Jack polynomials with prescribed symmetry and some of their clustering properties

Abstract: We study Jack polynomials in $N$ variables, with parameter $\alpha$, and having a prescribed symmetry with respect to two disjoint subsets of variables. For instance, these polynomials can exhibit a symmetry of type AS, which means that they are anti-symmetric in the first $m$ variables and symmetric in the remaining $N-m$ variables. One of our main goals is to extend recent works on symmetric Jack polynomials [arXiv:0711.3062, arXiv:1007.2692, arXiv:1303.4126] and prove that the Jack polynomials with prescribed symmetry also admit clusters of size $k$ and order $r$, that is, the polynomials vanish to order $r$ when $k+1$ variables coincide. We first prove some general properties for generic $\alpha$, such as their uniqueness as triangular eigenfunctions of operators of Sutherland type, and the existence of their analogues in infinity many variables. We then turn our attention to the case with $\alpha=-(k+1)/(r-1)$. We show that for each triplet $(k,r,N)$, there exist admissibility conditions on the indexing sets, called superpartitions, that guaranty both the regularity and the uniqueness of the polynomials. These conditions are also used to establish similar properties for non-symmetric Jack polynomials. As a result, we prove that the Jack polynomials with arbitrary prescribed symmetry, indexed by $(k,r,N)$-admissible superpartitions, admit clusters of size $k=1$ and order $r\geq 2$. In the last part of the article, we find necessary and sufficient conditions for the invariance under translation of the Jack polynomials with prescribed symmetry AS. This allows to find special families of superpartitions that imply the existence of clusters of size $k>1$ and order $r\geq 2$.