Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

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, i.e., 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 $\left(m,m,\dots ,m\right)$ with $2k$ components and the constructed singular polynomials are of isotype $\left(mk,mk\right)$ for the parameter $\kappa =$ $1/\left(m+2\right)$ . This 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.