Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra

TL;DR

This paper constructs a Heisenberg algebra action on the equivariant K-theory of Hilbert schemes of points on C^2 using shuffle algebra techniques, connecting geometric representation theory with Macdonald polynomials.

Contribution

It introduces a novel construction of Heisenberg algebra actions via shuffle algebra on the K-theory of Hilbert schemes, linking algebraic and geometric frameworks.

Findings

Heisenberg algebra acts through vertex operators on K-groups.

Commutative elements of shuffle algebra correspond to Macdonald polynomial basis.

Normalized fixed point basis aligns with Macdonald polynomials in Fock space.

Abstract

In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].