemma: Class 4S}, we~have $N_{j_2''}(t)\geq 2$, and since $\bt(j_2,3)\!\leq\! \bt(j_2',3)\!\leq\! \bt(j_1,4)$, we~have $N_{j_2}(t)\!\geq\! 3$, $N_{j_2'}(t)\!\geq\! 3$. Therefore $N_{j_2}(t)+N_{j_2'}(t)+N_{j_2''}(t)\!\geq\! 8$, so~$-1=\alpha_{j_1}(\gamma_t^\vee)\leq -2$, a contradiction. Hence in this case, $\bt(j_1,4)=\infty$ and $j_1\notin I_4$. Suppose now that $\bt(j_1,4)\neq \infty$ and $\bt(j_2',3)=\infty$ and let $t\geq \bt(j_1,4)+1$. Then $N_{j_1}(t)=4$ so $N_{j_2}(t)\leq 4$ and $N_{j_2''}(t)\leq 2$ by Lemma~\ref{l_relation_Nk-1_Nk}. Moreover, by~assumption $N_{j_2'}(t)\leq 2$ and thus $\alpha_{j_1}(\gamma_t^\vee)\geq 8-4-2-2=0$ cannot be equal to $-1$, so~$N_{j_1}$ is stuck at $4$: $j_1\notin I_5$. Moreover, in~these two cases, $N_{j_1}\leq 4$ and thus by Lemma~\ref{l_relation_Nk-1_Nk}, $N_{j_2''}\leq 2$. Finally, assume that $\bt(j_1,4)<\bt(j_2',3)\neq\infty$. Note that $\bt(j_2,3) < \bt(j_1,4)$ is necessary in order to have $\alpha_{j_1}(\gamma^\vee_{\bt(j_1,4)})=-1$. Let $t=\bt(j_2',3)$, by~assumption $N_{j_1}(t)\geq 4$ and $N_{j_3'}(t)\geq 1$. So $\alpha_{j_2'}(\gamma^\vee_t)=-1$ forces $N_{j_1}(t)=4$ and $N_{j_3'}(t)= 1$. The first consequence is that $\bt(j_2',3)<\bt(j_3',2)$. Since we suppose $\supp(j_2')\subset I_2$, a computation at time $\bt(j_3',2)$ then proves that \[ \supp(j_3')=\{j_2',j_3'\}. \] We then have \[ j_3'\in I_3 \implies j_2'\in I_5 \implies j_1\in I_7. \] Indeed, the first implication is a direct consequence of Lemma~\ref{l_relation_Nk-1_Nk}\eqref{l_relation_Nk-1_Nk2}. Now assume that $t:=\bt(j_2',5)<\infty$. Then by Lemma~\ref{l_relation_Nk-1_Nk}, we~have $N_{j_3'}(t)\leq 2$ and thus $\alpha_{j_2'}(\gamma_t^\vee)=-1=8-N_{j_1}(t)-N_{j_3'}(t)$ and so $N_{j_1}(t)\geq 7$ and \begin{equation}\label{e_ineq_j1_j2'} \bt(j_1,7)\leq \bt(j_2',5). \end{equation} We now show that, under the assumptions of the statement, $j_1\notin I_7$, which is enough to conclude. We computed $N_{j_1}(\bt(j_2',3))=4$ and $N_{j_3'}(\bt(j_2',3))=1$, so~$\bt(j_2',3)<\bt(j_1,5)$. Let $t=\bt(j_1,5)$ and suppose it is finite. For $\bt(j_2',3)\!t$ we have $\alpha_{j_2'}(\gamma^\vee_{t'})\!\leq\! 6-6-2\!<\!-1$ by Lemma~\ref{l_relation_Nk-1_Nk}), so~$\bt(j_2',4)=\infty$. Therefore if $t'=\bt(j_1,7)$ is finite, necessarily $(N_{j_1}(t'),N_{j_2''}(t'))=(6,4)$, which contradicts Lemma~\ref{l_relation_Nk-1_Nk}\eqref{l_relation_Nk-1_Nk2}. Hence $\bt(j_1,7)$ is infinite and $j_1\notin I_7$, which implies $j_2'\notin I_5$, $j_3'\notin I_3$. \item Suppose that $(N_{j_2}(t),N_{j_2'}(t),N_{j_2''}(t))=(5,4,2)$. The situation is similar: in this case $N_{j_2''}$ is stuck at $2$, so~$\bt(j_2',5)\leq \bt(j_1,7)$. However by Lemma~\ref{l_relation_Nk-1_Nk}\eqref{l_relation_Nk-1_Nk2}, $\bt(j_2',5) \leq \bt(j_3',3)$. We~deduce that, for any $\bt(j_1,6)< t'\leq \bt(j_2',5)$, $\alpha_{j_2'}(\gamma^\vee_{t'})=0$, so~$\bt(j_2',5)=\bt(j_1,7)=\bt(j_3',3)=\infty$. \item Suppose that $(N_{j_2}(t),N_{j_2'}(t),N_{j_2''}(t))=(4,4,3)$. Then by \eqref{e_ineq_j1_j2'} and Lem\-ma~\ref{l_relation_Nk-1_Nk}, we~have $$\bt(j_2,6)\leq \bt(j_1,7)\leq \min(\bt(j_2',5),\bt(j_2'',4)).$$ Hence if $t:=\bt(j_1,7)<\infty$, we~have $N_{j_2}(t)+N_{j_2'}(t)+N_{j_2''}(t)=13$, with $N_{j_2'}(t)+N_{j_2''}(t)\leq 7$. Consequently $\bt(j_2,6)<\bt(j_1,7)<\infty$. But this contradicts Lemma~\ref{l_impossible_scheme} below. Therefore $j_1\notin I_7$, $j_2'\notin I_5$ and $j_3'\notin I_3$.\qedhere \end{enumerate} \end{proof} \begin{Lemma}\label{l_impossible_scheme} Let $j_1,j_2$ be the first terms of a maximal chain for $\leq_2$. Then there exists no pair $(x,y)\in \llbracket 1,L\rrbracket^2$ such that $(N_{j_1}(x),N_{j_2}(x))=(6,4)$ and $(N_{j_1}(y),N_{j_2}(y))=(6,6)$. \end{Lemma} \begin{proof} We assume that there exists $(x,y)\in \llbracket 1,L\rrbracket^2$ such that $(N_{j_1}(x),N_{j_2}(x))=(6,4)$ and $(N_{j_1}(y),N_{j_2}(y))=(6,6)$. Let $z\in \llbracket 1,L\rrbracket$. Assume $(N_{j_1}(z),N_{j_2}(z))=(6,4)$. If~we had $\supp(j_2)=\{j_1,j_2\}$, then $j_2$ would be maximal for $\leq_2$ and Lemma~\ref{l_relation_Nk-1_Nk}\eqref{l_relation_Nk-1_Nk2} would lead to a contradiction. Therefore we have $\supp(j_2)=\{j_1,j_2,j_3\}$, for some $j_3\in I_1$, and by Lemma~\ref{l_description_chains_I2}, we~have $a_{j_1,j_2}=a_{j_3,j_1}=-1$. Let $t=\bt(j_2,6)$. We~have $N_{j_1}(t)=6$ by assumption. We~have $10-6-N_{j_3}(t)=-1$ and thus \begin{equation}\label{e_ineq_t_t} \bt(j_3,4)< \bt(j_3,5)<\bt(j_2,6). \end{equation} Let $t=\bt(j_2,5)$. We~have $t<\bt(j_1,7)$ and thus we have $N_{j_3}(t)=3$ (in order to have $8-6-3=-1$). Therefore $(N_{j_1}(t+1),N_{j_2}(t+1),N_{j_3}(t+1))=(6,5,3)$. Now let $t\in \llbracket 1,L\rrbracket$ be such that $(N_{j_1}(t),N_{j_2}(t),N_{j_3}(t))=(6,5,3)$. As $\bt(j_3,4)<\infty$ and $6-5\neq -1$, we~have $\supp(j_3)\neq \{j_2,j_3\}$. So by Lemma~\ref{l_description_chains_I2}, we~can write $\supp(j_3)=\{j_2,j_3,j_4\}$, for some $j_4\in I_1$, and we have $a_{j_2,j_3}=a_{j_4,j_2}=-1$. If~$t=\bt(j_3,4)$, we~have $N_{j_4}(t)=2$ (since $6-5-2=-1$). So we have \begin{equation}\label{e_configuration1} (N_{j_1}(t),N_{j_2}(t),N_{j_3}(t),N_{j_4}(t))=(6,5,3,2) \end{equation} and \begin{equation}\label{e_configuration2} (N_{j_1}(t+1),N_{j_2}(t+1),N_{j_3}(t+1),N_{j_4}(t+1))=(6,5,4,2). \end{equation} Let $t=\bt(j_3,5)$. By~\eqref{e_ineq_t_t}, we~have $\gamma_t^\vee=6\alpha_{j_1}^\vee+5\alpha_{j_2}^\vee+4\alpha_{j_3}^\vee+a\alpha_{j_4}^\vee+\delta^\vee$, for some $a\in \llbracket 1,4\rrbracket$ and $\delta^\vee\in \bigoplus_{i\in I\setminus \{j_1,j_2,j_3,j_4\}}\N\alpha_i^\vee$. We~have $\alpha_{j_3}(\gamma_t^\vee)=8-5+a\alpha_{j_3}(\alpha_{j_4}^\vee)=-1$ and thus $a \alpha_{j_3}(\alpha_{j_4}^\vee)=aa_{j_4,j_3}=-4$ and thus $a=4$. Consequently \[ \bt(j_4,4)<\bt(j_3,5)<\infty. \] Let $\tilde{t}=\bt(j_4,4)$. We~have $2N_{j_4}(\tilde{t})=6> N_{j_3}(\tilde{t})+1$ since $N_{j_3}(\tilde{t})\leq 4$. By~Lemma~\ref{l_relation_Nk-1_Nk}, this implies $\supp(j_4)\neq \{j_3,j_4\}.