A characteristic-free approach to the representation theory of \(\mathfrak S_n\)
The permutation module of the symmetric group \(\sigma_n\) on a Young subgroup for the partition \(\lambda\) is denoted by \(M^\lambda\), and this is viewed as an \(F\sigma_n\)-module for an arbitrary field \(F\). The paper describes many submodules of \(M^\lambda\) as intersections of kernels of \(F\sigma_n\)-homomorphisms. These submodules lie between \(M^\lambda\) and the Specht module \(S^\lambda\); in particular, \(S^\lambda\) is described as a kernel intersection. By removing one kernel at a time from the intersection, a series for \(M^\lambda\) is obtained, where each factor in the series is isomorphic to a Specht module. The order of the factors in this series Is independent of the field, and the special case where \(\text{char } F = 0\) gives Young's rule for calculating the composition factors of \(M^\lambda\) . Eases are given for the kernel intersections considered, and the difficult part of the construction centres on a combinatorial theorem 4.11, which is used to prove that the proposed bases span the spaces concerned. Precisely the same combinatorial theorem gives the Littlewood-Richardson rule from Young's rule. In view of the difficulties encountered in filling in the details of some of the earlier proofs of the Littlewood-Richardson rule, it is interesting to see that by adopting this approach, Young's rule and the Littlewood-Richardson rule are virtually equivalent. The paper does not describe a characteristic- free series for the modules involved in the Littlewood-Richardson rule, but an article being prepared by \textit{M. H. Peel} and the author will deal with this. The paper concludes with a necessary and sufficient condition for \(S^\lambda\) to contain a trivial \(F\sigma\)-module.
scientific article; zbMATH DE number 3558049