A bijection between Proctor's and Sundaram's odd orthogonal tableaux
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's odd orthogonal tableaux respectively, and \(F\) and \(G\) come (roughly speaking) from a Jacobi-Trudi-style determinantal expression for the characters of \(O(2n+1)\). Some of the sign-reversing involutions cancelling \(F\) down to \(f\) and \(G\) down to \(g\) come from a previous paper by the same authors, while others are new. A nice feature of the way the authors have set things up is that \(F=G\) so that no bijection between the terms of \(F,G\) needs to be supplied.
symplectic tableaux of Proctor and of Sundaram
scientific article; zbMATH DE number 1018476
scientific article