$ Let $j_5$ be such that $\supp(j_4)=\{j_3,j_4,j_5\}$. Then $6-N_{j_3}(\tilde{t})-N_{j_5}(\tilde{t})=-1=6-4-N_{j_5}(\tilde {t})$ and thus \begin{equation}\label{e_majoration_t5} \bt(j_5,3)<\bt(j_4,4)<\infty. \end{equation} Let $t=\bt(j_4,3)$. By~\eqref{e_configuration2}, we~have $4-4-N_{j_5}(t)=-1$ and thus $N_{j_5}(t)=1$. We~thus have\vspace*{-3pt} \[ (N_{j_1}(t),N_{j_2}(t),N_{j_3}(t),N_{j_4}(t),N_{j_5}(t))=(6,5,4,2,1) \] and then\vspace*{-3pt} \[ (N_{j_1}(t+1),N_{j_2}(t+1),N_{j_3}(t+1),N_{j_4}(t+1),N_{j_5}(t+1))=(6,5,4,3,1). \] By \eqref{e_majoration_t5}, $t:=\bt(j_5,2)<\infty$. Then $\gamma_t^\vee= z \alpha_{j_4}^\vee+\alpha_{j_5}^\vee+\delta^\vee,$ for some $z\in \Z_{\geq 3}$ and $\delta^\vee\in Q^\vee_\N=\bigoplus_{i\in I\setminus \{j_4,j_5\}}\N \alpha_i^\vee$. By~Lemma~\ref{l_description_chains_I2} $a_{j_4,j_5}=-1$, thus $2-z-\alpha_{j_5}(\delta^\vee)=-1$ and hence $z=3$ and $\alpha_{j_5}(\delta^\vee)=0$. This proves that $\supp(j_5)=\{j_4,j_5\}$. Then if $t=\bt(j_5,2)$, we~have\vspace*{-3pt} \[ (N_{j_1}(t),N_{j_2}(t),N_{j_3}(t),N_{j_4}(t),N_{j_5}(t))=(6,5,4,3,1) \] and\vspace*{-3pt} \[ (N_{j_1}(t+1),N_{j_2}(t+1),N_{j_3}(t+1),N_{j_4}(t+1),N_{j_5}(t+1))=(6,5,4,3,2). \] But then $2N_{j_i}(t+1)-N_{j_{i+1}}(t+1)-N_{j_{i-1}}(t+1)=0$ for every $i\in \llbracket 2,4\rrbracket$, and $2N_{j_5}(t+1)-N_{j_4}(t+1)=1$. Therefore the situation can no longer evolve, which proves the lemma. \end{proof} \subsubsection{Conclusion: Finiteness of the set of quantum roots} With the preceding results, we~are already able to give an upper bound on the number of quantum roots in a given Kac--Moody root system. In~particular, we~prove that the set $\cQ(\Phi_+)$ is finite for any Kac--Moody root system. \begin{Proposition}\label{p_majoration_coefficients_AS_roots} Let $k_{\max}$ be the length of a maximal chain for $\leq_2$ and $n=|I|$ (we have $k_{\max}\leq n$). Let $\beta$ be a quantum root. Write $\beta^\vee=\sum_{i\in I} x_i \alpha_i^\vee$. Then we have $x_i\leq\max(k_{\max}+1,6)$ for every $i\in I$. In~particular, $\htt(\beta^\vee)\leq n(\max(6,k_{\max}+1))$. \end{Proposition} \begin{proof} Let $i\in I$. If~$i\notin I_2$, we~have $x_i\leq 1$. Assume $i\in I_2$. Let $j\in I_2$ be such that $j\leq_2 i$ and such that $j$ is minimal for $\leq_2$. Then by \eqref{e_inequalities_N}, we~have $x_i\leq x_j$. Moreover, by~Lemma \ref{Lemma: class 3}, Lemma \ref{l_bounding_supp=2} and Proposition \ref{Proposition: supp=4 kmax}, $x_j\leq \max(6,k_{\max}+1)$. Therefore $\htt(\beta^\vee)=\sum_{i'\in I_1} x_{i'}\leq |I_1| \max(k_{\max}+1,6)\leq n\max(k_{\max}+1,6).$ \end{proof} \begin{Theorem}\label{t_finiteness_almost_simple_roots} Let $n\in \Z_{\geq 2}$. Then for every generalized Cartan matrix $A$ of size~$n$, if~$\Phi$ is the real root system associated to $A$, then the set of quantum roots is finite, with cardinality at most $(n+6)^n$. \end{Theorem} \begin{proof} Let $\cQ(\Phi_+)$ be the set of quantum roots. By Proposition~\ref{p_majoration_coefficients_AS_roots}, for every $\beta\in \cQ(\Phi_+)$, we~can write $\beta^\vee=\sum_{i\in I} x_i \alpha_i^\vee$, with $x_i\in \llbracket 0,n+5\rrbracket$ for every $i\in I$. Therefore $|\cQ(\Phi_+)|=|\{\beta^\vee\mid \beta\in \cQ(\Phi_+)\}|\leq (n+6)^n$. \end{proof} \skpt \begin{Remark} \begin{enumerate} \item Note that the bound given in Theorem~\ref{t_finiteness_almost_simple_roots} is very crude and is never sharp. A way to improve it would be to first count the number of possible $1$-star-convex subtrees of $I$ (whose cardinality is at most $2^{|I|}$). Then for a given such subtree $I_1$ of $I$, the constraints of Lemma~\ref{l_description_chains_I2}, Lemma~\ref{l_description_chains_In} and Proposition~\ref{p_majoration_coefficients_AS_roots} should yield a much better upper bound on the number of quantum $\beta$ satisfying $I_1(\beta)=I_1$. Moreover if a generalized Cartan matrix has few $-1$ coefficients, then the associated root system will have few quantum roots, see Lemma~\ref{l_no_-1_coefficient} for example. \item Note that in the reductive case, in~ADE type, the set of quantum roots is the entire set $\Phi_+$ (see Proposition~\ref{p_quantum_ADE}). By contrast, when $\Phi_+$ is infinite, ``almost all'' the roots are not quantum, by~the theorem above. \end{enumerate} \end{Remark} \Subsection{Classification of quantum roots through Dynkin sequences}\label{subsection: classification} In Sections~\ref{subsection: supp < 3} and~\ref{subsection: supp = 3} we have obtained finer structure results on quantum roots than those we used to prove the finiteness of $\cQ(\Phi_+)$. In~this section, we~reformulate these results in terms of Dynkin sequences. We~then obtain a complete classification of quantum roots through Dynkin sequences (see Theorem~\ref{Theorem: Classification quantum roots}). This section is independent of Section~\ref{s_almost_simple_roots_covers}. \subsubsection{Standard Dynkin diagrams} Certain classes of Dynkin diagrams have standard names, which we now recall. Most of our notation follows the standard classification of Dynkin diagrams (see e.g., \cite{bourbaki1981elements}). In~this section, $\Gamma$ is a Dynkin diagram with set of edges $E$ and weight function $w$. Recall that, if~$A=(a_{i,j})_{i,j\in I}$ is a generalized Cartan matrix, then the associated Dynkin diagram is $\Gamma=(I,E,w)$ with \[ E=\{(i,j)\in I^2 \mid a_{i,j}< 0\}\quand w(i,j)=-a_{i,j}\text{ for all }(i,j)\in E. \] We~identify a Dynkin diagram $\Gamma$ with its set of vertices. Among Dynkin segments, we~have the following classes: \begin{itemize} \item[(A)] For $n\geq 1$, we~say that $\Gamma$ is a Dynkin diagram of type $A_n$ if it is a segment with $n$ vertices such that $w(i,j)=1$ for all $(i,j)\in E$. \item[(C)] For $n\geq 2$, we~say that $\Gamma$ is a Dynkin diagram of type $C_n$ if it is a segment with a leaf $j_0$ such that $\Gamma \setminus \{j_0\}$ is a diagram of type $A_{n-1}$ and, for $j_1$ the unique vertex adjacent to $j_0$, $(w(j_0,j_1),w(j_1,j_0))=(1,2)$. We~call $j_0$ the \emph{special vertex} of $\Gamma$. \item[(F)] For $n\geq 4$, we~say that $\Gamma$ is a Dynkin diagram of type $F_n$ if it is a Dynkin segment with a leaf $j_{-1}$ such that $\Gamma \setminus \{j_{-1}\}$ is a Dynkin diagram of type $C_{n-1}$ and its special vertex $j_0$ is the unique neighbor of $j_{-1}$. \item[(G)] For $n\geq 2$, we~say that $\Gamma$ is a Dynkin diagram of type $G_n$ if it is a segment with a leaf $j_0$ such that $\Gamma \setminus \{j_0\}$ is a diagram of type $A_{n-1}$ and, for $j_1$ the unique vertex adjacent to $j_0$, $(w(j_0,j_1),w(j_1,j_0))=(1,3)$. \end{itemize} We also have classes of Dynkin trees which are not Dynkin segments, but Dynkin trees with a unique branching point: \begin{itemize} \item[(D)] For $n\geq 4$, we~say that $\Gamma$ is a Dynkin diagram of type $D_n$ if it has two distinct leaves $j_2'$, $j_2''$ sharing their unique neighbor $j_1$, such that $\Gamma \setminus \{j_2',j_2''\}$ is a Dynkin diagram of type $A_{n-2}$ with leaf $j_1$. \item[(E)] For $n\geq 6$, we~say that $\Gamma$ is a Dynkin diagram of type $E_n$ if it has a unique branching point $j_1$ with three neighbors, it is not of type $D_n$ but there exists $j_3'\in \Gamma$ such that $\Gamma \setminus \{j_3'\}$ is a Dynkin diagram of type $D_{n-1}$. \end{itemize} Note that the Dynkin diagrams associated to finite root systems are exactly the Dynkin diagrams of type $(A_n)_{n\geq 1}$, $(B_n)_{n\geq 2}$ (which is dual to type $C$), $(C_n)_{n\geq 2}$, $(D_n)_{n\geq 4}$ and the exceptional types $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$. \begin{figure}[htb] \centering \includegraphics[width=.7\linewidth]{Dynkin diagrams classical types} \caption{The different types of Dynkin diagrams} \label{fig: Type of Dynkin diagrams} \end{figure} \subsubsection{Dynkin sequences associated to quantum roots}\label{subsection: Dynkin sequences quantum roots} In this section, we~fix a quantum root $\beta$. For $n\geq 2$ and $j\in I_2(\beta)$ such that $\deg(j)=3$, let $I_n(\beta,j)$ denote the connected component of $I_n(\beta)$ containing $j$ (if $j\notin I_n(\beta)$ then we set $I_n(\beta,j)=\emptyset$). We~have the following classification for $(I_n(\beta,j))_{n\geq 2}$. \begin{Proposition}\label{Classification: supp=2} Let $j\in I_2(\beta)$ be such that $\deg(j)=3$ and $|\supp(j)|=2$. Then \begin{itemize} \item[(2G)] $I_2(\beta,j)$ is a Dynkin diagram of type $G$, $j$ is the only leaf of $I_2(\beta,j)$ which is a leaf in $I_1(\beta)$. Moreover $I_3(\beta,j)$ is empty. \end{itemize} \end{Proposition} \begin{figure}[htb] \centering \includegraphics[width=.7\linewidth]{Dynkin Type 2.png} \caption{Dynkin diagram of type (2G)} \end{figure} \begin{proof} By Lemma~\ref{l_bounding_supp=2}, $j\notin I_3$, so~$I_3(\beta,j)$ is empty. Moreover, since $|\supp(j)|=2$, by~Lemma \ref{l_description_chains_I2} $I_2(\beta,j)$ is a Dynkin segment of type $G$, with $j$ as special vertex. If~$I_2(\beta,j)\neq \{j\}$, by~Lemma \ref{l_description_chains_In}\eqref{l_description_chains_In4}, the other leaf of $I_2(\beta,j)$ has support of size three, hence is not a leaf in $I_1(\beta)$. \end{proof} \begin{Proposition}\label{Classification: supp=3} Let $j\in I_2(\beta)$ be such that $\deg(j)=3$ and $|\supp(j)|=3$. Then, one of the following is satisfied: \begin{itemize} \item[(3S)] $I_3(\beta,j)=\emptyset$ and $I_2(\beta,j)\setminus \{j\}$ is the disjoint union of two non-empty Dynkin diagrams of type $A$. Moreover the leaves of $I_2(\beta,j)$ are not leaves in $I_1(\beta)$. \item[(3C)] $(I_n(\beta,j))_{n\geq 2}$ is a decreasing sequence of Dynkin diagrams of type $C$ with leaf~$j$. The leaves of $I_2(\beta,j)$ are not leaves of $I_1(\beta)$. \item[(3F)] Let $j_2'\in \supp(j)$ be such that $a_{j_2',j}=-1$. Then $j_2'$ is a leaf of both $I_1(\beta)$ and $I_2(\beta,j)$, $I_2(\beta,j)\setminus \{j'_2\}$ is of type $C$, $I_3(\beta,j)$ is of type $C$ and $I_4(\beta,j)=\emptyset$. The leaves of $I_3(\beta,j)$ (\resp $I_2(\beta,j)\setminus \{j_2'\}$) are not leaves in $I_2(\beta,j)$ (\resp $I_1(\beta)$). \end{itemize} \end{Proposition} \begin{figure}[htb] \centering \includegraphics[width=.7\linewidth]{Dynkin Type 3.png} \caption{Dynkin sequences of type (3)} \label{fig: Dynkin diagram Type 3} \end{figure} \begin{proof} This is a reformulation of Lemma \ref{Lemma: class 3}. More precisely, write $\supp(j)=\{j,j_2,j_2'\}$ with $a_{j_2,j_1}=-2$ and $a_{j_2',j_1}=-1$. By~Proposition \ref{p_connected_components_I2}, $I_2(\beta,j)\setminus \{j\}$ is the disjoint union of two diagrams of type $A$. The case \eqref{Lemma: class 31} of Lemma \ref{Lemma: class 3} therefore corresponds to type (3S), the case \eqref{Lemma: class 32} corresponds to the type (3F), and the case~\eqref{Lemma: class 33} corresponds to type (3C). Moreover if $j_2\in I_2(\beta,j)$, in~every case we can apply Lem\-ma~\ref{l_description_chains_In}\,\eqref{l_description_chains_In4}\,\&\,\eqref{l_description_chains_In5} to the chain starting with $j_1 \lhd_2 j_2$ to deduce that, for every $n\geq 2$, the leaves of $I_n(\beta,j)$ are not leaves of $I_{n-1}(\beta,j)$ (except $j_2'$ for $n=2$ in type (3F), and $j$ for $n\geq 3$ in type (3C)). \end{proof} \begin{Proposition}\label{Classification supp=4} Let $j_1\in I_2(\beta)$ be such that $\deg(j_1)=3$ and $|\supp(j_1)|=4$. Then the Dynkin sequence $(I_n(\beta,j_1))_{n\geq 2 }$ is of one of the following forms. \begin{itemize} \item[(4S)] The diagram $I_2(\beta,j_1)\setminus \{j_1\}$ is the disjoint union of three non-empty diagrams of type $A$. Moreover the leaves of $I_2(\beta,j_1)$ are not leaves in $I_1(\beta)$, and $I_3(\beta,j_1)=\emptyset$. \item[(4A)] The Dynkin sequence $(I_n(\beta,j_1))_{n\geq 2}$ is a decreasing sequence of diagrams of type $A$ such that, for all $n \geq 2$, the leaves of $I_n(\beta,j_1)$ are not leaves of $I_{n-1}(\beta,j_1)$. \item[(4D)]\label{Class 4D} Let $K=N_{j_1}(\beta)=\max \{n\in \Z_{\geq 0}\mid j_1\in I_K(\beta)\}$. Then $K\geq 5$ and there exist $j_2',j_2''\in \supp(j_1)$ such that: \begin{itemize} \item $j_2''$ is a leaf in $I_1(\beta)$ and $j_2'$ is a leaf of $I_2(\beta,j_1)$ but not of $I_1(\beta)$ \item $(I_n(\beta,j_1))_{n \in \llbracket 1,K\rrbracket}$ is a decreasing sequence of diagrams. If~$K$ is even, then $I_{n}(\beta,j_1)$ is of type $D$ for $n\in \llbracket 1,K/2\rrbracket$ and $I_{n}(\beta,j_1)$ is of type $A$ for \hbox{$n\in \llbracket K/2+1,K\rrbracket$}. If~$K$ is odd, there exists $\eta\in \{0,1\}$ such that $I_{n}(\beta,j_1)$ is of type $D$ for $n\in \llbracket 1,(K-1)/2+\eta\rrbracket$ and $I_{n}(\beta,j_1)$ is of type $A$ for $n\in \llbracket (K-1)/2+\eta+1,K\rrbracket$. \item $j_2'$ and $j_2''$ are leaves of the diagrams containing them. For $n\in \llbracket 1,K-1\rrbracket$, if~$\tilde{j}$ is a leaf of $I_{n}(\beta,j_1)$ and $\tilde{j}\notin\{j_1,j_2',j_2''\}$, then $\tilde{j}\notin I_{n+1}(\beta,j_1)$. \item $I_{K-1}(\beta,j_1)\neq \{j_1\}.$ \end{itemize} \item[(4EA)] $I_2(\beta,j_1)$ is a Dynkin diagram of type $E$, and there exist $j_2',j_2''\in \supp(j_1)$ such that: \begin{itemize} \item $j_2''$ is a leaf in $I_1(\beta)$ and does not lie in $I_3(\beta,j_1)$ \item $\supp(j_2')=\{j_1,j_2',j_3'\}$ for some leaf $j_3'$ of both $I_1(\beta)$, $I_2(\beta)$ and $j_3'\notin I_3(\beta,j_1)$ \item $(I_n(\beta,j_1))_{n\in \{3,4,5\}}$ is a decreasing sequence of diagrams of type $A$, $j_2'$ is a leaf of $I_3(\beta,j_1)$ and $j_2'\notin I_5(\beta,j_1)$ \item $I_7(\beta,j_1)=\emptyset$ and, if~$I_5(\beta,j_1)=\emptyset$ then $j_2'\notin I_4(\beta,j_1)$ \item For $n\geq 1$, if~$j$ is a leaf of $I_n(\beta,j_1)$ and $j\notin \{j_1,j_2',j_3',j_2''\}$ then $j\notin I_{n+1}(\beta)$. \end{itemize} \item[(4ED)] $I_2(\beta,j_1)$ is a Dynkin diagram of type $E$, and there exist $j_2',j_2''\in \supp(j_1)$ such that: \begin{itemize} \item $\supp(j_2')=\{j_1,j_2',j_3'\}$ for some leaf $j_3'$ of $I_1(\beta)$ and $j_3'\notin I_3(\beta,j_1)$ \item $I_3(\beta,j_1)$ is a diagram of type $D$, two of its leaves are $j_2',j_2''$ and $j_2''\notin I_4(\beta,j_1)$ \item $I_4(\beta,j_1)$, $I_5(\beta,j_1)$ are non-empty diagrams of type $A$, $j_1$ is a leaf of $I_5(\beta,j_1)$ \item $I_7(\beta,j_1)=\emptyset$ \item Except in the case (4ED2) below, for $n\geq 1$ if $j$ is a leaf of $I_n(\beta,j_1)$ and $j\notin \{j_1,j_2',j_3',j_2''\}$ then $j\notin I_{n+1}(\beta,j_1)$. \end{itemize} Moreover, exactly one of the following holds: \begin{itemize} \item[(4ED1)] $j_2' \in I_4(\beta,j_1)$, $I_6(\beta,j_1)=\emptyset$. \item[(4ED2)] $j_2' \in I_4(\beta,j_1)$, $I_6(\beta,j_1)=\{j_1\}$ and there exists $n_0\in \llbracket 2,6\rrbracket$ such that $|I_{n-1}(\beta,j_1)\setminus \big(I_{n}(\beta,j_1)\cup \{j_2',j_2'',j_3'\}\big)|=1$ for $n\in \llbracket n_0+1,5\rrbracket$, and \[ I_{n_0-1}(\beta,j_1) \setminus \{j_2',j_2'',j_3'\}=I_{n_0}(\beta,j_1)\setminus \{j_2',j_2'',j_3'\}. \] Moreover for $n0$ (see \cite[Cor.