The multiplicative structure of the K-theoretical McKay correspondence for the Hilbert scheme of points in the complex plane

TL;DR

This paper explores the K-theory of the Hilbert scheme of points in the complex plane, establishing a formula for endomorphisms related to Adams powers and describing multiplication structure constants.

Contribution

It proves a conjectured formula for endomorphisms in K-theory and details the multiplication structure constants induced by tensor product.

Findings

Proved a conjectured formula for endomorphisms in K-theory.

Described the structure constants for multiplication in K-theory.

Connected K-theory of Hilbert schemes to symmetric functions.

Abstract

We consider the K-theory of the Hilbert scheme of points in the complex plane, which under McKay correspondence is isomorphic to the space of symmetric functions $\Lambda^n$. We prove a formula conjectured by Boissi\`ere for the endomorphism of $\Lambda^n$ induced by multiplication by the classes of the Adams powers of the tautological bundle. We describe the structure constants for the multiplication on $\Lambda^n$ induced by the tensor product in K-theory.