Decomposition numbers for weight three blocks of symmetric groups and Iwahori–Hecke algebras

Fayers, Matthew

doi:10.1090/S0002-9947-07-04156-6

Decomposition numbers for weight three blocks of symmetric groups and Iwahori–Hecke algebras
HTML articles powered by AMS MathViewer

by Matthew Fayers PDF

Trans. Amer. Math. Soc. 360 (2008), 1341-1376 Request permission

Abstract:

Let $\mathbb {F}$ be a field, $q$ a non-zero element of $\mathbb {F}$ and $\mathcal {H}_{n}=\mathcal {H}_{\mathbb {F},q}(\mathfrak {S}_n)$ the Iwahori–Hecke algebra of the symmetric group $\mathfrak {S}_n$. If $B$ is a block of $\mathcal {H}_{n}$ of $e$-weight $3$ and the characteristic of $\mathbb {F}$ is at least $5$, we prove that the decomposition numbers for $B$ are all at most $1$. In particular, the decomposition numbers for a $p$-block of $\mathfrak {S}_n$ of defect $3$ are all at most $1$.

References

Jonathan Brundan and Alexander Kleshchev, Representation theory of symmetric groups and their double covers, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 31–53. MR 1994959, DOI 10.1142/9789812564481_{0}003
Joseph Chuang and Kai Meng Tan, Some canonical basis vectors in the basic $U_q(\widehat {\mathfrak {s}\mathfrak {l}}_n)$-module, J. Algebra 248 (2002), no. 2, 765–779. MR 1882121, DOI 10.1006/jabr.2001.9030
Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, DOI 10.1112/plms/s3-52.1.20
Matthew Fayers, Weight two blocks of Iwahori-Hecke algebras in characteristic two, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 3, 385–397. MR 2177166, DOI 10.1017/S0305004105008637
M. Fayers & K. M. Tan, Adjustment matrices for weight three blocks of Iwahori–Hecke algebras, J. Algebra 306 (2006), 76–103.
G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics, vol. 682, Springer, Berlin, 1978. MR 513828
Gordon James, The decomposition matrices of $\textrm {GL}_n(q)$ for $n\le 10$, Proc. London Math. Soc. (3) 60 (1990), no. 2, 225–265. MR 1031453, DOI 10.1112/plms/s3-60.2.225
Gordon James, Sinéad Lyle, and Andrew Mathas, Rouquier blocks, Math. Z. 252 (2006), no. 3, 511–531. MR 2207757, DOI 10.1007/s00209-005-0863-0
Gordon James and Andrew Mathas, A $q$-analogue of the Jantzen-Schaper theorem, Proc. London Math. Soc. (3) 74 (1997), no. 2, 241–274. MR 1425323, DOI 10.1112/S0024611597000099
Gordon James and Andrew Mathas, Equating decomposition numbers for different primes, J. Algebra 258 (2002), no. 2, 599–614. MR 1943936, DOI 10.1016/S0021-8693(02)00644-0
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
Bernard Leclerc and Hyohe Miyachi, Some closed formulas for canonical bases of Fock spaces, Represent. Theory 6 (2002), 290–312. MR 1927956, DOI 10.1090/S1088-4165-02-00136-X
Stuart Martin and Lee Russell, Defect $3$ blocks of symmetric group algebras, J. Algebra 213 (1999), no. 1, 304–339. MR 1674687, DOI 10.1006/jabr.1998.7649
Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316, DOI 10.1090/ulect/015
G. Mullineux, Bijections of $p$-regular partitions and $p$-modular irreducibles of the symmetric groups, J. London Math. Soc. (2) 20 (1979), no. 1, 60–66. MR 545202, DOI 10.1112/jlms/s2-20.1.60
Matthew J. Richards, Some decomposition numbers for Hecke algebras of general linear groups, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 3, 383–402. MR 1357053, DOI 10.1017/S0305004100074296
Joanna Scopes, Cartan matrices and Morita equivalence for blocks of the symmetric groups, J. Algebra 142 (1991), no. 2, 441–455. MR 1127075, DOI 10.1016/0021-8693(91)90319-4