  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.
