Some remarks on branching rules and tensor products for algebraic groups
Let \(\mathbb{F}\) be an algebraically closed field of characteristic \(p>0\). If \(L\) is some irreducible rational \(\text{GL}_n(\mathbb{F})\)-module and \(V\) is the natural \(\text{GL}_n(\mathbb{F})\)-module, there is a close relationship between the highest weight vectors (relative to \(\text{GL}_{n-1}(\mathbb{F})\)) in the restriction \(L \downarrow_{\text{GL}_{n-1}(\mathbb{F})}\) and the highest weight vectors (relative to \(\text{GL}_n(\mathbb{F})\)) in the tensor product \(L\otimes V^*\). The authors obtain more results on various relations between the branching rules from \(\text{GL}_n(\mathbb{F})\) to its Levi subgroups and on decompositions of tensor products over \(\text{GL}_n(\mathbb{F})\) itself. Some of these results are valid for arbitrary type.
scientific article; zbMATH DE number 1340616