Singular nonsymmetric Jack polynomials for some rectangular tableaux

Abstract

In the intersection of the theories of nonsymmetric Jack polynomials in $N$ variables and representations of the symmetric groups $\mathcal{S}_{N}$ one finds the singular polynomials. For certain values of the parameter $\kappa$ there are Jack polynomials which span an irreducible $\mathcal{S}_{N}$-module and are annihilated by the Dunkl operators. The $\mathcal{S}_{N}$-module is labeled by a partition of $N$, called the isotype of the polynomials. In this paper the Jack polynomials are of the vector-valued type, that is, elements of the tensor product of the scalar polynomials with the span of reverse standard Young tableaux of the shape of a fixed partition of $N$. In particular this partition is of shape $\left( m,m,\ldots,m\right) $ with $2k$ components and the constructed singular polynomials are of isotype $\left( mk,mk\right) $ for the parameter $\kappa=$ $1/\left( m+2\right) $. The paper contains the necessary background on nonsymmetric Jack polynomials and representation theory and explains the role of Jucys-Murphy elements in the construction. The main ingredient is the proof of uniqueness of certain spectral vectors, namely, the list of eigenvalues of the Jack polynomials for the Cherednik-Dunkl operators, when specialized to $\kappa=1/\left( m+2\right) $. The paper finishes with a discussion of associated maps of modules of the rational Cherednik algebra and an example illustrating the difficulty of finding singular polynomials for arbitrary partitions.