\,1.10]{philippe2023grading} for example) and thus $\langle \alpha^\vee,\beta\rangle=1$ for all $\alpha\in \Inv(s_\beta)$, which proves that $\beta$ is quantum. \end{proof} \begin{Remark} Proposition~\ref{p_quantum_ADE} can also be proved using Theorem~\ref{Theorem: Classification quantum roots}: Dynkin diagrams of type ADE are trees which are $1$-star-convex at any given vertex. Fix a root system $\Phi$ of type ADE, and $\beta\in \Phi_+$. Then the Dynkin diagram $I_1(\beta)$ is of type ADE as well. If~it is of type A, it has no vertex of degree 3 or larger, so~$I_2(\beta)=\emptyset$. Hence $(I_n(\beta))_{n\geq 1}$ is quantum and $\beta\in \cQ(\Phi_+)$. If~$I_1(\beta)$ is of type $D$ or $E$, it admits a unique vertex $j$ of degree 3 and no vertex of higher degree, so~$I_2(\beta)$ is non-empty if and only if it contains $j$. We~check that if this is the case, $(I_n(\beta))_{n\geq 1}$ must be of the form (4A), with $I_3(\beta)=\emptyset$, when $I_1(\beta)$ is of type D; or of the forms (4A), (4D),(4EA), (4ED) when $I_1(\beta)$ is of type E. \end{Remark} However, for every other indecomposable Cartan matrix (of finite type), there exists a positive root which is not quantum. Indeed, there exist $i,j\in I$ such that $\langle \alpha_i^\vee,\alpha_j\rangle\leq-2$. Set $\beta=r_j(\alpha_i)$. Then $\alpha_i^\vee\in \Inv(s_\beta)$ and $\langle \beta^\vee,\alpha_j\rangle=-\langle \alpha_i^\vee,\alpha_j\rangle\geq 2$ and thus $r_j(\alpha_i)$ is not quantum. The set of quantum roots can in fact be explicitly described using the notions of long and short roots \cite[Lem.\,4.2]{lenart2012uniform}. Note that when $\Phi$ is infinite, then $\Phi_+$ strictly contains the set of quantum roots, which is finite. \Subsubsection{Case where the generalized Cartan matrix has no coefficient $-1$}\label{ss_no_minus_one} \begin{Lemma}\label{l_no_-1_coefficient} Let $\beta\in \Phi_+$ be a quantum root. Write $\beta=r_{i_L}\ldots r_{i_{2}}(\alpha_{i_1})$, with $i_1,\ldots,i_L\in I$ and $L\in \N$ minimal. Then we have $a_{i_1,i_2}=-1$ (where $A=(a_{i,j})_{i,j\in I}$ is the generalized Cartan matrix of the Kac--Moody datum). In~particular, if~$a_{i,j}\neq -1$ for every $i,j\in I$, the set of quantum roots is exactly the set of simple roots. \end{Lemma} \begin{proof} We have $s_{\beta}=r_{i_L}\ldots r_{i_{2}} r_{i_1} r_{i_{2}} \ldots r_{i_L}$. We~have $r_{i_L}\ldots r_{i_{3}}\cdot\alpha_{i_{2}}\in \Inv(s_{\beta})\setminus \{\beta\}$ and \[ \langle r_{i_L}\ldots r_{i_2}(\alpha_{i_1}^\vee),r_{i_L}\ldots r_{i_3}(\alpha_{i_2})\rangle=-\langle \alpha_{i_1}^\vee,\alpha_{i_2}\rangle=-a_{i_1,i_2}=1.\qedhere \] \end{proof} \Subsubsection{Type $A_n^{(1)}$, $n\geq 2$} \begin{Proposition} We assume that the generalized Cartan matrix $A$ is of type $A_n^{(1)}$, with $n\in \Z_{\geq 2}$ (see \cite[Ch.\,4]{kac1994infinite} for the classification). We~regard $I$ as a diagram as in Section~\ref{ss_gen_def_not} and we denote by $\mathrm{Seg}(I)$ the set of non-empty subsets of $I$ which are a segment when regarded as a subdiagram of $I$. Then $\cQ(\Phi_+)=\{\sum_{i\in J} \alpha_i\mid J\in \mathrm{Seg}(I)\}$. \end{Proposition} \begin{proof} Let $\beta\in \cQ(\Phi_+)$. We~have $\deg(j)=2$ for every $j\in I$ (we defined the degree in \eqref{e_definition_degree}). By~Lemma~\ref{l_description_chains_I2}, we~deduce that $I_2(\beta)=\emptyset$. Therefore by Proposition~\ref{p_characterization_I1}, $\{\beta^{\vee}\mid \beta\in \cQ(\Phi_+)\}=\{\sum_{j\in J} \alpha_j^\vee\mid J\in \mathrm{Seg}(I)\}$. As the generalized Cartan matrix only contains $0$ or $-1$ coefficients off the diagonal, we~have $\beta^\vee=\sum_{i\in I} n_i \alpha_i^\vee$, for all $\beta=\sum_{i\in I} n_i\alpha_i\in \Phi$, and the proposition follows. \end{proof} Note that the case $A_1^{(1)}$ is a particular case of Section~\ref{ss_no_minus_one}. \subsubsection{Type $D_n^{(1)}$, $n\geq 4$} Let $n\in \Z_{\geq 5}$. Let $I=I_{D_n^{(1)}}$ be the Dynkin diagram of type $D_n^{(1)}$ (see Figure~\ref{f_Dynkin_diagram_Dn_aff}). \begin{figure}[htb] \centering \includegraphics[scale=0.4]{Graphe_Dn_aff_3.png} \caption{Dynkin diagram of type $D_n^{(1)}$}\label{f_Dynkin_diagram_Dn_aff} \end{figure} It admits two vertices of valency $3$, which we denote $j_1, \tilde{j}_1$. Let $j_2,j_2'$ (\resp $\tilde j_2,\tilde j_2'$) denote the two leaves of $I$ in $\supp(j_1)$ (\resp $\supp(\tilde{j}_1)$). The Dynkin diagram $I$ contains four distinct Dynkin diagrams of type $D_n$, obtained by removing one of the vertices among $j_2,j_2',\tilde{j}_2,\tilde{j}_2'$. Let $\Phi\subset \oplus_{i\in I} \Z\alpha_i$ be the root system associated to $I$. From Theorem~\ref{Theorem: Classification quantum roots} we deduce that $\beta\in \oplus_{i\in I} \mathbb N \alpha_i$ is a quantum root if and only if one of the following conditions is satisfied: \begin{enumerate} \item The element $\beta$ is a root in one of the four standard root subsystems of type~$D_n$ (in particular $I_1(\beta)\neq I$ and $I_3(\beta)=\emptyset$). \item We have that $I_1(\beta)=I$ and $I_2(\beta)\subset I\setminus\{j_2,j_2',\tilde j_2,\tilde j_2'\}$ admits at most two connected components, each of them containing exactly one element among $\{j_1,\tilde j_1\}$. Moreover $I_3(\beta)=\emptyset$. \end{enumerate} Note that both of these cases correspond to Dynkin sequences of type (4A). If $n=4$, then $I_{D_4^{(1)}}$ consists of a unique vertex of valency $4$ and four leaves. Similarly a root $\beta$ is quantum if and only if either it is a root supported on a subdiagram of type $D_4$, or if $I_1(\beta)=I_{D_4^{(1)}}$ and $I_2(\beta)=\emptyset$. \section{From quantum roots to covers for the affinized Bruhat order}\label{s_almost_simple_roots_covers} In this section, we~use the finiteness of the set of quantum roots to obtain finiteness results on the affinized Bruhat order on the Weyl semi-group associated to a Kac--Moody datum. We start this section by introducing this object and of the affinized