Contemporary Mathematics Volume 194, 1996 COMMUTING ACTIONS - A Tale of Two Groups GEORGIA BENKART ABSTRACT. When two groups or two algebras have commuting actions on the same space, their representation theories and combinatorics are inti- mately connected. In this article we will concentrate on three examples to illustrate the power of commuting actions and will discuss generalizations of these examples to other settings and various applications of them to such subjects as the theory of symmetric functions and knot theory. I. Three Examples (1.1) Let V = pn where F is a field of characteristic zero, and identify the vectors in V with t:D.e n x 1 matrices over F. The general linear group GLn of n x n invertible matrices with entries in F acts on V by matrix multiplication and on the k-fold tensor product V®k of V by the diagonal action, g(v1 ® · · · ® vk) = gvl ® ... ® 9Vk, making them GLn-modules. The symmetric group sk also acts on y®k as place permutations. Since these two actions commute, the space EndaLn (V181k) of transformations commuting with the action of the general linear group on y®k contains a homomorphic image of the group algebra FSk· When F has characteristic zero, the group algebra FSk is a semisimple associative algebra, and it decomposes into a sum, FSk = EBAf-klA, of simple ideals JA indexed by the partitions . of k. Let t'(.) denote the number of nonzero parts of.. 1991 Mathematics Subject Classification. Primary 17B10, 22E46, 81R50 Secondary 05E05, 05E10. Partially supported by National Science Foundation Grant #DMS-9300523. © 1996 American Mathematical Society 1
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