Jack Littlewood-Richardson Coefficients and the Nazarov-Sklyanin Lax Operator

TL;DR

This paper explores the properties of polynomial eigenfunctions of the Nazarov-Sklyanin quantum Lax operator, deriving new constraints on Jack Littlewood-Richardson coefficients through a novel generalization of Kerov's formula.

Contribution

It introduces a new system of constraints on Jack Littlewood-Richardson coefficients based on the multiplication of partitions, extending previous work on eigenfunctions.

Findings

Derived a system of constraints on Jack Littlewood-Richardson coefficients.

Generalized Kerov's formula relating coefficients and residues.

Connected eigenfunction products to partition multiplication.

Abstract

We continue the work begun by Mickler-Moll investigating the properties of the polynomial eigenfunctions of the Nazarov-Sklyanin quantum Lax operator. By considering products of these eigenfunctions, we produce a novel generalization of a formula of Kerov relating Jack Littlewood-Richardson coefficients and residues of certain rational functions. Precisely, we derive a system of constraints on Jack Littlewood-Richardson coefficients in terms of a simple multiplication operation on partitions.