Generalised Cartan invariants of symmetric groups

Evseev, Anton

doi:10.1090/S0002-9947-2014-06176-X

Generalised Cartan invariants of symmetric groups
HTML articles powered by AMS MathViewer

by Anton Evseev PDF

Trans. Amer. Math. Soc. 367 (2015), 2823-2851 Request permission

Abstract:

Külshammer, Olsson, and Robinson developed an $\ell$-analogue of modular representation theory of symmetric groups where $\ell$ is not necessarily a prime. They gave a conjectural combinatorial description for invariant factors of the Cartan matrix in this context. We confirm their conjecture by proving a more precise blockwise version of the conjecture due to Bessenrodt and Hill.

References

Masanori Ando, Takeshi Suzuki, and Hiro-Fumi Yamada, Combinatorics for graded Cartan matrices of the Iwahori-Hecke algebra of type $A$, Ann. Comb. 17 (2013), no. 3, 427–442. MR 3090170, DOI 10.1007/s00026-013-0197-2
Susumu Ariki, On the decomposition numbers of the Hecke algebra of $G(m,1,n)$, J. Math. Kyoto Univ. 36 (1996), no. 4, 789–808. MR 1443748, DOI 10.1215/kjm/1250518452
Christine Bessenrodt and David Hill, Cartan invariants of symmetric groups and Iwahori-Hecke algebras, J. Lond. Math. Soc. (2) 81 (2010), no. 1, 113–128. MR 2580456, DOI 10.1112/jlms/jdp060
Christine Bessenrodt and Jørn B. Olsson, A note on Cartan matrices for symmetric groups, Arch. Math. (Basel) 81 (2003), no. 5, 497–504. MR 2029709, DOI 10.1007/s00013-003-0799-8
Jonathan Brundan and Alexander Kleshchev, Cartan determinants and Shapovalov forms, Math. Ann. 324 (2002), no. 3, 431–449. MR 1938453, DOI 10.1007/s00208-002-0346-0
Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), no. 3, 451–484. MR 2551762, DOI 10.1007/s00222-009-0204-8
Stephen Donkin, Representations of Hecke algebras and characters of symmetric groups, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 49–67. MR 1985722
A. Evseev, Character correspondences for symmetric groups and wreath products, ArXiv: 1208.2380 (2012).
I. Grojnowski, Affine $\widehat {\mathfrak {sl}_p}$ controls the modular representation theory of the symmetric group and related Hecke algebras, ArXiv: math/9907129 (1999).
David Hill, Elementary divisors of the Shapovalov form on the basic representation of Kac-Moody algebras, J. Algebra 319 (2008), no. 12, 5208–5246. MR 2423824, DOI 10.1016/j.jalgebra.2007.11.009
I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge Tracts in Mathematics, vol. 163, Cambridge University Press, Cambridge, 2005. MR 2165457, DOI 10.1017/CBO9780511542800
Burkhard Külshammer, Jørn B. Olsson, and Geoffrey R. Robinson, Generalized blocks for symmetric groups, Invent. Math. 151 (2003), no. 3, 513–552. MR 1961337, DOI 10.1007/s00222-002-0258-3
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
Jørn B. Olsson, Lower defect groups in symmetric groups, J. Algebra 104 (1986), no. 1, 37–56. MR 865885, DOI 10.1016/0021-8693(86)90232-2
Raphaël Rouquier, Isométries parfaites dans les blocs à défaut abélien des groupes symétriques et sporadiques, J. Algebra 168 (1994), no. 2, 648–694 (French). MR 1292784, DOI 10.1006/jabr.1994.1248
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Shunsuke Tsuchioka, Graded Cartan determinants of the symmetric groups, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2019–2040. MR 3152721, DOI 10.1090/S0002-9947-2013-05916-8