Composition Factors of Tensor Products of Symmetric Powers

We determine the composition factors of the tensor product $S(E)\otimes S(E)$ of two copies of the symmetric algebra of the natural module $E$ of a general linear group over an algebraically closed field of positive characteristic. Our main result may be regarded as a substantial generalisation of the tensor product theorem of Krop and Sullivan, on composition factors of $S(E)$. We earlier answered the question of which polynomially injective modules are infinitesimally injective in terms of the "divisibility index". We are now able to give an explicit description of the divisibility index for polynomial modules for general linear groups of degree at most $3$.