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 End Sym ( 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.