A combinatorial correspondence for walks in Weyl chambers
type: Item
instance of: scholarly article
title: A combinatorial correspondence for walks in Weyl chambers
zbMATH Open document ID: 0840.05100
DOI: 10.1016/0097-3165(95)90004-7
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-06-27T00:00:00Z
review text: Let \(t_l\) be a sequence satisfying \(t_{-l}= t_l\). Then there is a determinantal identity \[ \text{det}|t_{i- j}|= \text{det}|t_{i- j}+ t_{i+ j+ 1}|\text{det} |t_{i- j}+ t_{i+ j- 1}|, \] which can be proved using elementary row and column operations. The same ideas can be used to factor determinants of the form \(\text{det}|t_{a_i- b_j}|\). In the paper under review, these determinantal identities are given combinatorial bijective proofs in the case where \(t_l\) is the hyperbolic Bessel function \[ I_l(2x)= \sum^\infty_{k= 0} x^{2k+ l}/ k!(k+ l)!. \] A similar treatment is given in case \(t_l= (\begin{smallmatrix} k\\ k/2+ l\end{smallmatrix})\). The counting argument is based on a relation between unconstrained walks in \(W\)-inariant lattices in \(\mathbb{R}^n\), \(W\) a Weyl group, and walks that are constrained to lie in the interior of a Weyl chamber. Exponential generating functions for the constrained walk numbers are shown to be determinants of hyperbolic Bessel functions. Terms on either side of the identities are paired up using symmetries from \(W\). The details are somewhat involved. Applications of the results of this paper to representation theory are given, using a relation in some cases between the numbers of constrained walks and the weights which occur in irreducible decompositions of tensor products of representations of Lie groups.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 802359
zbMATH Keywords: determinantal identity
zbMATH Keywords: hyperbolic Bessel function
zbMATH Keywords: Weyl group
zbMATH Keywords: Weyl chamber
zbMATH Keywords: generating functions
zbMATH Keywords: constrained walk
zbMATH Keywords: tensor products of representations
author: David Grabiner
reviewed by: Michael Falk
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90004-7
OpenAlex ID: W1999297235
cites work: Random Walk in a Weyl Chamber
cites work: Random walks in Weyl chambers and the decomposition of tensor powers
cites work: Reflection and algorithm proofs of some more Lie group dual pair identities
cites work: Q3748279
cites work: Ascending subsequences of permutations and the shapes of tableaux
description: scientific article; zbMATH DE number 802359