Application of Lagrange inversion to wall-crossing for Quot schemes on surfaces

TL;DR

This paper proves a new combinatorial identity related to Quot schemes on surfaces by applying Lagrange inversion, establishing the equality of two power series derived from geometric and algebraic methods.

Contribution

It introduces a novel proof technique for the equality of power series in Quot schemes, independent of prior literature, using Lagrange inversion.

Findings

Established equality of two power series via Lagrange inversion

Proved a new combinatorial identity

Made Quot scheme analysis independent of previous results

Abstract

Motivated by my work on enumerative invariants for Quot schemes, I related two power series obtained by two different means. One of them was computed using geometric arguments via virtual localization methods and the other one came from working with representation theoretic objects called vertex algebras. In this note, I give proof of the equality of the two power series by relying only on techniques related to Lagrange inversion. This makes my work on Quot schemes independent of the previous results in the literature and proves a new combinatorial identity.