# Connections between vector-valued and highest weight Jack and Macdonald polynomials

## TL;DR

This paper investigates the relationships and projection properties between vector-valued and scalar Jack and Macdonald polynomials, focusing on their algebraic symmetries and representation-theoretic aspects.

## Contribution

It establishes conditions for projections that preserve symmetry actions and explores the connection between nonsymmetric and highest weight symmetric polynomials, including clustering conjectures.

## Key findings

- Identified conditions for projections commuting with symmetric group and Hecke algebra actions.
- Analyzed the relation between nonsymmetric and highest weight symmetric polynomials.
- Studied the quasistaircase partition in the context of clustering conjectures.

## Abstract

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively, and with the Cherednik operators for which these polynomials are eigenfunctions. In the framework of the representation theory of the symmetric group and the Hecke algebra, we study the relation between singular nonsymmetric Jack and Macdonald polynomials and highest weight symmetric Jack and Macdonald polynomials. Moreover, we study the quasistaircase partition as a continuation of our study on the conjectures of Bernevig and Haldane on clustering properties of symmetric Jack polynomials.