Olga Azenhas

Matrix realizations of littlewood—richardson sequences

In this paper we consider the problem of characterizing the invariant factors of three matrices A B, and C, such that AB — C Our matrices have entries over a principal ideal domain or over a local domain. In Section 2 we show that this problem is localizablc The above problem lias a well-known solution in terms of Littlewood-Richardson sequences. We introduce the concept of a matrix realization of a Littlewood-Richardson sequence. The main result is an explicit construction of a sequence of matrices which realizes a previously given Littlewood Richardson sequence. Our methods offer a matrix theoretical proof of a well-known result of T, Klein on extensions of p-modules.

OPPOSITE LITTLEWOOD-RICHARDSON SEQUENCES AND THEIR MATRIX REALIZATIONS

Abstract We introduce the concept of opposite Littlewood-Richardson sequences . Given the partitions a 0 , a 1 ,…, a t and the nonnegative integers m 1 ⩽ … ⩽ m t , that concept gives a necessary and sufficient condition for the existence of a sequence of matrices A 0 , B 1 ,…, B t with invariant partitions a 0 , (1 m 1 ),…, (1 m t ) such that a i is the invariant partition of A 0 B 1 … B i for i = 1,…, t , and (1 m 1 ) + ··· + (1 m t ) is the invariant partition of B 1 B 2 … B t . We also present an explicit construction of a sequence of matrices which realizes a previously given opposite Littlewood-Richardson sequence. This result is a generalization of a well-known result of T. Klein and J. A. Green on extensions of p -modules.

The admissible interval for the invariant factors of a product of matrices

It is well known that Littlewood-Richardson sequences give a combinatorial characterization for the invariant factors of a product of two matrices over a principal ideal domain. Given partitions a and c, let LR(a,c) be the set of partitions b for which at least one Littlewood - Richardson sequence of type (a,b,c) exists. I. Zaballa has shown in [20] that LR(a, c) has a minimal element w and a maximal element n, with respect to the order bf majorization, depending only on a and c;. In generalLR(a, c) is not the whole interval [w, n]. Here a combinatorial algorithm is provided for constructing all the elements of LR(a, c). This algorithm consists in starting with the minimal Littlewood-Richardson sequence of shape c/a and successively modifying it until the maximal Littlewood - Richardson sequence of shape c/a is achieved. Also explicit bijections between Littlewood - Richardson sequences of conjugate shape and weight and between Littlewood - Richardson sequences of dual shape and equal weight are presented...

An analogue of the Robinson–Schensted–Knuth correspondence and non-symmetric Cauchy kernels for truncated staircases

Abstract We prove a restriction of an analogue of the Robinson–Schensted–Knuth correspondence for semi-skyline augmented fillings, due to Mason, to multisets of cells of a staircase possibly truncated by a smaller staircase at the upper left end corner, or at the bottom right end corner. The restriction to be imposed on the pairs of semi-skyline augmented fillings is that the pair of shapes, rearrangements of each other, satisfies an inequality in the Bruhat order, w.r.t. the symmetric group, where one shape is bounded by the reverse of the other. For semi-standard Young tableaux the inequality means that the pair of their right keys is such that one key is bounded by the Schutzenberger evacuation of the other. This bijection is then used to obtain an expansion formula of the non-symmetric Cauchy kernel, over staircases or truncated staircases, in the basis of Demazure characters of type A , and the basis of Demazure atoms. The expansion implies Lascoux expansion formula, when specialized to staircases or truncated staircases, and make explicit, in the latter, the Young tableaux in the Demazure crystal by interpreting Demazure operators via elementary bubble sorting operators acting on weak compositions.

Growth Diagrams and Non-symmetric Cauchy Identities on NW (SE) Near Staircases

The Robinson-Schensted-Knuth (RSK) correspondence is an important combinatorial bijection between two line arrays of positive integers (or non-negative integer matrices) and pairs of semi-standard Young tableaux (SSYTs). One of its applications, in the theory of Schur polynomials, is a bijective proof of the well known Cauchy identity. An interesting analogue of this bijection was given by Mason, where SSYTs are replaced by semi-skyline augmented fillings (SSAFs), originated in the Haglund-Haiman-Loehr formula for non-symmetric Macdonald polynomials. The latter object SSAF has the advantage of detecting the key of a SSYT which is easily read off from the SSAF shape. Using this analogue, we have previously considered the restriction of RSK correspondence to multisets of cells in a (truncated) staircase. The image is described by a Bruhat order inequality between the keys of the recording and the insertion fillings. This has allowed to derive a (truncated) triangular version of the Cauchy identity, due to Lascoux, where Schur polynomials are replaced by key polynomials or Demazure characters. We now consider the restriction of RSK to a near staircase, in French convention, where the top leftmost and the bottom rightmost cells and also possibly some cells in the diagonal layer are deleted. The image is described by additional Bruhat order inequalities, specified by the cells in the diagonal layer. The bijection is then used to extend the triangular version to near staircases, also a version due to Lascoux, where Demazure characters are now under the action of Demazure operators specified by the cells in the diagonal layer. Our analysis is made in the framework of Fomin’s growth diagrams where a formulation of the Mason’s analogue is given. This is then used to show how to pass from a triangular shape to a near staircase, via the action crystal operators, and how this affects the keys in the image of the RSK.