# Vector-valued Jack Polynomials and Wavefunctions on the Torus

## TL;DR

This paper introduces a new class of eigenfunctions for the Calogero-Sutherland model using vector-valued Jack polynomials, expanding the mathematical framework for quantum many-body systems on the torus.

## Contribution

It develops a novel construction of eigenfunctions employing generalized Jack polynomials valued in symmetric group modules, applicable to all irreducible representations.

## Key findings

- Constructed eigenfunctions using matrix solutions of differential systems.
- Generated symmetric probability densities on the N-torus from these eigenfunctions.
- Extended the theory of Jack polynomials to vector-valued cases for quantum models.

## Abstract

The Hamiltonian of the quantum Calogero-Sutherland model of $N$ identical particles on the circle with $1/r^{2}$ interactions has eigenfunctions consisting of Jack polynomials times the base state. By use of the generalized Jack polynomials taking values in modules of the symmetric group and the matrix solution of a system of linear differential equations one constructs novel eigenfunctions of the Hamiltonian. Like the usual wavefunctions each eigenfunction determines a symmetric probability density on the $N$-torus. The construction applies to any irreducible representation of the symmetric group. The methods depend on the theory of generalized Jack polynomials due to Griffeth, and the Yang-Baxter graph approach of Luque and the author.