Sydney University Algebra Seminar

    University of Sydney

    School of Mathematics and Statistics

    Algebra Seminar

    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.