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.
Categories: math.RT math.CA