Young modules for symmetric groups (Q2765551)

Suppose that \(K\) is a field of characteristic \(p\). Let \(\Sigma_n\) denote the symmetric group on \(n\) letters. Then, for each partition, \(\lambda\), of \(n\), there is a permutation module, \(M^\lambda\), for \(\Sigma_n\), arising from the cosets of any Young subgroup of \(\Sigma_n\) of type \(\lambda\). \textit{G. D. James} [Arch. Math. 41, 294-300 (1983; Zbl 0506.20004)], has parametrized the indecomposable direct summands of these modules (via those partitions \(\mu\) which satisfy \(\mu\geq\lambda\), where \(\geq\) is the dominance order on partitions); the modules arising are now known as Young modules.NEWLINENEWLINENEWLINE\textit{J. Grabmeier} [Bayreuther Math. Schr. 20, 9-152 (1985; Zbl 0683.20015)] and \textit{A. A. Klyachko} [Sel. Math. Sov. 3, 45-55 (1984; Zbl 0588.20002)] have developed a ``Green correspondence'' for Young modules, giving the Green correspondent of each Young module as a tensor product of Young modules over the quotient of the normaliser of a Young subgroup by the Young subgroup. However, their parametrization used Schur algebras; the present paper provides a proof of the parametrization involving only the representation theory of the symmetric groups, and also proves the correspondence result using only the Brauer construction (which is valid for arbitrary finite groups).