Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Abstract

Let M be the set of all rearrangements of t fixed integers in {1, . . . , n}. We consider those Young tableaux T, of weight (m 1 , . . . , m t ) in M, arising from a sequence of products of matrices over a local principal ideal domain, with maximal ideal (p), a , a U(pI m 1 ⊕ I n−m 1 ), a U 2 k=1 (pI m k ⊕ I n−m k ),where a is an n × n nonsingular diagonal matrix, with invariant partition a, and U is an n × n unimodular matrix. Given a partition a and an n × n unimodular matrix U , we consider the set T (a,M) (U ) of all sequences of matrices, as above, with (m 1 , . . . , m t ) running over M. The symmetric group acts on T (a,M) (U ) by place permutations of the tuples in M. When t = 2, 3, the action of the symmetric group on the set of Young tableaux, having the set T (a,M) (U ) as matrix realization, is described by a decomposition of the indexing sets of ୋ Work supported by CMUC/FCT. the Littlewood-Richardson tableau in T (a,M) (U ), afforded by the matrix U . This description, in cases t = 2, 3, gives necessary and sufficient conditions for the existence of an unimodular matrix U such that T (a,M) (U ) is a matrix realization of a set of Young tableaux, with given shape c/a and weight running over M. If H is the tableau arising from the sequence of matrices, above, when a = 0, it is shown that the words of the tableaux T and H are Knuth equivalent. The relationship between this action of the symmetric group and the one described by A. Lascoux and M.P. Schutzenberger [Noncommutative structures in algebra and geometric combinatorics