Bruhat order. \Subsection{Definition and notations: Affine Weyl semi-group}\label{ss_def_not_aff} Let $\mathcal D=(A,X,Y,(\alpha_i)_{i\in I},(\alpha_i^\vee)_{i\in I})$ be a Kac--Moody root datum. Let $W^v$ be the vectorial Weyl group and $\Phi$ the root system attached to it. \subsubsection*{Dominant coweights and Tits cone} Elements of $Y$ are called coweights; a coweight~$\lambda$ is \emph{regular} if $\langle \lambda,\beta\rangle\neq 0$ for all $\beta\in \Phi$, or equivalently if its fixator in $W^v$ is trivial. We~say that a coweight $\lambda\in Y$ is \emph{dominant} if $\langle \lambda,\alpha_i\rangle\geq 0$ for all $i\in I$, or equivalently if $\langle \lambda,\beta\rangle \geq 0$ for all $\beta\in\Phi_+$. The set of dominant coweights is denoted by \index{y@$Y^{++}$, $Y^+$} $Y^{++}=\{\lambda \in Y\mid \langle \lambda,\alpha_i\rangle \geq 0\; \forall i\in I\}$ and the set of dominant regular coweights is denoted \index{Y@$Y^{++}_{\reg}$}$Y^{++}_{\reg}=\{\lambda \in Y^{++}\mid \langle \lambda,\alpha_i\rangle >0\; \forall i \in I\}$. The orbit of $Y^{++}$ by $W^v$ is the \emph{integral Tits cone}, $Y^+=\bigcup_{w\in W^v} w.Y^{++}$; it is a sub-cone of $Y$ and it is a proper sub-cone if and only if $W^v$ is infinite, if~and only if~$A$ is not of finite type (see \cite[1.4.2]{kumar2002kac}). The set $Y^{++}$ is a fundamental domain for the action of $W^v$ on $Y^+$, and for any coweight $\lambda \in Y^+$, we~denote by $\lambda^{++}$ to be the unique element of $Y^{++}$ in its $W^v$-orbit. For $\lambda,\mu\in Y$, we~set \[ [\lambda,\mu]_{Y}=\{\nu\in Y\mid \exists t\in \Q, \nu=(1-t)\lambda+t\mu\}\quand ]\lambda,\mu[_{Y}=[\lambda,\mu]_Y\setminus \{\lambda,\mu\}. \] \subsubsection*{Parabolic subgroups and minimal coset representatives} Let $\lambda\in Y^{++}$ be a dominant coweight. Then the fixator of $\lambda$ in $W^v$ is a standard parabolic subgroup, which we denote by \index{w@$W_\lambda$, $W_J$, $W^\lambda$} $W_\lambda$. Explicitly, if~$J=\{i\in I \mid \langle \lambda,\alpha_i\rangle=0\}$, then $W_\lambda =W_J:=\langle r_j\mid j\in J\rangle$. Any class $[w]\in W^v/W_\lambda$ admits a unique element of minimal length, and we denote by $W^\lambda$ the set of coset representatives of minimal length. Therefore \[ w\in W^\lambda \iff \ell(w)\leq \ell(\tilde w)\ \forall \tilde w \in wW_\lambda. \] Suppose now that $\lambda\in Y^+$ (not necessarily dominant), we~denote by \index{v@$v^\lambda$}$v^\lambda$ the unique element of minimal length such that $\lambda=v^\lambda \lambda^{++}$; it is an element of $W^{\lambda^{++}}$. The coweight $\lambda$ also admits a parabolic fixator in $W^v$, which is $W_\lambda = v^\lambda W_{\lambda^{++}} (v^\lambda)^{-1}$. A coweight $\lambda$ is \emph{spherical} if its fixator $W_\lambda$ is finite, and it is \emph{regular} if $W_\lambda$ is trivial. These notions only depend on the dominant part $\lambda^{++}$ of $\lambda$. We~denote by \index{Y@$Y^+_{\sph}$} $Y^+_{\sph}$ the set of spherical coweights. \subsubsection*{Affine Weyl semi-group} Through the left action $W^v \curvearrowright Y$ we can form the semidirect product $Y\rtimes W^v$; for $(\lambda,w)\in Y\rtimes W^v$ we denote by $\qp^\lambda w$ the corresponding element of $Y\rtimes W^v$. If~$W^v$ is finite, this semi-direct product is (a finite extension of) a Coxeter group, hence admits a Bruhat order, a Bruhat length and a set of simple reflections. However, when $W^v$ is infinite, it does not have a Coxeter structure, and there is no known analog of the Bruhat order on it. By definition, $Y^+\subset Y$ is stable under the action of $W^v$, therefore we can form \index{w@$W^+$}$W^+=Y^+ \rtimes W^v$ which is a sub semi-group of $Y\rtimes W^v$. This semi-group is called \emph{the affine Weyl semi-group}; it is sometimes referred to as ``the affine Weyl group'' in the literature, even though it is not a group. We denote by \index{p@$\pr^{Y^+}$, $\pr^{Y^{++}}$} $\pr^{Y^+}$ the projection function $W^+\rightarrow Y^+$ and $\pr^{Y^{++}}$ the projection on the dominant coweight, so~if $\bx=\qp^\lambda w \in W^+$, then $\pr^{Y^+}(\bx)=\lambda$ and $\pr^{Y^{++}}(\bx)=\lambda^{++}$. We~say that an element $\bx\in W^+$ is \emph{spherical} if its coweight $\pr^{Y^+}(\bx)$ is spherical. \subsubsection*{Affine roots} Let \index{p@$\Phi^a$}$\Phi^a=\Phi \times \mathbb Z$; it is \emph{the set of affine roots} of $\mathcal D$. Let $(\beta,n)\in \Phi^a$, we~say that $(\beta,n)$ is positive if $n>0$ or ($n=0$ and $\beta \in \Phi_+$); we write $\Phi^a_+$ for the set of positive affine roots. We~also have $\Phi^a = \Phi^a_+ \sqcup -\Phi^a_+$. We~refer to \cite[1.2]{philippe2023grading} for a geometric interpretation of affine roots and of the affinized Bruhat order. The semidirect product $Y\rtimes W^v$ acts on $\Phi^a$ by: \begin{equation}\label{eq: W^+ action} \qp^\lambda w\cdot(\beta,n)=(w\beta,n+\langle \lambda,w\beta\rangle). \end{equation} For any $n \in \mathbb Z$, its sign is denoted $\sgn(n)\in \{-1,+1\}$, with the convention that $\sgn(0)=+1$. For $n\in \Z$ and $\beta\in \Phi_+$, we~set:\begin{align}\label{eq: affine_roots} \index{b@$\beta[n]$} &\beta[n]=(\sgn(n) \beta,|n|\qp)\in \Phi_+^a \\ \index{s@$s_{\beta[n]}$}&s_{\beta[n]}=\qp^{n\beta^\vee} s_\beta \in Y\rtimes W^v. \end{align} Note that the element $s_{\beta[n]}$ does not belong to $W^+$ in general, but it is an element of order $2$ in $Y\rtimes W^v$, the \emph{affine reflection associated to $\beta[n]$}. \subsubsection*{Affinized Bruhat order} Even though $Y\rtimes W^v$ is a priori not a Coxeter group when~$W^v$ is infinite, Braverman, Kazhdan and Patnaik have defined a preorder on~$W^+$ in \cite[B.2]{braverman2016iwahori} using the action of $Y\rtimes W^v$ on $\Phi^a$. It is called the \emph{affinized Bruhat order on $W^+$}. It is defined as the transitive closure of the relation $<$ defined by, for $\bx\in W^+$ and $\beta[n]\in \Phi^a_+$: $s_{\beta[n]}\bx<\bx \iff \bx^{-1}(\beta[n]) \notin \Phi^a_+$. In~some sense this means that $\beta[n]$ belongs to the inversion set of $\bx^{-1}$. Note in particular that if $\bx^{-1}(\beta[n]) \notin \Phi_+^a$ and $\bx$ lies in $W^+$, then so does $s_{\beta[n]}\bx$ . Explicitly, if~$\bx\!=\!\qp^\lambda w \!\in\! W^+$, then $\bx^{-1}(\beta[n])\!\notin\! \Phi_+^a$ if and only if $|n|\!<\!\sgn(n)\langle \lambda,\beta\rangle$, or $|n|=\sgn(n)\langle \lambda,\beta\rangle$ and $\sgn(n)w^{-1}\beta \in\Phi_-$. D.\,Muthiah (see \cite{muthiah2018iwahori}) has proved that the transitive closure of $<$ is anti-symmetric, hence it is an order. Moreover, together with D.