Sydney University Algebra Seminar

# Algebra Seminar

### 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.