Decomposition numbers for the principal Φ 2n -block of Sp 4n (q) and SO 4n+1 (q)
[Nombres de décomposition pour le bloc principal Φ 2n de Sp 4n (q) et SO 4n+1 (q)]
Dudas, Olivier1 ; Norton, Emily2
1 Aix-Marseille Université, CNRS, Institut de Mathématiques de Marseille – I2M Campus de Luminy, Avenue de Luminy, Case 930, 13288 Marseille Cedex 9 (France)
2 School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NZ (United Kingdom)
Annales de l'Institut Fourier, Online first, 44 p.

Nous calculons les nombres de décomposition des caractères unipotents du ℓ-bloc unipotent d’un groupe fini de type de Lie B 2n (q) ou C 2n (q) lorsque q est une puissance d’un nombre premier impair et ℓ est un nombre premier impair avec q d’ordre 2n dans 𝔽 ℓ . En cours de route, nous étendons à ces groupes finis les résultats de [12] sur le graphe de branchement pour l’induction et la restriction de Harish-Chandra.

We compute the decomposition numbers of the unipotent characters lying in the principal ℓ-block of a finite group of Lie type B 2n (q) or C 2n (q) when q is an odd prime power and ℓ is an odd prime number such that the order of q mod ℓ is 2n. Along the way, we extend to these finite groups the results of [12] on the branching graph for Harish-Chandra induction and restriction.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3659
Classification : 20C20, 20C33
Keywords: Finite groups of Lie type, Decomposition numbers.
Mot clés : Groupes finis de type Lie, nombres de décomposition.

Dudas, Olivier 1 ; Norton, Emily 2

1 Aix-Marseille Université, CNRS, Institut de Mathématiques de Marseille – I2M Campus de Luminy, Avenue de Luminy, Case 930, 13288 Marseille Cedex 9 (France)
2 School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NZ (United Kingdom) 
@unpublished{AIF_0__0_0_A110_0,
     author = {Dudas, Olivier and Norton, Emily},
     title = {Decomposition numbers for the principal $\Phi _{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3659},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Dudas, Olivier
AU  - Norton, Emily
TI  - Decomposition numbers for the principal $\Phi _{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3659
LA  - en
ID  - AIF_0__0_0_A110_0
ER  - 
%0 Unpublished Work
%A Dudas, Olivier
%A Norton, Emily
%T Decomposition numbers for the principal $\Phi _{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3659
%G en
%F AIF_0__0_0_A110_0
Dudas, Olivier; Norton, Emily. Decomposition numbers for the principal $\Phi _{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$. Annales de l'Institut Fourier, Online first, 44 p.

[1] Bonnafé, Cédric; Dat, Jean-François; Rouquier, Raphaël Derived categories and Deligne–Lusztig varieties II, Ann. Math., Volume 185 (2017) no. 2, pp. 609-670 | DOI | MR | Zbl

[2] Broué, Michel Isométries de caractères et équivalences de Morita ou dérivées, Publ. Math., Inst. Hautes Étud. Sci. (1990) no. 71, pp. 45-63 | DOI | Numdam | MR | Zbl

[3] Broué, Michel; Malle, Gunter; Michel, Jean Generic blocks of finite reductive groups, Représentations unipotentes génériques et blocs des groupes réductifs finis (Astérisque), Société Mathématique de France, 1993 no. 212, pp. 7-92 | Numdam | MR | Zbl

[4] Brunat, Olivier; Dudas, Olivier; Taylor, Jay Unitriangular shape of decomposition matrices of unipotent blocks, Ann. Math., Volume 192 (2020) no. 2, pp. 583-663 | DOI | MR | Zbl

[5] Chuang, Joseph; Turner, Will Cubist algebras, Adv. Math., Volume 217 (2008) no. 4, pp. 1614-1670 | DOI | MR | Zbl

[6] Deligne, Pierre; Lusztig, George Representations of reductive groups over finite fields, Ann. Math., Volume 103 (1976) no. 1, pp. 103-161 | DOI | MR | Zbl

[7] Dudas, Olivier A note on decomposition numbers for groups of Lie type of small rank, J. Algebra, Volume 388 (2013), pp. 364-373 | DOI | MR | Zbl

[8] Dudas, Olivier; Malle, Gunter Decomposition matrices for low-rank unitary groups, Proc. Lond. Math. Soc., Volume 110 (2015) no. 6, pp. 1517-1557 | DOI | MR | Zbl

[9] Dudas, Olivier; Malle, Gunter Decomposition matrices for exceptional groups at d=4, J. Pure Appl. Algebra, Volume 220 (2016) no. 3, pp. 1096-1121 | DOI | MR | Zbl

[10] Dudas, Olivier; Malle, Gunter Decomposition matrices for groups of Lie type in non-defining characteristic (2020) (https://arxiv.org/abs/2001.06395)

[11] Dudas, Olivier; Varagnolo, Michela; Vasserot, Eric Categorical actions on unipotent representations of finite classical groups, Categorification and higher representation theory (Contemporary Mathematics), Volume 683, American Mathematical Society, 2017, pp. 41-104 | DOI | MR | Zbl

