Positivity properties for spherical functions of maximal Young subgroups

TL;DR

This paper introduces a new basis for the coset space of a symmetric group subgroup, exhibiting positivity properties similar to root systems, and proves nonnegativity of associated spherical functions.

Contribution

It constructs a basis for the coset space with root-system-like positivity and demonstrates nonnegativity of spherical functions in the Gelfand pair.

Findings

Basis ${ mf B}_{n,k}$ parametrized by Young tableaux with constraints.

Positivity properties akin to root systems.

Spherical functions are nonnegative linear combinations of the basis.

Abstract

Let $S_k \times S_{n-k}$ be a maximal Young subgroup of the symmetric group $S_n$. We introduce a basis ${\mathcal B}_{n,k}$ for the coset space $S_n/S_k \times S_{n-k}$ that is naturally parametrized by the set of standard Young tableaux with $n$ boxes, at most two rows, and at most $k$ boxes in the second row. The basis ${\mathcal B}_{n,k}$ has positivity properties that resemble those of a root system, and there is a composition series of the coset space in which each term is spanned by the basis elements that it contains. We prove that the spherical functions of the associated Gelfand pair are nonnegative linear combinations of the ${\mathcal B}_{n,k}$.