Endomorphism rings of permutation modules over maximal Young subgroups

Abstract

Let K be a field of characteristic two, and let λ be a two-part partition of some natural number r. Denote the permutation module corresponding to the (maximal) Young subgroup Σ λ in Σ r by M λ . We construct a full set of orthogonal primitive idempotents of the centraliser subalgebra S K ( λ ) = 1 λ S K ( 2 , r ) 1 λ = End K Σ r ( M λ ) of the Schur algebra S K ( 2 , r ) . These idempotents are naturally in one-to-one correspondence with the 2-Kostka numbers.