Year: 2025
Type: article
Abstract: A Young subgroup of the symmetric group \(\mathcal{S}_{N}\) with three factors, is realized as the stabilizer \(G_{n}\) of a monomial \(x^{\lambda}\) ( \(=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2} }\cdots x_{N}^{\lambda_{N}}\)) with \(\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right)\) (meaning \(d_{j}\) is repeated \(n_{j}\) times, \(1\leq j\leq3\)), thus is isomorphic to the direct product \(\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times\mathcal{S}_{n_{3}}\). The orbit of \(x^{\lambda}\) under the action of \(\mathcal{S}_{N}\) (by permutation of coordinates) spans a module \(V_{\lambda}\), the representation induced from the identity representation of \(G_{n}\). The space \(V_{\lambda}\) decomposes into a direct sum of irreducible \(\mathcal{S}_{N}\)-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group \(G_{n}\). This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals \(I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3\). These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for \(\mathcal{S}_{N}\)-modules \(V\) of two-row tableau type, corresponding to Young tableaux of shape \(\left[ N-k,k\right]\). The method is based on analyzing the effect of a cycle on \(G_{n}\)-invariant elements of \(V\). These are constructed in terms of Hahn polynomials in two variables.
Author: Charles F. Dunkl
Institution: University of Virginia
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4411586346