Generalized Clustering Conditions of Jack Polynomials at Negative Jack Parameter $\alpha$

Understand

We present several conjectures on the behavior and clustering properties of Jack polynomials at \emph{negative} parameter $\alpha=-\frac{k+1}{r-1}$, of partitions that violate the $(k,r,N)$ admissibility rule of Feigin \emph{et.

• al.} [\onlinecite{feigin2002}].
• We find that "highest weight" Jack polynomials of specific partitions represent the minimum degree polynomials in $N$ variables that vanish when $s$ distinct clusters of $k+1$ particles are formed, with $s$ and $k$ positive integers.
• Explicit counting formulas are conjectured.