On Young modules of defect \(2\) blocks of symmetric group algebras
Let \(\Sigma_n\) denote the symmetric group of degree \(n\) and let \(k\) denote a field of prime characteristic \(p\). Given a partition \(\lambda\) of \(n\), let \(\Sigma_\lambda\) denote the Young subgroup of \(\Sigma_n\) corresponding to \(\lambda\). It is a direct product of symmetric groups that correspond to the parts of \(\lambda\). Let \(M^\lambda\) be the permutation module for \(\Sigma_n\) over \(k\) that arises from the permutation action on the cosets of \(\Sigma_\lambda\). The Specht module \(S^\lambda\) is a submodule of \(M^\lambda\) whose head \(D^\lambda\) is simple if \(\lambda\) is \(p\)-regular, that is, if no part of \(\lambda\) is repeated more than \(p-1\) times. As \(\lambda\) runs over the \(p\)-regular partitions of \(n\), the corresponding \(D^\lambda\) run over all the non-isomorphic simple \(k\Sigma_n\)-modules. The Young submodule \(Y^\lambda\) is the unique indecomposable direct summand of \(M^\lambda\) that contains \(S^\lambda\) as a submodule. \(Y^\lambda\) may be lifted to a characteristic zero module, unique up to isomorphism, and we may speak of the (ordinary) character \(\text{ch}(Y^\lambda)\), which contains the irreducible character \(\chi^\lambda\) according to the usual parametrization of the irreducible characters of \(\Sigma_n\). In the paper under review, the authors investigate \(Y^\lambda\) when it belongs to a \(p\)-block of \(k\Sigma_n\) of defect 2. Such blocks only occur when \(p\) is odd. Much is known about these blocks following work of K.~Erdmann, S.~Martin, M.~J.~Richards and J.~Scopes. For example, there are bounds for the entries of the Cartan matrix of such a block. Under this defect 2 hypothesis, the authors determine the irreducible constituents of \(\text{ch}(Y^\lambda)\), which occur with multiplicity at most 1. They also describe the Loewy and socle layers of \(Y^\lambda\). The third section of the paper contains additional results relating to Young modules.
scientific article; zbMATH DE number 1400114