# Clustering Properties of Jack Polynomials at negative rationals

I'm trying to confirm or deny the truthfulness of a statement. Also I'm asking about a reference for the proof of the statement if it is true.

Following Feigin et al, we say a partition $\lambda$ is $(k,r,n)$-admissible if $$\lambda_i - \lambda_{i+k}\geq r, \quad 1\leq i\leq n-k$$ Suppose $k+1$ and $r-1$ are relatively prime and define $\beta(k,r)=-\frac{r-1}{k+1}$. Define the $\mathbb{C}$-vector space $$I_n^{(k,r)}=\mathrm{span}_{\mathbb{C}}\{P_\lambda^{\beta(k,r)}(x_1, \cdots, x_n)\mid \lambda\text{ is }(k,r,n)\text{-admissible}\}$$ where $P_{\lambda}^{\beta(k,r)}$ is the specialization of Jack polynomial to $\beta=\beta(k,r)$. Feigin et al. proved $P_{\lambda}^{\beta(k,r)}$ is well-defined and furthermore that $I^{(k,r)}_n\subset \mathbb{C}[x_1, \cdots, x_n]^{S_n}$ is an ideal.

I have read in a physics talk-note about quantum Hall effect that the ideal $I^{(k,r)}_n$ is equal to the $\mathbb{C}$-span of all symmetric polynomials $f(x_1, \cdots, x_n)$ with the so called $(k,r)$-clustering property: $$f(\underbrace{Z,Z,\cdots, Z}_{k\text{ times}}, x_{k+1}, x_{k+2}, \cdots, x_n)= \prod_{i=k+1}^n (Z-x_i)^r g(Z,x_{k+1}, \cdots, x_n)$$ for some polynomial $g$. Can someone give me a proper reference for the proof of this statement? Is this even true?

Let's call the above space $J_n^{(k,r)}$. In here the authors prove (among other things) that $I_n^{(k,r)}\subset J^{(k,r)}_n$. So basically what I'm asking is the proof for the reverse inclusion. This also boils down to a study of dimension the space of homogeneous polynomials of degree $d$ in $I^{(k,r)}_n$. One needs to show that $$\dim\left[I^{(k,r)}_n\right]_d \geq \dim\left[J^{(k,r)}_n\right]_d$$ The left hand side is equal to the number $(k,r,n)$-admissible partitions $\lambda$ with $|\lambda|=d$.

For my specific puposes, I only need a much more relaxed version of this statement. Let $n=km$, the minimal degree a $P_\lambda^{\beta(k,r)}\in I^{(k,r)}_n$ can have is $D=m(m-1)kr/2$ and there is exactly one $(k,r,n)$-admissible partition with $|\lambda|=D$. It is actually enough for me to know whether $$\dim\left[J^{(k,r)}_n\right]_D=1$$ is true.