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), Δ a , Δ a U ( pI m 1 ⊕ I n - m 1 ) , Δ a U ∏ k = 1 2 ( pI m k ⊕ I n - m k ) , … , Δ a U ∏ k = 1 t ( pI m k ⊕ I n - m k ) , where Δ 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 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, (Naples, 1978), Quaderni de La Ricerca Scientifica, vol. 109, CNR, Rome, 1981; M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002], on words, is discussed.