A bijection between Proctor's and Sundaram's odd orthogonal tableaux
type: Item
instance of: scholarly article
title: A bijection between Proctor's and Sundaram's odd orthogonal tableaux
zbMATH Open document ID: 0874.05059
DOI: 10.1016/0012-365X(95)00278-5
author: Markus Fulmek
author: C. Krattenthaler
published in: Discrete Mathematics
publication date: 1997-10-28T00:00:00Z
review text: It is well known that the irreducible polynomial representations of the complex general linear groups are indexed by partitions \(\lambda\), and the weights occurring in the irreducible representations correspond to column-strict tableaux of shape \(\lambda\). When one replaces the general linear group by the group \(O(2n+1)\) of orthogonal \((2n+1) \times (2n+1)\) matrices (the classical Lie groups of type B), the irreducible representations are again indexed by partitions \(\lambda\). However, there are competing notions of odd orthogonal tableaux of shape \(\lambda\) defined by different authors: King and Welsh, Proctor, and Sundaram. Since all of these odd orthogonal tableaux of shape \(\lambda\) correspond to the weights in the irreducible \(O(2n+1)\)-representation indexed by \(\lambda\), it follows that there must exist, in principle, bijections between all of these sets of tableaux, preserving the weight. This paper provides an ``explicit'' such bijection between the symplectic tableaux of Proctor and of Sundaram, and mentions how similar bijections can be used together with known bijections to tie all of the different notions of odd orthogonal tableau together by bijections. The bijection uses Garsia and Milne's involution principle, which roughly speaking says that if one has (a) two generating functions \(f,g\) that can be obtained by cancellation according to a sign-reversing involution on some larger generating functions \(F,G\) involving signs, and (b) a bijection preserving weights and signs between the terms of the larger generating functions \(F\) and \(G\), then one can produce a weight-preserving bijection between the terms of \(f\) and \(g\). This bijection is perhaps not completely explicit, as it involves iterating the bijections an unspecified number of times, and for this reason we have left the word ``explicit'' in quotes in the previous paragraph. In the set-up of this paper, \(f, g\) are the generating functions for Proctor's and Sundaram'…
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 1018476
zbMATH Keywords: irreducible representations
zbMATH Keywords: column-strict tableaux
zbMATH Keywords: orthogonal tableaux
zbMATH Keywords: symplectic tableaux of Proctor and of Sundaram
zbMATH Keywords: involution principle
zbMATH Keywords: generating functions
zbMATH Keywords: bijection
reviewed by: Victor Reiner
MaRDI profile type: Publication
cites work: Symplectic standard tableaux
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cites work: Method for constructing bijections for classical partition identities
cites work: Binomial determinants, paths, and hook length formulae
cites work: Standard Young tableaux and weight multiplicities of the classical Lie groups
cites work: Construction of orthogonal group modules using tableaux
cites work: A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras
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cites work: A Schensted Algorithm Which Models Tensor Representations of the Orthogonal Group
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cites work: A symplectic jeu de taquin bijection between the tableaux of King and of De Concini
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cites work: Orthogonal tableaux and an insertion algorithm for \(SO(2n+1)\)
full work available at URL: 0012-365x(95)00278-5
OpenAlex ID: W2009681959
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description: scientific article; zbMATH DE number 1018476