[Paper Review] Action of Coxeter groups on m-harmonic polynomials and KZ equations

Giovanni Felder, А. П. Веселов|ArXiv.org|Aug 2, 2001

Algebraic structures and combinatorial models24 references18 citations

TL;DR

This paper studies m-harmonic polynomials—generalizations of classical harmonic polynomials—via the Matsuo-Cherednik correspondence applied to the degenerate case of zero spectral parameters in KZ equations. It computes the Poincaré polynomials of the space $ H_m $ and its isotypical components for Coxeter groups, and provides an explicit formula for the lowest-degree $ m $-harmonic polynomials in the $ S_n $ case.

ABSTRACT

The Matsuo-Cherednik correspondence is an isomorphism from solutions of Knizhnik-Zamolodchikov equations to eigenfunctions of generalized Calogero-Moser systems associated to Coxeter groups G and a multiplicity function m on their root systems. We apply this correspondence to the most degenerate case of zero spectral parameters. The space of eigenfunctions is then the space H_m of m-harmonic polynomials, recently introduced in math-ph/0105014. We compute the Poincare' polynomials for the space H_m and of its isotypical components corresponding to each irreducible representation of the group G. We also give an explicit formula for m-harmonic polynomials of lowest positive degree in the S_n case.

Motivation & Objective

• To generalize classical harmonic polynomials to m-harmonic polynomials associated with Coxeter groups and multiplicity functions.
• To investigate the structure of the space $ H_m $ of m-harmonic polynomials, particularly its Poincaré polynomial and isotypical components.
• To apply the Matsuo-Cherednik correspondence in the degenerate case (zero spectral parameters) to relate solutions of KZ equations to m-harmonic polynomials.
• To compute explicit formulas for m-harmonic polynomials of lowest positive degree in the symmetric group $ S_n $ case.
• To explore the distribution of degrees of m-harmonic polynomials using asymptotic representation theory and Kerov's results on Plancherel measure.

Proposed method

• Utilizes the Matsuo-Cherednik isomorphism between solutions of Knizhnik–Zamolodchikov (KZ) equations and eigenfunctions of generalized Calogero–Moser systems.
• Applies a modified version of the KZ equations with values in $ S(V)/I(\lambda) $, valid for all $ \lambda \in V $, including $ \lambda = 0 $, to handle the degenerate case.
• Employs the Matsuo-Cherednik map $ \mu $ to relate solutions of the KZ equations to solutions of the generalized Calogero–Moser system with zero spectral parameters.
• Computes the Poincaré polynomial $ P(H_m, t) $ for the space $ H_m $ and its isotypical components via representation-theoretic decomposition.
• Uses the action of the Coxeter group $ G $ on $ H_m $, decomposing it into irreducible representations and computing the Poincaré series for each isotypical component.
• Applies Kerov’s asymptotic results on Plancherel measure of $ S_n $ to analyze the distribution of degrees of m-harmonic polynomials for large $ n $.

Experimental results

Research questions

• RQ1 What is the structure of the space $ H_m $ of m-harmonic polynomials for a Coxeter group $ G $ with multiplicity function $ m $, particularly its Poincaré polynomial?
• RQ2 How do the isotypical components of $ H_m $, corresponding to irreducible representations of $ G $, behave under the action of the group?
• RQ3 Can an explicit formula be derived for the lowest-degree m-harmonic polynomials in the case of the symmetric group $ S_n $?
• RQ4 What is the asymptotic distribution of the degrees of m-harmonic polynomials for large $ n $, particularly in the $ S_n $ case?
• RQ5 To what extent does the Gorenstein property of the algebra of quasiinvariants $ Q_m $ reflect in the palindromic nature of $ P(H_m, t) $?

Key findings

• The Poincaré polynomial of the space $ H_m $ of m-harmonic polynomials is palindromic, reflecting the Gorenstein property of the algebra of quasiinvariants $ Q_m $.
• For the symmetric group $ S_n $, an explicit formula is derived for m-harmonic polynomials of lowest positive degree.
• The Poincaré polynomial of $ H_m $ is computed for each isotypical component corresponding to an irreducible representation of $ G $, revealing the graded structure of $ H_m $.
• As $ n \to \infty $, the normalized degree distribution $ \frac{1}{n}(d^{-} - \frac{n(n-1)}{2}) $ converges in distribution to $ N(0, 1/2) $, indicating Gaussian fluctuations around the mean.
• The Matsuo-Cherednik isomorphism is extended to the degenerate case $ \lambda = 0 $ via a module $ S(V)/I(\lambda) $, enabling the study of m-harmonic polynomials as solutions of KZ equations at zero spectral parameters.
• The result confirms that the Poincaré polynomial of $ H_m $ is independent of the choice of basis and depends only on the group $ G $ and multiplicity function $ m $, with a structure that generalizes the classical case $ m = 0 $.