Please use this identifier to cite or link to this item:
Title: On an involution on the set of Littlewood-Richardson tableaux and the hidden commutativity
Authors: Azenhas, Olga 
Keywords: Young tableaux; Littlewood-Richardson rule; Projection; Involution
Issue Date: 2000
Publisher: Centro de Matemática da Universidade de Coimbra
Citation: Pré-Publicações DMUC. 00-27 (2000)
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.
Rights: openAccess
Appears in Collections:FCTUC Matemática - Artigos em Revistas Nacionais

Show full item record

Page view(s) 50

checked on Aug 11, 2022


checked on Aug 11, 2022

Google ScholarTM


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.