[Paper Review] Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

Boris Feigin, M. Jimbo+2 more|ArXiv.org|Sep 4, 2002

Algebraic structures and combinatorial models2 references18 citations

TL;DR

This paper introduces a q-deformation of the ideal of symmetric polynomials vanishing on shifted diagonals, characterized by Macdonald polynomials at $ t^{k+1}q^{r-1} = 1 $. It proves that the quotient algebra retains the monomial basis of level $ k $ integrable representations of $ \widehat{\mathfrak{sl}}_r $, providing a deformation of the current algebra relations and linking them to correlation functions via Macdonald polynomials.

ABSTRACT

For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at t^{k+1}q^{r-1}=1, and is a deformed version of the one studied earlier in the context of Jack polynomials. In this paper we give a characterization of I^(k,r)_n in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x_{2}=tq^{s_1}x_1,...,x_{k+1}=tq^{s_k}x_k. The ideal I^(k,r)_n may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra \hat{sl_r}. We give a brief discussion about this connection.

Motivation & Objective

• To characterize the ideal $ I_n^{(k,r)} $ of symmetric polynomials vanishing on $ k $-codimensional shifted diagonals using Macdonald polynomials.
• To establish a $ q $-deformation of the current algebra relations of $ \widehat{\mathfrak{sl}}_r $ at level $ k $, preserving the monomial basis structure.
• To connect the deformed algebra to correlation functions of abelian currents in integrable representations.
• To generalize the zero-condition characterization of matrix elements of currents in $ \widehat{\mathfrak{sl}}_r $-representations to the $ q $-deformed setting.

Proposed method

• Define the ideal $ I_n^{(k,r)} $ as the span of Macdonald polynomials $ P_\lambda(x_1,\dots,x_n;q,t) $ at $ t^{k+1}q^{r-1} = 1 $.
• Characterize $ I_n^{(k,r)} $ via explicit vanishing conditions on shifted diagonals: $ x_{i+1} = t q^{s_i} x_i $ for $ 1 \leq i \leq k $, with $ 0 \leq s_1 \leq \cdots \leq s_k \leq r-2 $.
• Construct a deformed abelian current $ e(z) = \sum e_i z^i $ satisfying $ e(z) e(tq^{s_1}z) \cdots e(t^k q^{s_k}z) = 0 $ for all such $ s_i $.
• Show that the Fourier coefficients of this product generate an ideal $ \tilde{\mathcal{J}}(q,t) $ in $ \mathbb{C}[\{e_i\}_{i \in \mathbb{Z}}] $, and prove that the quotient has the same monomial basis as the vacuum representation of $ \widehat{\mathfrak{sl}}_r $ at level $ k $.
• Use the structure of Macdonald polynomials and their specialization at $ t^{k+1}q^{r-1} = 1 $ to analyze the vanishing conditions and basis properties.
• Relate the deformed algebra $ E_{k,r}(q,t) $ to the Weyl group action and show that irreducible representations of $ \widehat{\mathfrak{sl}}_r $ remain irreducible upon restriction to $ E_{k,r}(q,t) $ for generic $ q,t $.

Experimental results

Research questions

• RQ1 What is the precise set of zero conditions on shifted diagonals that characterize the ideal $ I_n^{(k,r)} $ of symmetric polynomials?
• RQ2 How can the level $ k $ integrable representations of $ \widehat{\mathfrak{sl}}_r $ be deformed via $ q,t $-parameters while preserving the monomial basis?
• RQ3 What is the role of Macdonald polynomials in characterizing the deformed ideal and its relation to current algebra relations?
• RQ4 How does the $ q $-deformation of the current algebra $ e(z) $, satisfying $ e(z)e(tq^{s_1}z)\cdots e(t^k q^{s_k}z) = 0 $, relate to the structure of correlation functions?
• RQ5 Can the matrix elements of the current $ e(z) $ in integrable representations be characterized by $ q $-deformed zero conditions on diagonals of codimension $ k+1 $?

Key findings

• The ideal $ I_n^{(k,r)} $ is characterized by the vanishing of symmetric polynomials on all shifted diagonals of the form $ x_{i+1} = t q^{s_i} x_i $ for $ 1 \leq i \leq k $, with $ 0 \leq s_1 \leq \cdots \leq s_k \leq r-2 $.
• At $ t^{k+1}q^{r-1} = 1 $, the Macdonald polynomials $ P_\lambda(x_1,\dots,x_n;q,t) $ form a basis of $ I_n^{(k,r)} $, providing a deformed version of the Jack polynomial ideal.
• For generic $ q,t $ satisfying $ t^{k+1}q^{r-1} = 1 $, the quotient algebra $ \mathbb{C}[\{e_i\}_{i \geq 1}] / \mathcal{J}(q,t) $ has the same monomial basis as the principal subspace of the level $ k $ vacuum representation of $ \widehat{\mathfrak{sl}}_r $.
• The deformed current $ e(z) $ satisfies $ e(z)e(tq^{s_1}z)\cdots e(t^k q^{s_k}z) = 0 $ for all $ 0 \leq s_1 \leq \cdots \leq s_k \leq r-2 $, generalizing the undeformed relation $ e(z)^{k+1} = 0 $ at $ t=1, q=\tau $.
• The algebra $ E_{k,r}(q,t) = \mathbb{C}[D,D^{-1}] \ltimes \mathbb{C}[\{e_i\}_{i \in \mathbb{Z}}] / \tilde{\mathcal{J}}(q,t) $ has irreducible representations that are deformations of those of $ \widehat{\mathfrak{sl}}_r $, preserving irreducibility upon restriction.
• The $ q $-deformation provides an explicit, $ q $-dependent characterization of integrability conditions for the current $ e(z) $, which is implicit in the undeformed case.