Cellularity of endomorphism algebras of Young permutation modules

Abstract

Let E be an n-dimensional vector space. Then the symmetric group Sym(n) acts on E by permuting the elements of a basis and hence on the r-fold tensor product E⊗r. Bowman, Doty and Martin ask, in [1], whether the endomorphism algebra EndSym(n)(E⊗r) is cellular. The module E⊗r is the permutation module for a certain Young Sym(n)-set. We shall show that the endomorphism algebra of the permutation module on an arbitrary Young Sym(n)-set is a cellular algebra. We determine, in terms of the point stabilisers which appear, when the endomorphism algebra is quasi-hereditary.