A combinatorial correspondence for walks in Weyl chambers
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.
scientific article; zbMATH DE number 802359