Factorisability in K-theory

TL;DR

This paper provides a detailed proof of a lemma relating the Euler characteristic generating functions of factorisable sequences of sheaves on symmetric powers of a scheme with torus action, expressed via plethystic exponential.

Contribution

It offers a comprehensive proof of Okounkov's lemma connecting factorisable sheaves on symmetric powers to plethystic exponential formulas in K-theory.

Findings

Euler characteristic generating functions are expressed as plethystic exponentials.

The proof clarifies the structure of factorisable sequences of sheaves.

Connections between symmetric powers and K-theoretic generating functions are established.

Abstract

We revisit and give a detailed proof of a lemma of Okounkov showing that, for a scheme X with a torus action, the Euler characteristic generating function associated with a "factorisable" sequence of torus-equivariant coherent sheaves on the symmetric powers $\operatorname{Sym}^n X$ equals the plethystic exponential of the generating function of Euler characteristics of some sequence of sheaves on X.