[Dualité d’Ennola pour les décompositions de produits tensoriels]
Ennola duality relates the character table of the finite unitary group $\mathrm{GU}_n(\mathbb{F}_q)$ to that of $\mathrm{GL}_n(\mathbb{F}_q)$ where we replace $q$ by $-q$ (see [5] for the original observation and [21] for its proof). The aim of this paper is to investigate Ennola duality for the decomposition of tensor products of irreducible characters. It does not hold just by replacing $q$ by $-q$. The main result of this paper is the construction of a family of two-variable polynomials $\mathcal{T}_{\mu }(u,q)$ indexed by triples of partitions of $n$ which interpolates between multiplicities in decompositions of tensor products of unipotent characters for $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{GU}_n(\mathbb{F}_q)$. We give a module theoretic interpretation of these polynomials and deduce that they have non-negative integer coefficients. We also deduce that the coefficient of the term of highest degree in $u$ equals the corresponding Kronecker coefficient for the symmetric group and that the constant term in $u$ give multiplicities in tensor products of generic irreducible characters of unipotent type (i.e., unipotent characters twisted by linear characters of $\mathrm{GL}_1(\mathbb{F}_q)$).
La dualité d’Ennola relie la table des caractères du groupe unitaire fini $\mathrm{GU}_n(\mathbb{F}_q)$ à celle de $\mathrm{GL}_n(\mathbb{F}_q)$ en remplaçant $q$ par $-q$ (voir [5] pour l’observation originale et [21] pour sa preuve). L’objectif de cet article est d’étudier la dualité d’Ennola pour les décompositions des produits tensoriels de caractères irréductibles. Les multiplicités des caractères irréductibles dans le produit tensoriel de deux caractères irréductibles sont des polynômes en $q$ à coefficients entiers. Ces polynômes ne vérifient pas la dualité en remplaçant simplement $q$ par $-q$. Le résultat principal de cet article est la construction d’une famille de polynômes à deux variables $\mathcal{T}_\mu (u, q)$ indexés par des triplets de partitions de $n$ qui déforment simultanément les multiplicités pour les caractères unipotents de $\mathrm{GL}_n(\mathbb{F}_q)$ et celles pour les caractères unipotents de $\mathrm{GU}_n(\mathbb{F}_q)$. Nous donnons une interprétation de ces polynômes en terme de modules gradués pour les groupes symétriques et en déduisons que ces polynômes sont à coefficients entiers positifs. Nous en déduisons également que le coefficient du terme de plus haut degré en $u$ est égal au coefficient de Kronecker correspondant pour le groupe symétrique et que le terme constant en $u$ donne les multiplicités dans les produits tensoriels de caractères irréductibles génériques de type unipotent (c’est-à-dire les caractères unipotents tordus par des caractères linéaires de $\mathrm{GL}_1(\mathbb{F}_q)$).
Accepté le :
Publié le :
Keywords: Ennola duality, tensor products of irreducible characters, quiver varieties
Mots-clés : Dualité d’Ennola, produits tensoriels de caractères irréductibles, variétés de carquois
Emmanuel Letellier 1 ; Fernando Rodriguez Villegas 2
CC-BY 4.0
@article{JEP_2026__13__73_0,
author = {Emmanuel Letellier and Fernando Rodriguez Villegas},
title = {Ennola duality for decomposition of tensor~products},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {73--110},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.323},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.323/}
}
TY - JOUR AU - Emmanuel Letellier AU - Fernando Rodriguez Villegas TI - Ennola duality for decomposition of tensor products JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 73 EP - 110 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.323/ DO - 10.5802/jep.323 LA - en ID - JEP_2026__13__73_0 ER -
%0 Journal Article %A Emmanuel Letellier %A Fernando Rodriguez Villegas %T Ennola duality for decomposition of tensor products %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 73-110 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.323/ %R 10.5802/jep.323 %G en %F JEP_2026__13__73_0
Emmanuel Letellier; Fernando Rodriguez Villegas. Ennola duality for decomposition of tensor products. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 73-110. doi: 10.5802/jep.323
[1] - “On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero”, Duke Math. J. 118 (2003) no. 2, p. 339-352 | DOI | Zbl | MR
