A GL-Equivariant Complex Inducing Character Identities for Schur Modules

TL;DR

This paper constructs a GL-equivariant complex of Schur modules over rings of positive characteristic, providing new tools to derive classical identities and explicitly realize Adams operations in algebraic K-theory.

Contribution

It introduces a novel GL-equivariant complex that generalizes existing identities and offers an explicit construction for Adams operations in algebraic K-theory.

Findings

Provides a new complex for Schur modules in positive characteristic

Reproves Adams operations identities explicitly

Globalizes to complexes of vector bundles

Abstract

In this paper we construct a GL-equivariant complex of Schur modules over a ring of positive characteristic that can be used to deduce classical alternating sum identities for Schur polynomials. This complex globalizes to a complex of vector bundles and can also be used to give an explicit construction of an exact sequence predicted by work of Grayson involving Adams operations identities on the algebraic K-theory of a given scheme $X$. The more general complex gives an explicit construction that reproves the aforementioned Adams operations identities in full generality.