\,Orr, he defines in \cite{muthiah2019bruhat} a $\mathbb R$-valued length function on $W^+$ strictly compatible with the affinized Bruhat order which we now introduce. \subsubsection*{Affinized Bruhat length} If $\qp^\lambda w \in W^+$, the \emph{affinized Bruhat length} is defined by: \index{l@$\ell^a$} \begin{equation}\label{eq: affine length formula}\ell^a(\qp^\lambda w)=2\htt(\lambda^{++})+\ell_{v^\lambda}(w). \end{equation} The function \index{l@$\ell_v$} $\ell_{v^\lambda}$ is the relative Bruhat length on $W^v$: $\ell_v(w)=\ell(v^{-1}w)-\ell(v)$ for all $v,w\in W^v$. The equivalence between this definition of the affinized Bruhat length and its definition in \cite{muthiah2019bruhat} is given by \cite[Prop.\,2.10]{philippe2023grading}. This length is strictly compatible with the affinized Bruhat order by \cite[Th.\,3.3]{muthiah2019bruhat}. Note that it depends on the choice of height function, and since $\htt$ may not take values in $\mathbb Z$, it is not integral in general. However, since two comparable elements $\bx,\by$ satisfy \[ \pr^{Y^{++}}(\bx)-\pr^{Y^{++}}(\by)\in Q^\vee, \] the length difference $\ell^a(\bx)-\ell^a(\by)$ is an integer independent of this choice of height function. \Subsubsection*{Covers and gradation of partially ordered sets} \begin{Definition}\label{Definition: covers}Let $(X,<)$ be a partially ordered set. \begin{itemize} \item For $x,y\in X$, we~say that $y$ \emph{covers} $x$ if: $xx$ and $\ell(y)=\ell(x)+1$. \end{itemize} \end{Definition} The Bruhat length on $W^v$ is an example of a grading (actually $\mathbb N$-valued) of \hbox{$(W^v,{<})$}, where $<$ is the Bruhat order; we~refer to \cite[Ch.\,2]{bjorner2005combinatorics} for general properties of the Bruhat order. The affinized Bruhat length is a $\mathbb Z$-grading of $(W^+,<)$ (by \cite[Th.\,6.2]{muthiah2019bruhat} in particular cases and \cite{philippe2023grading} in the general case): for all $\bx,\by\in W^+$ such that $\bx<\by$, \begin{equation}\label{e_characterization_cover} \bx \lhd \by \iff \ell^a(\by)=\ell^a(\bx)+1. \end{equation} The next lemma is a weaker reformulation of \cite[Prop.\,3.20]{philippe2023grading}, which is a more explicit classification of covers in $W^+$. In~the reductive case, this is a reformulation of the classification of covers given by Schremmer in terms of quantum Bruhat graph (see \cite[Prop.\,4.5]{schremmer2024affinebruhat}). \begin{Lemma}\label{Lemma: form of covers} Let $\bx,\by \in W^+$ and suppose that $\by$ covers $\bx$. Then there are unique elements $\lambda \in Y^{++},\,v, w \in W^v,\, \beta\in \Phi_+$ and $n\in \mathbb Z$ such that: \begin{itemize} \item $v$ is a minimal coset representative: $v\in W^\lambda$ \item $\bx = \qp^{v\lambda} w$ \item $\by=s_{v(\beta)[n]}\bx = \qp^{v(s_\beta \lambda +n\beta^\vee)}s_{v(\beta)} w$. \end{itemize} Moreover $n \in \{0,\langle \lambda,\beta\rangle,-1, \langle \lambda,\beta\rangle +1\}$, and $\pr^{Y^{++}}(\by)=\pr^{Y^{++}}(\bx)$ if and only if $n \in \{0,\langle \lambda,\beta\rangle\}$. Furthermore: \begin{enumeratei} \item\label{Lemma: form of coversi} If $n=\langle \lambda,\beta\rangle$ (possibly equal to zero), then $\ell(s_\beta v^{-1}w)=\ell(v^{-1}w)+1$, so~$v^{-1}w \lhd s_\beta v^{-1}w$ \item\label{Lemma: form of coversii} If $n=0\neq\langle \lambda,\beta\rangle$, then $\ell(vs_\beta)=\ell(v)-1$, so~$vs_\beta \lhd v$. \item\label{Lemma: form of coversiii} If $n\in \{-1,\langle \lambda,\beta\rangle+1\}$ then $\beta$ is a quantum root and there exists an element $u \in W^v$ such that $\pr^{Y^+}(\by)=vu\cdot\pr^{Y^{++}}(\by)$ and $\ell(vu)=\ell(v)+\ell(u)$, in~particular $v\leq vu$. \end{enumeratei} \end{Lemma} The first part is a direct consequence of the definition of the affinized Bruhat order and of \cite[Prop.\,3.1]{philippe2023grading}, items \eqref{Lemma: form of coversi} and \eqref{Lemma: form of coversii} correspond to \cite[Prop.\,3.20(1)\,\&\,(2)]{philippe2023grading} whereas \eqref{Lemma: form of coversiii} corresponds to \cite[Prop.\,3.20(3)(a),(c)\,\&\,(4)(a),(c)]{philippe2023grading}. In this section, we~use the finiteness of the set of quantum roots to show that any element $\bx \in W^+$ has a finite number of covers, and if it is spherical, then it also has a finite number of co-covers. This extends to the general Kac--Moody case a result obtained by Welch in the affine ADE case (\cite[Th.\,1.1]{welch2022classification}). We~provide examples of non-spherical elements which admit an infinite number of co-covers, and we give a characterization of the irreducible Kac--Moody root systems for which every element admits a finite number of co-covers. At the end of this section, we~construct explicit covers in $W^+$ associated to any given quantum root, thereby showing that the notion of quantum root is the most refined one in this context. \Subsection{Finiteness results for covers and co-covers} \begin{Theorem}\label{Theorem: finiteness of covers} Suppose that $\bx$ is an element of an affine Weyl semi-group $W^+$. Then $\bx$ has a finite number of covers for the affinized Bruhat order. \end{Theorem} \begin{proof} For any $u\in W^v$, let $n_{\cov}(u)$ (\resp $n_{\cocov}(u)$) denote the number of covers (\resp cocovers) of $u$ in $W^v$. Since $W^v$ is a Coxeter group of finite rank, these numbers are finite. Let $\bx$ be an element of $W^+$. Then, by Lemma \ref{Lemma: form of covers}\eqref{Lemma: form of coversiii}, the number of covers~$\by$ of~$\bx$ such that $\pr^{Y^{++}}(\by)\neq \pr^{Y^{++}}(\bx)$ is bounded by twice the number $N_{\asr}(\Phi)$ of quantum roots of $\Phi$, which is finite by Theorem~\ref{t_finiteness_almost_simple_roots}. Moreover, write $\bx=\qp^{v\lambda}w$ with $\lambda = \pr^{Y^{++}}(\bx)$ and $v\in W^v$ minimal, then the number of covers $\by$ such that $\pr^{Y^{++}}(\by)= \pr^{Y^{++}}(\bx)$ is exactly $n_{\cov}(v^{-1}w)+n_{\cocov}(v)$ using items \eqref{Lemma: form of coversi} and \eqref{Lemma: form of coversii} of the same lemma. Hence $\bx$ has at most $2N_{\asr}(\Phi)+n_{\cov}(v^{-1}w)+n_{\cocov}(v)$ covers. \end{proof} \begin{Corollary}\label{c_finiteness_chains} Let $n\in \Z_{\geq 0}$ and $\bx\in W^+$. Then \[ \{\by\in W^+\mid \exists (\bx_1,\ldots,\bx_n)\in (W^+)^n, \bx=\bx_1\lhd \bx_2\lhd\ldots \bx_{n-1}\lhd \bx_{n}=\by\} \] is finite. \end{Corollary} \begin{Corollary} Let $\bx,\by\in W^+$. Then $[\bx,\by]:=\{\bz\in W^+\mid \bx\leq \bz\leq \by\}$ is finite. \end{Corollary} \begin{proof} Assume $\bx\leq \by$. As the Bruhat order is strictly compatible with $\ell^a$ (by~\cite[Th.