Connections between the representations of the symmetric groups and the symplectic group in characteristic \(2\)
Let \(K\) be a field and let \(\text{SP}_{2l}(K)\) be the symplectic group defined over a non-degenerate symplectic space \(E\) of dimension \(2l\) over \(K\). The group \(\text{SP}_{2l}(K)\) acts on the exterior powers \(\bigwedge^rE\) and the composition factors of these modules are known. These modules are labelled by the fundamental domain weight of \(\text{SP}_{2l}(K)\). The irreducible \(\text{SP}_{2l}(K)\)-module with highest weight \(\omega_r\) is denoted by \(L(\omega_r)\), where \(\omega_r\) is a fundamental weight. The paper under review is divided into three sections and throughout it is assumed that \(K\) has characteristic 2. In section one the authors consider the symmetric group \(S_{2l+1}\) of degree \(2l+1\) as a subgroup of \(\text{SP}_{2l}(K)\) and prove that each composition factor of the action of \(\text{SP}_{2l}(K)\) on \(\bigwedge^rE\) remains irreducible when restricted to \(S_{2l+1}\). In section two they consider the symmetric group on \(2l+2\) letters as a subgroup of \(\text{SP}_{2l}(K)\) and show that if \(l\) is even then the irreducible module \(L(\omega_1+\omega_{2r})\) with highest weight \(\omega_1+\omega_{2r}\) is irreducible under the action of \(S_{2l+2}\). In section three the authors obtain general results about tensor products of irreducible modules of \(\text{SP}_{2l}(K)\) and the spin module.
scientific article; zbMATH DE number 1384438
scientific article