[2] - “Absolutely indecomposable representations and Kac-Moody Lie algebras”, Invent. Math. 155 (2004) no. 3, p. 537-559, With an appendix by H. Nakajima | DOI | Zbl | MR
[3] - “Representations of reductive groups over finite fields”, Ann. of Math. (2) 103 (1976) no. 1, p. 103-161 | DOI | Zbl | MR
[4] - Representations of finite groups of Lie type, London Math. Soc. Student Texts, vol. 95, Cambridge University Press, Cambridge, 2020 | Zbl | DOI | MR
[5] - “On the characters of the finite unitary groups”, Ann. Acad. Sci. Fenn. Ser. A I Math. 323 (1963), p. 35 | Zbl | MR
[6] - “Mixed Hodge structures of configuration spaces”, 1995 | arXiv | Zbl
[7] - “Arithmetic harmonic analysis on character and quiver varieties”, Duke Math. J. 160 (2011) no. 2, p. 323-400 | DOI | Zbl | MR
[8] - “Arithmetic harmonic analysis on character and quiver varieties II”, Adv. Math. 234 (2013), p. 85-128 | DOI | Zbl | MR
[9] - “Positivity for Kac polynomials and DT-invariants of quivers”, Ann. of Math. (2) 177 (2013) no. 3, p. 1147-1168 | DOI | Zbl | MR
[10] - “Proof of Springer’s hypothesis”, Israel J. Math. 28 (1977) no. 4, p. 272-286 | DOI | Zbl | MR
[11] - “The space of invariant functions on a finite Lie algebra”, Trans. Amer. Math. Soc. 348 (1996) no. 1, p. 31-50 | DOI | Zbl | MR
[12] - “Fourier transforms, nilpotent orbits, Hall polynomials and Green functions”, in Finite reductive groups (Luminy, 1994), Progress in Math., vol. 141, Birkhäuser Boston, Boston, MA, 1997, p. 291-309 | Zbl
[13] - “Deligne-Lusztig induction for invariant functions on finite Lie algebras of Chevalley’s type”, Tokyo J. Math. 28 (2005) no. 1, p. 265-282 | DOI | Zbl | MR
[14] - Fourier transforms of invariant functions on finite reductive Lie algebras, Lect. Notes in Math., vol. 1859, Springer-Verlag, Berlin, 2005 | DOI | Zbl | MR
[15] - “Quiver varieties and the character ring of general linear groups over finite fields”, J. Eur. Math. Soc. (JEMS) 15 (2013) no. 4, p. 1375-1455 | DOI | Zbl | MR
[16] - “Tensor products of unipotent characters of general linear groups over finite fields”, Transform. Groups 18 (2013) no. 1, p. 233-262 | DOI | MR | Zbl
[17] - “Character varieties with Zariski closures of -conjugacy classes at punctures”, Selecta Math. (N.S.) 21 (2015) no. 1, p. 293-344 | DOI | Zbl | MR
[18] - “-character stacks and Langlands duality over finite fields”, 2024 | arXiv | Zbl
[19] - “Fourier transforms on a semisimple Lie algebra over ”, in Algebraic groups Utrecht 1986, Lect. Notes in Math., vol. 1271, Springer, Berlin, 1987, p. 177-188 | DOI | Zbl | MR
[20] - “Quiver varieties and Weyl group actions”, Ann. Inst. Fourier (Grenoble) 50 (2000) no. 2, p. 461-489 | DOI | Numdam | Zbl | MR
[21] - “The characters of the finite unitary groups”, J. Algebra 49 (1977) no. 1, p. 167-171 | DOI | Zbl | MR
[22] - Symmetric functions and Hall polynomials, Oxford Classic Texts in the Phys. Sci., The Clarendon Press, Oxford University Press, New York, 2015 | MR | Zbl
[23] - “A remark on quiver varieties and Weyl groups”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 1 (2002) no. 3, p. 649-686 | MR | Numdam | Zbl
[24] - “Generic decomposition of tensor products”, http://www.math.rwth-aachen.de/~Frank.Luebeck/preprints/tprdexmpl/tendec.html
[25] - “A computational criterion for the Kac conjecture”, J. Algebra 318 (2007) no. 2, p. 669-679 | DOI | Zbl | MR
[26] - “Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras”, Duke Math. J. 76 (1994) no. 2, p. 365-416 | DOI | Zbl | MR
[27] - “Reflection functors for quiver varieties and Weyl group actions”, Math. Ann. 327 (2003) no. 4, p. 671-721 | DOI | Zbl | MR
[28] - “PARI/GP version 13.1.6”, 2023, Univ. Bordeaux, http://pari.math.u-bordeaux.fr
[29] - “A generalization of Kac polynomials and tensor product of representations of ”, Transform. Groups (2024), online, arXiv:2306.08950 | DOI
[30] - “Trigonometric sums, Green functions of finite groups and representations of Weyl groups”, Invent. Math. 36 (1976), p. 173-207 | DOI | MR
[31] - “Caractères d’algèbres de Lie finies”, in Séminaire P. Dubreil, F. Aribaud et M.-P. Malliavin (1974/75), Secrétariat Math., Paris, 1975, p. Exp. No. 10, 6 p. | Numdam | MR
[32] et al. - “Sage Mathematics Software (version 10.0)”, 2023, The Sage Development Team, http://www.sagemath.org
[33] - “On the characteristic map of finite unitary groups”, Adv. Math. 210 (2007) no. 2, p. 707-732 | DOI | MR
[34] - “Realizability of a model in infinite statistics”, Comm. Math. Phys. 147 (1992) no. 1, p. 199-210 | DOI | MR
Cité par Sources :