# Schur Algebras for the Alternating Group and Koszul Duality

## TL;DR

This paper introduces the alternating Schur algebra, explores its structure via bipartite graphs, and develops a combinatorial approach to Koszul duality for related modules, linking algebraic and graphical methods.

## Contribution

It defines the alternating Schur algebra, provides a graphical basis, and connects Koszul duality with combinatorial techniques for modules over classical Schur algebras.

## Key findings

- Graphical basis of $AS_F(n,d)$ in characteristic not 2
- Characterization of $AS_F(n,d)$-modules via $S_F(n,d)$-modules
- Dependence of Koszul duality behavior on parameters $n$ and $d$

## Abstract

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$.