[Paper Review] Symmetric polynomials vanishing on the diagonals shifted by roots of unity

Boris Feigin, M. Jimbo+3 more|ArXiv.org|Sep 11, 2002

Advanced Mathematical Identities2 references4 citations

TL;DR

This paper characterizes symmetric polynomials vanishing on diagonals shifted by roots of unity when $k+1$ and $r-1$ are coprime. It constructs an explicit basis using Macdonald polynomials with $r-1$-slim partitions and admissible partitions, and derives the character of the space via a product formula involving $b_n^{(k)}(v^{r-1})$ and a generating function for slim partitions.

ABSTRACT

For a pair of positive integers (k,r) with r>1 such that k+1 and r-1 are relatively prime, we describe the space of symmetric polynomials in variables x_1,...,x_n which vanish at all diagonals of codimension k of the form x_i=tq^{s_i}x_{i-1}, i=2,...,k+1, where t and q are primitive roots of unity of orders k+1 and r-1.

Motivation & Objective

• To describe the space of symmetric polynomials in $n$ variables that vanish on diagonals of the form $x_i = t q^{s_i} x_{i-1}$, where $t$ and $q$ are primitive roots of unity of orders $k+1$ and $r-1$, respectively.
• To construct an explicit basis for this space under the condition that $\gcd(k+1, r-1) = 1$.
• To compute the character $\chi_{k,r}(z,v)$ of the space of such symmetric polynomials.
• To establish a connection between the vanishing conditions and the representation theory of $W_k$ algebra minimal series $(k+1, k+r)$ via the Jack limit.
• To generalize results from the generic $u$-case in [FJMM2] to the root of unity setting where $u$ is a primitive root of unity of order $(k+1)(r-1)$.

Proposed method

• Define the wheel set $\mathcal{S}_{r-1}(q,t) = \{t, tq, \dots, tq^{r-2}\}$ with $t^{k+1} = 1$, $q^{r-1} = 1$, and $t$, $q$ primitive roots of unity.
• Use Macdonald polynomials $P_\lambda(x;q,t)$ with $q = u^{-(k+1)}$, $t = u^{r-1}$, and $u$ a primitive root of unity of order $(k+1)(r-1)$.
• Introduce the notion of $r-1$-slim partitions: partitions where each part appears at most $r-1$ times.
• Define $F_n^{(k,r)}$ as the space of symmetric polynomials vanishing on all $k+1$-tuples satisfying $x_2 = t x_1, \dots, x_{k+1} = t q^{k} x_k$, $x_1 = t q^{k} x_{k+1}$, with $t^{k+1} = 1$, $q^{r-1} = 1$.
• Apply the Frobenius homomorphism $\mathcal{F}$ to lift functions from $F_n^{(k,2)}$ to $F_n^{(k,r)}$ via $f_\lambda = \mathcal{F}(\tilde{f}_\lambda)$.
• Use the Macdonald operator $D_n(X;q,\tilde{t})$ to verify that the vanishing condition is preserved under certain transformations, enabling the construction of a basis via eigenvalue separation.

Experimental results

Research questions

• RQ1 What is the structure of the space of symmetric polynomials vanishing on diagonals shifted by roots of unity when $\gcd(k+1, r-1) = 1$?
• RQ2 How can one construct an explicit basis for this space using Macdonald polynomials?
• RQ3 What is the character $\chi_{k,r}(z,v)$ of the space $F^{(k,r)}$ of such symmetric polynomials?
• RQ4 How does the root of unity case differ from the generic $u$-case in [FJMM2], particularly in terms of vanishing planes and resonance conditions?
• RQ5 What is the relationship between this space and the correlation functions of an abelian current in a vertex operator algebra associated with the $W_k$ algebra minimal series $(k+1, k+r)$?

Key findings

• The space $F_n^{(k,r)}$ of symmetric polynomials vanishing on the specified diagonals has a basis formed by products $f_\lambda g_\mu$, where $f_\lambda$ is a basis element of $F_n^{(k,2)}$ and $g_\mu$ is a Macdonald polynomial with $\mu$ an $r-1$-slim partition.
• The character $\chi_{k,r}(z,v)$ is given by $\sum_{n=0}^\infty \left( b_n^{(k)}(v^{r-1}) \prod_{s=1}^n \frac{1 - v^{s(r-1)}}{1 - v^s} \right) z^n$, where $b_n^{(k)}(v)$ is the generating function for $k$-admissible partitions.
• The basis construction relies on the Frobenius homomorphism $\mathcal{F}$, lifting functions from $F_n^{(k,2)}$ to $F_n^{(k,r)}$ via $f_\lambda = \mathcal{F}(\tilde{f}_\lambda)$.
• The vanishing condition is preserved under the action of Macdonald operators $D_n(X;q,\tilde{t})$, which ensures that the eigenvalue separation argument holds for distinct $\lambda$.
• The space $F_n^{(k,r)}$ is isomorphic to the tensor product of the space of $k$-admissible partitions and the space of $r-1$-slim partitions, with the character being a product of generating functions.
• In the Jack limit $u \to 1$, the space $F^{(k,r)}$ corresponds to Jack polynomials with parameter $\beta = -(r-1)/(k+1)$, and is expected to coincide with correlation functions of an abelian current in the $W_k$ algebra minimal series $(k+1, k+r)$.