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University of Sydney
School of Mathematics and Statistics
Andrew Mathas
University of Sydney
Equating decomposition numbers for different primes.
Friday 19th May, 12-1pm, Carslaw 175.
The main outstanding problem in the modular representation
theory of the symmetric groups is the determination of
their p-modular 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 3-modular
decomposition matrix of Sym(11) and the 5-modular 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
Iwahori-Hecke algebras of the symmetric group in characteristic
zero.
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