Composition factors of tensor products of truncated symmetric powers

Let G be the general linear group of degree n over an algebraically closed field K of characteristic p>0. We study the m-fold tensor product S¯(E)⊗m of the truncated symmetric algebra S¯(E) of the symmetric algebra S(E) of the natural module E for G. We are particularly interested in the set of partitions λ occurring as the highest weight of a composition factor of S¯(E)⊗m. We explain how the determination of these composition factors is related to the determination of the set of composition factors of the m-fold tensor product S(E)⊗m of the symmetric algebra. We give a complete description of the composition factors of S¯(E)⊗m in terms of “distinguished” partitions.Our main interest is in the classical case, but since the quantised version is essentially no more difficult we express our results in the general context throughout.