Some spherical function values for two-row tableaux and Young subgroups with three factors

<p>A Young subgroup of the symmetric group <span class="math inline">\(\mathcal{S}_{N}\)</span> with three factors, is realized as the stabilizer <span class="math inline">\(G_{n}\)</span> of a monomial <span class="math inline">\(x^{\lambda}\)</span> ( <span class="math inline">\(=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2} }\cdots x_{N}^{\lambda_{N}}\)</span>) with <span class="math inline">\(\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right)\)</span> (meaning <span class="math inline">\(d_{j}\)</span> is repeated <span class="math inline">\(n_{j}\)</span> times, <span class="math inline">\(1\leq j\leq3\)</span>), thus is isomorphic to the direct product <span class="math inline">\(\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times\mathcal{S}_{n_{3}}\)</span>. The orbit of <span class="math inline">\(x^{\lambda}\)</span> under the action of <span class="math inline">\(\mathcal{S}_{N}\)</span> (by permutation of coordinates) spans a module <span class="math inline">\(V_{\lambda}\)</span>, the representation induced from the identity representation of <span class="math inline">\(G_{n}\)</span>. The space <span class="math inline">\(V_{\lambda}\)</span> decomposes into a direct sum of irreducible <span class="math inline">\(\mathcal{S}_{N}\)</span>-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group <span class="math inline">\(G_{n}\)</span>. 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 <span class="math inline">\(I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3\)</span>. 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 <span class="math inline">\(\mathcal{S}_{N}\)</span>-modules <span class="math inline">\(V\)</span> of two-row tableau type, corresponding to Young tableaux of shape <span class="math inline">\(\left[ N-k,k\right]\)</span>. The method is based on analyzing the effect of a cycle on <span class="math inline">\(G_{n}\)</span>-invariant elements of <span class="math inline">\(V\)</span>. These are constructed in terms of Hahn polynomials in two variables.</p>