1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.