
School of Mathematics and Statistics
Andrew Mathas
University of Sydney
Equating decomposition numbers for different primes.
Friday 19th May, 121pm, Carslaw 175.
The main outstanding problem in the modular representation
theory of the symmetric groups is the determination of
their pmodular decomposition matrices.
As an experiment, Gordon James and I started to compute the
decomposition matrices of the symmetric groups in characteristic
5; the first problem that we were unable to resolve was the
multiplicity of the simple module D(12,9) in the Specht module
S(8,8,4,1); all that we could determine was that this multiplicity
was either 1 or 2. (Thanks to a computer calculation of Lübeck
and Müller, we now know the answer is 1.)
In the process of this investigation we noticed the striking
similarity between the following submatrices of the 3modular
decomposition matrix of Sym(11) and the 5modular decomposition
matrix of Sym(21).
18,3  1 10,1  1
17,4  1 1 9,2  1 1
13,8  . 1 1 7,4  . 1 1
13,4^2  1 1 1 1 7,2^2  1 1 1 1
12,9  . . 1 . 1 6,5  . . 1 . 1
12,4^2,1  1 1 1 1 1 1 6,2^2,1  1 1 1 1 1 1
8^2,5  . . 1 1 1 . 1 4^2,3  . . 1 1 1 . 1
8^2,4,1  . 1 1 1 1 1 1 1 4^2,2,1  . 1 1 1 2 1 1 1
n=21 and p=5 n=11 and p=3
In this talk we will give some explanation as to why these
decomposition matrices are are almost identical. The answer is
given by a general result about the decomposition matrices of the
IwahoriHecke algebras of the symmetric group in characteristic
zero.
