zbMATH — the first resource for mathematics

A Murnaghan-Nakayama rule for values of unipotent characters in classical groups. (English) Zbl 1376.20015
Represent. Theory 20, 139-161 (2016); correction ibid. 21, 1-3 (2017).
Summary: We derive a Murnaghan-Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai’s explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some \( \ell \)-singular element for certain primes \( \ell \).
As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes \( \ell \geq 3\) the first Cartan invariant in the principal \( \ell \)-block is larger than 2 unless Sylow \( \ell \)-subgroups are cyclic.

20C33 Representations of finite groups of Lie type
20C20 Modular representations and characters
Full Text: DOI
[1] Asai, Teruaki, Unipotent class functions of split special orthogonal groups \({\rm SO}^+_{2n}\) over finite fields, Comm. Algebra, 12, 5-6, 517-615 (1984) · Zbl 0545.20028
[2] Asai, Teruaki, The unipotent class functions on the symplectic groups and the odd orthogonal groups over finite fields, Comm. Algebra, 12, 5-6, 617-645 (1984) · Zbl 0559.20024
[3] Asai, Teruaki, The unipotent class functions of nonsplit finite special orthogonal groups, Comm. Algebra, 13, 4, 845-924 (1985) · Zbl 0574.20031
[4] Carter, Roger W., Finite groups of Lie type, Pure and Applied Mathematics (New York), xii+544 pp. (1985), John Wiley & Sons, Inc., New York
[5] Digne, Fran{\c{c}}ois; Michel, Jean, Representations of finite groups of Lie type, London Mathematical Society Student Texts 21, iv+159 pp. (1991), Cambridge University Press, Cambridge · Zbl 0815.20014
[6] Fong, Paul; Srinivasan, Bhama, Generalized Harish-Chandra theory for unipotent characters of finite classical groups, J. Algebra, 104, 2, 301-309 (1986) · Zbl 0606.20035
[7] Geck, Meinolf; Jacon, Nicolas, Representations of Hecke algebras at roots of unity, Algebra and Applications 15, xii+401 pp. (2011), Springer-Verlag London, Ltd., London · Zbl 1232.20008
[8] James, Gordon; Mathas, Andrew, The Jantzen sum formula for cyclotomic \(q\)-Schur algebras, Trans. Amer. Math. Soc., 352, 11, 5381-5404 (2000) · Zbl 0964.16015
[9] Koshitani, Shigeo; K{\"u}lshammer, Burkhard; Sambale, Benjamin, On Loewy lengths of blocks, Math. Proc. Cambridge Philos. Soc., 156, 3, 555-570 (2014) · Zbl 1329.20009
[10] Lassueur, Caroline; Malle, Gunter, Simple endotrivial modules for linear, unitary and exceptional groups, Math. Z., 280, 3-4, 1047-1074 (2015) · Zbl 1327.20008
[11] [LMS13] C. Lassueur, G. Malle, E. Schulte, Simple endotrivial modules for quasi-simple groups, J. Reine Angew. Math. DOI: 10.1515/crelle-2013-0100. · Zbl 1397.20022
[12] Lusztig, George, Characters of reductive groups over a finite field, Annals of Mathematics Studies 107, xxi+384 pp. (1984), Princeton University Press, Princeton, NJ · Zbl 0556.20033
[13] Malle, Gunter; Testerman, Donna, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Studies in Advanced Mathematics 133, xiv+309 pp. (2011), Cambridge University Press, Cambridge · Zbl 1256.20045
[14] Parker, Christopher; Rowley, Peter, A characteristic 5 identification of the Lyons group, J. London Math. Soc. (2), 69, 1, 128-140 (2004) · Zbl 1065.20028
[15] Pfeiffer, G{\"o}tz, Character tables of Weyl groups in GAP, Bayreuth. Math. Schr., 47, 165-222 (1994) · Zbl 0830.20023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.