Young Modules and Schur Subalgebras
نویسنده
چکیده
Let K r be the group algebra of the symmetric group on r symbols over a eld K of prime characteristic p. This work is motivated by the problem of comparing modules in blocks of K r of a xed core and for varying weight w, where r 2 N is arbitrary. We approach this via polynomial representations of the general linear groups GL n (K) or equivalently via Schur algebras. The main part of the thesis deals with n = 2. Let be a partition of r and be a Young subgroup of r. We determine the p-Kostka numbers for permutation modules M = (1) " r , where is a two-part partition of r. We compute explicitly the Cartan matrices of S(2; r) over a eld of any prime characteristic , using Doty's work on admissable decompositions. Ordering the columns and rows in a suitable way, we then show that the Cartan matrix of certain Schur algebras S(2; d) forms a submatrix of the Cartan matrix of S(2; r), where d r sat-isses a certain congruence relation. In the cases where the Cartan matrix of S(2; d) forms a submatrix of the Cartan matrix of S(2; r) we prove that the Schur algebra S(2; r) contains a subalgebra isomorphic to S(2; d). This subalgebra is of the form eS(2; r)e, where e is an idempotent of S(2; r). Under this embedding, the quiver of S(2; d) corresponds to a full subquiver of the quiver of S(2; r). We nally exploit the theory of quasi-hereditary algebras to prove that there exists a Morita equivalence between a quotient of K d and a quotient of K r. This equivalence gives rise to a correspondence between Specht modules and a correspondence between Young modules. The submodule lattices of such corresponding modules are isomorphic. The corresponding modules belong to blocks with the same p-core but with diierent weight. We conjecture an improvement of this result.
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تاریخ انتشار 1999