# Some Singular Vector-valued Jack and Macdonald Polynomials

## TL;DR

This paper investigates singular vector-valued Jack and Macdonald polynomials associated with symmetric groups and Hecke algebras, focusing on their singular values and the structure of polynomials with specific leading terms.

## Contribution

It characterizes the singular polynomials' leading terms and determines the singular values based on the Ferrers diagram edges of the Young tableau shape.

## Key findings

- Identifies conditions for singular polynomials with leading term $x_1^m ensor S$
- Determines singular values depend on Ferrers diagram edge properties
- Analyzes the structure of vector-valued Jack and Macdonald polynomials

## Abstract

For each partition $\tau$ of $N$ there are irreducible modules of the symmetric groups $\mathcal{S}_{N}$ or the corresponding Hecke algebra $\mathcal{H}_{N}\left( t\right) $ whose bases consist of reverse standard Young tableaux of shape $\tau$. There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules, respectively.The Jack polynomials are a special case of those constructed by Griffeth for the infinite family $G\left( n,p,N\right) $ of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For both the group $\mathcal{S}_{N}$ and the Hecke algebra $\mathcal{H}_{N}\left( t\right) $ there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by $\kappa$ and $\left( q,t\right) $ respectively. For certain values of the parameters (called singular values) there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is $x_{1}^{m}\otimes S$, where $S$ is an arbitrary reverse standard Young tableau of shape $\tau$. The singular values depend on properties of the edge of the Ferrers diagram of $\tau$.