Modular Littlewood-Richardson coefficients
The authors study the polynomial representations of the general linear group \(\text{GL}_n(F)\) and modular representations of the symmetric group \(\Sigma_r\) over \(F\), where \(F\) is an algebraically closed field of characteristic \(p>0\). If \(p=0\), then the classical Littlewood-Richardson coefficients describe the decomposition of all the following modules: (M1) the restriction of an irreducible \(\text{GL}_n(F)\)-module to a Levi subgroup of \(\text{GL}_n(F)\); (M2) the tensor product of irreducible \(\text{GL}_n(F)\)-modules; (M3) the restriction of an irreducible \(\Sigma_r\)-module to a Young subgroup of \(\Sigma_r\); (M4) the \(\Sigma_r\)-module induced from an irreducible module for a Young subgroup. In the modular case, one has tilting modules (for the general linear group) and Young modules (for the symmetric group). The goal of this paper is to deal with the case of positive characteristic. The authors describe three quite different sources of connections between them, and obtain connections between various `modular Littlewood-Richardson coefficients'. In particular, they compute certain modular Littlewood-Richardson coefficients exactly.
scientific article; zbMATH DE number 1384390