[12] Dudas, Olivier; Varagnolo, Michela; Vasserot, Eric Categorical actions on unipotent representations of finite unitary groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 129 (2019), pp. 129-197 | DOI | MR | Zbl

[13] Dunkl, Charles; Griffeth, Stephen Generalized Jack polynomials and the representation theory of rational Cherednik algebras, Sel. Math., New Ser., Volume 16 (2010) no. 4, pp. 791-818 | DOI | MR | Zbl

[14] Fayers, Matthew Weight two blocks of Iwahori–Hecke algebras of type B, J. Algebra, Volume 303 (2006) no. 1, pp. 154-201 | DOI | MR | Zbl

[15] Feit, Walter On large Zsigmondy primes, Proc. Am. Math. Soc., Volume 102 (1988) no. 1, pp. 29-36 | DOI | MR | Zbl

[16] Foda, Omar; Leclerc, Bernard; Okado, Masato; Thibon, Jean-Yves; Welsh, Trevor A. Branching functions of A n-1 (1) and Jantzen–Seitz problem for Ariki–Koike algebras, Adv. Math., Volume 141 (1999) no. 2, pp. 322-365 | DOI | MR | Zbl

[17] Fong, Paul; Srinivasan, Bhama Generalized Harish-Chandra theory for unipotent characters of finite classical groups, J. Algebra, Volume 104 (1986) no. 2, pp. 301-309 | DOI | MR | Zbl

[18] Fong, Paul; Srinivasan, Bhama The blocks of finite classical groups, J. Reine Angew. Math., Volume 396 (1989), pp. 122-191 | MR | Zbl

[19] Fong, Paul; Srinivasan, Bhama Brauer trees in classical groups, J. Algebra, Volume 131 (1990) no. 1, pp. 179-225 | DOI | MR | Zbl

[20] Geck, Meinolf Basic sets of Brauer characters of finite groups of Lie type. II, J. Lond. Math. Soc., Volume 47 (1993) no. 2, pp. 255-268 | DOI | MR | Zbl

[21] Geck, Meinolf; Hiss, Gerhard Basic sets of Brauer characters of finite groups of Lie type, J. Reine Angew. Math., Volume 418 (1991), pp. 173-188 https://eudml.org/doc/153337 | MR | Zbl

[22] Geck, Meinolf; Pfeiffer, Götz Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs. New Series, 21, Clarendon Press, 2000, xvi+446 pages | DOI | MR | Zbl

[23] Gerber, Thomas Triple crystal action in Fock spaces, Adv. Math., Volume 329 (2018), pp. 916-954 | DOI | MR | Zbl

[24] Griffeth, Stephen; Norton, Emily Character formulas and Bernstein–Gelfand–Gelfand resolutions for Cherednik algebra modules, Proc. Lond. Math. Soc., Volume 113 (2016) no. 6, pp. 868-906 | DOI | MR | Zbl

[25] Gruber, Jochen; Hiss, Gerhard Decomposition numbers of finite classical groups for linear primes, J. Reine Angew. Math., Volume 485 (1997), pp. 55-91 | DOI | MR | Zbl

[26] Himstedt, Frank; Noeske, Felix Decomposition numbers of SO 7 (q) and Sp 6 (q), J. Algebra, Volume 413 (2014), pp. 15-40 | DOI | MR | Zbl

[27] Jimbo, Michio; Misra, Kailash C.; Miwa, Tetsuji; Okado, Masato Combinatorics of representations of U q (𝔰𝔩 ^(n)) at q=0, Commun. Math. Phys., Volume 136 (1991) no. 3, pp. 543-566 | DOI | MR | Zbl

[28] Lusztig, George Characters of reductive groups over a finite field, Annals of Mathematics Studies, 107, Princeton University Press, 1984, xxi+384 pages | DOI | MR | Zbl

[29] Michel, Jean The development version of the CHEVIE package of GAP3, J. Algebra, Volume 435 (2015), pp. 308-336 | DOI | MR | Zbl

[30] Okuyama, Tetsuro; Waki, Katsushi Decomposition numbers of Sp(4,q), J. Algebra, Volume 199 (1998) no. 2, pp. 544-555 | DOI | MR | Zbl

[31] Rouquier, Raphaël Modular representations of finite groups and Lie theory (2022) (https://arxiv.org/abs/2202.08451)

[32] Spaltenstein, N. A property of special representations of Weyl groups, J. Reine Angew. Math., Volume 343 (1983), pp. 212-220 | DOI | MR | Zbl

[33] Springer, Tonny A. Regular elements of finite reflection groups, Invent. Math., Volume 25 (1974), pp. 159-198 | DOI | MR | Zbl

[34] Uglov, Denis Canonical bases of higher-level q-deformed Fock spaces and Kazhdan–Lusztig polynomials, Physical combinatorics (Kyoto, 1999) (Progress in Mathematics), Volume 191, Birkhäuser, 2000, pp. 249-299 | DOI | MR | Zbl

Cité par Sources :