Some Singular Vector-Valued Jack and Macdonald Polynomials

For each partition $\tau$ of N, there are irreducible modules of the symmetric groups ${\mathcal{S}}_N$ and of the corresponding Hecke algebra ${\mathcal{H}}_N\left(t\right)$ whose bases consist of the reverse standard Young tableaux of shape $\tau$ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials 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 each of the groups ${\mathcal{S}}_N$ 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 these 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 the properties of the edge of the Ferrers diagram of $\tau$ .