On an involution on the set of Littlewood-Richardson tableaux and the hidden commutativity

Abstract

The original definition of the Littlewood-Richardson (LR) rule for composing partitions is exclusively considered, i. e., the classical combinatorial device for calculating the Littlewood-Richardson coefficients. The main result is an explicit involution on the set of LR tableaux which transforms an LR tableau of type [a, b, c] into one of type [b, a, c]. On the basis of the involution definition it is a projection of LR tableaux of order r into those of order r - 1, for r ~ 1. The main feature of this projection is the decomposition of an LR tableau of order r and type [a, b, c] into a nested sequence of LR tableaux of order s and type [a(s), (b1, ... , bs); (Cr-s+l, ... , cr )], s == 1, ... , r, where (a(s))~==l is a sequence of interlacing partitions which defines a decomposition of an LR tableau of type [b, a, c] into a nested sequence of LR tableaux of order s and type [(b1, ... ,bs);a(s); (Cr-s+l, ... ,cr )], s == 1, ... ,r. This projection is accomplished introducing a combinatorial deletion and insertion operation on a LR tableau preserving the LR conditions. This involution yields a self-contained and direct combinatorial interpretation of the well-known commutative property of the original LR rule, as well as of the symmetry of the Littlewood-Richardson coefficients given by the equality Ngb == N ba . It is known that the LR rule describes the Smith invariants of a product of integral matrices. It has been proven that this rule is also describing the eigenvalues of a sum of Hermitian matrices [13, 14, 17]. With the present involution we aim to a deeper understanding of the structure the LR rule and its relationship with these two problems in matrix theory.