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 $\bar{S}(E)^{\otimes m}$ of the truncated symmetric algebra $\bar{S}(E)$ of the symmetric algebra $S(E)$ of the natural module $E$ for $G$. We are particularly interested in the set of partitions $\lambda$ occurring as the highest weight of a composition factor of $\bar{S}(E)^{\otimes m}$. 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 $S(E)^{\otimes m}$ of the symmetric algebra. We give a complete description of the composition factors of $\bar{S}(E)^{\otimes m}$ 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.