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

Let M be the set of all rearrangements of t fixed integers in {1, … , n}. We consider those Young tableaux image, of weight (m1, … , mt) in M, arising from a sequence of products of matrices over a local principal ideal domain, with maximal ideal (p),imagewhere Δa 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 (m1, … , mt) 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 image is the tableau arising from the sequence of matrices, above, when a = 0, it is shown that the words of the tableaux image and image 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.