\,3.3]{muthiah2019bruhat}), for every $\bz\in [\bx,\by]$, there exist $k\in \llbracket 0,\ell^a(\by)-\ell^a(\bx)\rrbracket$ and $\bx_1,\ldots,\bx_k\in W^+$ such that $\bx=\bx_1\lhd\bx_2\lhd \ldots \lhd \bx_k=\bz$ and thus the result follows from Corollary~\ref{c_finiteness_chains}. \end{proof} \begin{Lemma}\label{l_characterization_bruhat_order_Yin} Let $Y_\mathrm{in}=\{\lambda\in Y\mid \langle \lambda,\alpha_i\rangle=0,\forall i \in I\}$. Let $\lambda\in Y_\mathrm{in}$, $\mu\in Y^+$ and $v,w\in W^v$. Then $\qp^{\mu}w \leq \qp^{\lambda}v$ if and only if $\mu=\lambda$ and $w\leq v$ (for the Bruhat order on $W^v$). \end{Lemma} \begin{proof} Let $\beta\in \Phi_+$ and $n\in \Z$ be such that $s_{\beta[n]} \qp^\lambda v<\qp^\lambda v$, note that $s_{\beta[n]} \qp^\lambda v=\qp^{\lambda+n\beta^\vee} s_\beta v$. Then by \cite[Lem.\,1.7]{philippe2023grading} we have $\lambda+n\beta^\vee\in [\lambda,s_\beta\cdot\lambda]_Y=\{\lambda\}$ and hence $n=0$. As $\lambda\in Y_\mathrm{in}$, we~have $W^\lambda=\{1\}$. Now $s_\beta \qp^\lambda v=\qp^\lambda s_\beta v< \qp^\lambda v$ if and only if $\ell^a(\qp^\lambda s_\beta v)<\ell^a(\qp^\lambda v)$ if and only if $\ell(s_\beta v)< \ell(v)$ if and only if $s_\beta v< v$ for the Bruhat order on $W^v$. The lemma follows. \end{proof} \begin{Theorem}\label{thm_finiteness_co-covers} Suppose that $\by\in W^+$ is spherical, or that $\pr^{Y^+}(\by)\in Y_\mathrm{in}$. Then it admits a finite number of co-covers. \end{Theorem} \begin{proof} Suppose that $\by$ is spherical, let us write $\by=\qp^\mu w$ with $\mu\in Y^+_{\sph}$. Let $C(\by)$ denote the set of co-covers of $\by$ in $W^+$. We~have $C(\by)=C_1(\by)\sqcup C_2(\by)$, where $C_1(\by)=\{\bx \in C(\by)\mid \pr^{Y^{++}}(\bx)=\pr^{Y^{++}}(\by)=\mu^{++}\}$ and $C_2(\by)=C(\by)\setminus C_1(\by)$. We~show separately that both of these sets are finite. Recall that $v^\mu$ denotes the minimal element of $\{v\in W^v \mid \mu=v(\mu^{++})\}$. \begin{itemize} \item Suppose that $\bx \in C_1(\by)$. Then, by Lemma \ref{Lemma: form of covers}, we~can write $\bx=\qp^{v(\mu^{++})}\tilde w$ and $\by=s_{v(\beta)[n]}(\bx)$ for some $\beta \in \Phi_+$, $v\in W^{\mu^{++}},\tilde w\in W^v$ and $n\in \{0,\langle \mu^{++},\beta\rangle\}$. The element $\tilde w$ is entirely determined by $(v,\beta)$ since $\tilde w =s_{v(\beta)}w$. So we need to prove that there is a finite number of viable choices for $(v,\beta)$. We~have: \begin{itemize} \item If $n=0\neq \langle \mu^{++},\beta\rangle$, then $\mu=vs_\beta(\mu^{++})$, so~$vs_\beta$ lies in $v^\mu.W_{\mu^{++}}$, where $W_{\mu^{++}}$ is the stabilizer of $\mu^{++}$ in $W^v$. By~assumption, $\mu$ is spherical so $W_{\mu^{++}}$ is finite, and so is $v^\mu.W_{\mu^{++}}$. Moreover, by Lemma \ref{Lemma: form of covers}\eqref{Lemma: form of coversii}, $v$ covers $vs_\beta$. Since $W^v$ is a Coxeter group of finite rank, any element of $v^\mu W_{\mu^{++}}$ admits a finite number of covers, hence the pair $(v,\beta)$ can only take a finite number of values. \item If $n\neq 0$, then $\bx$ and $\by$ have the same co-weight $\mu$, so~$v=v^\mu$. Moreover, $\tilde w = s_{v(\beta)}w=vs_\beta v^{-1} w$ and therefore by Lemma \ref{Lemma: form of covers}\eqref{Lemma: form of coversi}, $v^{-1}w=s_\beta v^{-1}\tilde w$ covers $v^{-1}\tilde w = s_\beta v^{-1} w$. So the number of viable choices of $\beta$ is equal to the number of co-covers of $v^{-1}w$, which is finite. \end{itemize} Either way, we~thus have shown that $C_1(\by)$ is finite; let us now turn to $C_2(\by)$. \item Suppose that $\bx \in C_2(\by)$. By~Lemma \ref{Lemma: form of covers} we can write $\bx=\qp^{v\lambda}\tilde w$ with $\lambda \in Y^{++}$, $\tilde w\in W^v, \,v\in W^\lambda$ and $\by=s_{v(\beta)[n]}\bx$ for $\beta \in \Phi_+$ and $n\in \{-1,\langle \lambda,\beta\rangle+1\}$. The coweight $\lambda$ is either equal to $v^{-1}\mu-\beta^\vee$ (if $n\neq-1$) or to $s_\beta v^{-1}\mu -\beta^\vee$ (if $n=-1$). Moreover, $\tilde w=s_{v(\beta)}w$ and thus it suffices to show that the set of pairs $(v,\beta)$ such that $\by\in \{\qp^{v(\lambda+\beta^\vee)}w,\qp^{vs_\beta(\lambda+\beta^\vee)}w\}$ and is a cover of $\qp^{v\lambda}s_{v(\beta)}w$ (for one of these two choices of $\lambda$) is finite. Note that, since $\mu$ is spherical, the set \[ \{\tilde v \in W^v\mid \mu=\tilde v \mu^{++}\}=v^\mu.W_{\mu^{++}} \] is finite, so~we can define \[ L=\max\{\ell(\tilde v)\mid \tilde v\in W^v,\tilde v\mu^{++}=\mu\}. \] Then by Lemma \ref{Lemma: form of covers}\eqref{Lemma: form of coversiii}, we~have $\ell(v)0,\quad \alpha_i(\lambda)=\alpha_i(\nu)-2>0 \] and for every $i'\in I\setminus \{j\}$, we~have \[ \alpha_{i'}(\lambda)=\alpha_{i'}(\nu)-\alpha_{i'}(\alpha_j^\vee)\geq \alpha_{i'}(\nu)\geq 0. \] Therefore $\lambda\in Y^{++}$ and $\{k\in I\mid \alpha_k(\lambda)=0\}\subset J'\setminus\{j\}$. Thus $W_{\lambda}\subsetneq W_\nu$ and by assumption on $J'$, $W_{\lambda}$ is finite. Let $w\in W_\nu\cap W^\lambda$ (we prove below that this set is non-empty). Set $\beta=w(\alpha_i)$ and $\bx_w=\qp^{w\cdot\lambda}s_{\beta}$. We~have: \[ s_{\beta[\langle w\cdot\lambda,\beta\rangle+1]}\bx_{w}=\qp^{(\langle w\cdot\lambda,\beta\rangle+1)\beta^\vee} \qp^{s_\beta w\cdot\lambda}=\qp^\nu \] and thus $\bx_w<\qp^\nu$ if and only if $\ell^a(\bx_w)<\ell^a (\qp^\nu)$. Moreover we have: \[ \begin{aligned} \ell^a(\bx_w)&=2\htt(\lambda)+\ell_{w}(s_{w(\alpha_i)})\\ &=2(\htt(\nu)-1)+\ell(w^{-1} wr_iw^{-1})-\ell(w)\\ &=\ell^a(\qp^\nu)-2+\ell(r_iw^{-1})-\ell(w). \end{aligned} \] Therefore \begin{equation}\label{e_difference_length} \ell^a(\bx_w)\in \{\ell^a(\qp^\nu)-1,\ell^a(\qp^{\nu})-3\}, \end{equation} for every $w\in W^\lambda\cap W_\nu$. Let $w\in W_\nu$. Then we can write uniquely $w=vu$, with $v\in W^\lambda$, $u\in W_\lambda$. But then $u\in W_\nu$ and hence $v\in W^\lambda\cap W_\nu$. Therefore $W_\nu\cdot\lambda=(W_\nu\cap W^\lambda)\cdot\lambda$. As $W_\lambda$ is finite, $W_\nu\cdot\lambda$ is infinite and thus $(W^\lambda\cap W_\nu)\cdot\lambda$ and $\{\bx_w\mid w\in W^\lambda\cap W_\nu\}$ are infinite. For $\by\in W^+$, denote by $C'(\by)$ the union of the set of co-covers of $\by$ with $\{\by\}$. By~compatibility of $\ell^a$ with $<$ and by \eqref{e_difference_length}, for every $w\in W_\nu\cap W^\lambda$, we~have $\bx_w\in C'(C'(C'(\qp^\nu)))$. Therefore \[ C'(C'(C'(\qp^\nu)))\supset \{\bx_w\mid w\in W^\lambda\cap W_\nu\}, \] and the right-hand side is infinite. Consequently there exists $\bx\in W^+$ for which $C'(\bx)$ is infinite. \end{proof} \begin{Remark} The situation of Lemma~\ref{l_reciprocal_thm_cocovers} can happen only when $A$ is of indefinite type. Indeed, if~$A$ has finite type, $W^v$ is finite and if $A$ has affine type, then every element of $Y^{++}$ with infinite fixator is fixed by $W^v$, by~\cite[Prop.\,5.8(b) \& Prop.\,3.12(f)]{kac1994infinite}. \end{Remark} \begin{Example} Assume that $I$ has cardinality $3$ and write $I=\{i,j,k\}$. Assume that $\alpha_m(\alpha_{m'}^\vee)\alpha_{m'}(\alpha_{m}^\vee)\geq 4$ for every $m,m'\in \{i,j,k\}$. By~\cite[1.3.21\,Prop.]