Cauchy identities for genus 2 Schur polynomials

TL;DR

This paper introduces genus 2 Schur polynomials as a generalization of classical Schur functions, establishing Cauchy identities for these polynomials at the special case when parameters t and q equal 1.

Contribution

It develops a genus 2 analog of Cauchy identities specifically for Schur polynomials, extending the classical theory to a higher-genus setting.

Findings

Established Cauchy identities for genus 2 Schur polynomials at t=q=1

Connected the algebraic structure to genus 2 mapping class group actions

Extended classical symmetric function identities to higher genus context

Abstract

Genus 2 Macdonald polynomials $\Psi^{(q,t)}_{j_1,j_2,j_3}$ generalize ordinary Macdonald polynomials in several aspects. First, they provide common eigenfunctions for commuting difference operators that generalize the Macdonald difference operators of type $A_1$. Second, the algebra generated by these difference operators together with multiplication operators admits an action of genus 2 mapping class group (MCG) that generalizes the well-known action of $SL(2,{\mathbb Z})$ for ordinary Macdonald polynomials. In this paper, one more important aspect of Macdonald theory is considered: the Cauchy identities. We construct a genus 2 generalization of Cauchy identities in the particular case when $t=q=1$, i.e. for genus 2 Schur polynomials.