Modular irreducible representations of the symmetric group as linear codes

Abstract

We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht polynomials, is very easy to explain, and it follows directly from Clausen’s theorem by replacing the indeterminate x ij of the letter place algebra by x j i . Our approach is implemented in SYMMETRICA. It was used in order to obtain computational results on code theoretic properties of the p-modular irreducible representation [ λ] p corresponding to a p-regular partition λ via embedding it into representation spaces obtained from ordinary irreducible representations. The first embedding is into the permutation representation induced from the column group of a standard Young tableau of shape λ. The second embedding is the embedding of [ λ] p into the space of [λ] , the p-modular representation obtained from the ordinary irreducible representation [ λ] by reducing the coefficients modulo p. We include a few tables with dimensions and minimum distances of these codes; others can be found via our home page.