{kumar2002kac}, this implies that $\langle r_i,r_j\rangle$, $\langle r_j,r_k\rangle$ and $\langle r_i,r_k\rangle$ are infinite. Take $\nu\in Y$ such that $\alpha_i(\nu)\geq 3$, $\alpha_j(\nu)=\alpha_k(\nu)=0$. Set $\lambda=\nu-\alpha_i^\vee\in Y^{++}$. Then $W_\lambda=\{1\}$ and with the same proof as in Lemma~\ref{l_reciprocal_thm_cocovers}, we~can prove that $\qp^{w\cdot\lambda}wr_iw^{-1}$ is covered by $\qp^\nu$ for every $w\in \langle r_j,r_k\rangle$. \end{Example} Let $\A=Y\otimes \R$\index{a@$\A$} be the \emph{standard apartment}. The elements of $\Phi$ can naturally be regarded as elements of the dual $\A^*$ of $\A$. The Tits cone $\sT$\index{t@$\sT$} is then $W^v\cdot\overline{C^v_f}$, where $\overline{C^v_f}=\{x\in \A \mid \langle x,\alpha_i\rangle\geq 0,\forall i\in I\}$. \begin{Proposition}\label{Proposition: nonspherical infinite cocovers} We assume that $(\langle \alpha_j^\vee,\alpha_i\rangle)_{i,j\in I}$ is indecomposable. Let $\A_\mathrm{in}=\bigcap_{i\in I} \ker(\alpha_i)_\R$, where $\ker(\alpha_i)_\R=\{x\in \A\mid \alpha_i(x)=0\}$, for $i\in I$. The following properties are equivalent: \begin{enumerate} \item\label{Proposition: nonspherical infinite cocovers1} The set $\sT\setminus \A_\mathrm{in}$ is open. \item\label{Proposition: nonspherical infinite cocovers2} Every proper parabolic subgroup of $W^v$ is finite. \item\label{Proposition: nonspherical infinite cocovers3} For every $\bx\in W^+$, the set of co-covers of $\bx$ is finite. \end{enumerate} \end{Proposition} \skpt \begin{proof} $\eqref{Proposition: nonspherical infinite cocovers1}\Rightarrow\eqref{Proposition: nonspherical infinite cocovers2}$. Assume \eqref{Proposition: nonspherical infinite cocovers1}. Let $J$ be a proper subset of $I$. Let $\lambda\in Y$ be such that $\alpha_j(\lambda)=0$ for all $j\in J$ and $\alpha_j(\lambda)>0$ for all $j\in I\setminus J$. Then $\lambda\in \overline{C^v_f}\setminus \A_\mathrm{in}\subset \sT\setminus \A_\mathrm{in}\subset \mathring{\sT}$. Then $W_\lambda=W_J$ and by \cite[Prop.\,3.12(f)]{kac1994infinite}, $W_J$ is finite. $\eqref{Proposition: nonspherical infinite cocovers2}\Rightarrow\eqref{Proposition: nonspherical infinite cocovers1}$. Assume \eqref{Proposition: nonspherical infinite cocovers2}. Let $x\in \sT\setminus \A_\mathrm{in}$. We~can write $x=w.y$, for some $w\in W^v$ and $y\in \overline{C^v_f}\setminus \A_\mathrm{in}$. Let $J=\{i\in I\mid \alpha_i(y)=0\}$. Then $J\subsetneq I$ and thus $W_y=W_J$ is finite. By~\cite[Prop.\,3.12(f)]{kac1994infinite}, there exists an open neighborhood $V$ of $y$ contained in $\sT$. Then the fixator of every element of $V$ is finite and thus $V\subset \sT\setminus \A_\mathrm{in}$. Then $x\in w^{-1}\cdot V\subset \sT\setminus \A_\mathrm{in}$, which proves that $\sT\setminus \A_\mathrm{in}$ is open. $\eqref{Proposition: nonspherical infinite cocovers3}\Rightarrow\eqref{Proposition: nonspherical infinite cocovers1}$. Assume \eqref{Proposition: nonspherical infinite cocovers3}. Then by Lemma~\ref{l_reciprocal_thm_cocovers}, for every $\mu\in Y^+$, we~have either $W_\mu$ is finite or $W_\mu=W^v$. Let $x\in \sT$. Write $x=w.y$, with $w\in W^v$ and $y\in \overline{C^v_f}$. Let $J=\{i\in I\mid \alpha_i(y)=0\}$. Let $\mu\in Y$ be such that for $i\in I$, $\alpha_i(\mu)=1$ if $i\in I\setminus J$ and $\alpha_i(\mu)=0$ if $i\in J$. Then $W_\mu=W_x$. If~$W_x=W^v$, then $x\in \A_\mathrm{in}$ and if $W_x$ is finite, then $x\in \mathring{\sT}$ by \cite[Prop.\,3.12\,f)]{kac1994infinite}, which proves that $\sT=\A_\mathrm{in}\sqcup \mathring{\sT}$. Thus we have~\eqref{Proposition: nonspherical infinite cocovers1}. $\eqref{Proposition: nonspherical infinite cocovers1}\Rightarrow\eqref{Proposition: nonspherical infinite cocovers3}$ is a consequence of Theorem~\ref{thm_finiteness_co-covers}. \end{proof} \subsection{Explicit covers for a given quantum root} In this section, we~construct, for any quantum root, explicit covers in which it appears; hence, the restriction on $\beta\in \Phi_+$ appearing in Lemma~\ref{Lemma: form of covers} is optimal. \begin{Proposition}\label{p_almost_simple_roots_covers} Let $\beta$ be a quantum root, and let $\lambda \in Y^{++}_{\reg}$ be such that $\lambda+\beta^\vee \in Y^{++}_{\reg}$ (explicitly, $\langle \lambda,\alpha_i\rangle > -\langle \beta^\vee,\alpha_i\rangle$ for all $i\in I$). Then: \begin{enumerate} \item\label{p_almost_simple_roots_covers1} Let $v$ be any element of $W^v$ and let $w\in W^v$ be such that $\ell(s_\beta w)=\ell(s_\beta)+\ell(w)$ (in particular, this always holds for $w=1_{W^v}$). Then $\qp^{v(\lambda+\beta^\vee)}vw$ covers $\qp^{v(\lambda)}vs_\beta w$. \item\label{p_almost_simple_roots_covers2} Let $v\in W^v$ be such that $\ell(vs_\beta)=\ell(v)+\ell(s_\beta)$ and let $w$ be any element of $W^v$. Then $\qp^{vs_\beta(\lambda+\beta^\vee)}s_{v(\beta)}w$ covers $\qp^{v(\lambda)}w$. \end{enumerate} In particular, for any quantum root $\beta$ there exist covers of the form given in Lem\-ma~\ref{Lemma: form of covers} with varying dominance class. \end{Proposition} \begin{proof} Recall the expression of the affinized Bruhat length obtained in \cite[Prop.\,2.10]{philippe2023grading}: for any $\lambda \in Y^{++}_{\reg}$, $v,w\in W^v$: \begin{equation} \ell^a(\qp^{v(\lambda)}w)=2\htt(\lambda)+\ell(v^{-1}w)-\ell(v). \end{equation} Thus, for any $\lambda \in Y^{++}_{\reg}$ such that $\lambda+\beta^\vee \in Y^{++}_{\reg}$: \begin{align*} \ell^a(s_{v(\beta)[\langle \lambda,\beta\rangle +1]}\qp^{v(\lambda)}v s_\beta w)&=\ell^a(\qp^{v(\lambda+\beta^\vee)}v w)\\ &=2\htt(\lambda+\beta^\vee)+\ell(w)-\ell(v)\\ &=\ell^a(\qp^{v(\lambda)}vs_\beta w)+2 \htt(\beta^\vee)+\ell(w)-\ell(s_\beta w).\\ \ell^a(s_{v(\beta)[-1]}\qp^{v(\lambda)}w) &= \ell^a(\qp^{vs_\beta(\lambda+\beta^\vee)}vs_\beta v^{-1}w)\\ &=2\htt(\lambda+\beta^\vee)+\ell(v^{-1}w)-\ell(vs_\beta)\\ &=\ell^a(\qp^{v(\lambda)}w)+2\htt(\beta^\vee)+\ell(v)-\ell(vs_\beta). \end{align*} Therefore, since quantum roots are characterized by the fact that $2\htt(\beta^\vee)=\ell(s_\beta)+1$ (by Lemma~\ref{l_characterization_almost_simpleness}), we~obtain that the length difference $ \ell^a(s_{v(\beta)[\langle \lambda,\beta\rangle +1]}\qp^{v(\lambda)}v s_\beta w)-\ell^a(\qp^{v(\lambda)}vs_\beta w)$ (\resp $\ell^a(s_{v(\beta)[-1]}\qp^{v(\lambda)}w)-\ell^a(\qp^{v(\lambda)}w)$) is exactly one if and only if condition \eqref{p_almost_simple_roots_covers1} (\resp condition \eqref{p_almost_simple_roots_covers2}) is satisfied. \end{proof} \backmatter \begin{center} \smaller \printindex \end{center} \bibliographystyle{jepalpha+eid} \bibliography{hebert-philippe} \end{document}