Schur algebras for the alternating group and Koszul duality

Thangavelu Geetha, Amritanshu Prasad and Shraddha Srivastava

Vol. 306 (2020), No. 1, 153–184

DOI: 10.2140/pjm.2020.306.153

Abstract

We introduce the alternating Schur algebra ${AS}_F\left(n,d\right)$ 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 ${AS}_F\left(n,d\right)$ 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 $Z∕2Z$-graded algebra. The algebra ${AS}_F\left(n,d\right)$ is $Z∕2Z$-graded, having the classical Schur algebra $S_F\left(n,d\right)$ as its even part. This leads to an approach to Koszul duality for $S_F\left(n,d\right)$-modules that is amenable to combinatorial methods. We characterize the category of ${AS}_F\left(n,d\right)$-modules in terms of $S_F\left(n,d\right)$-modules and their Koszul duals. We use the graphical basis of ${AS}_F\left(n,d\right)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$.

Keywords

Schur algebra, Koszul duality, Schur–Weyl duality, alternating group

Mathematical Subject Classification 2010

Primary: 05E10, 20G05, 20G43

Milestones

Received: 15 May 2019

Revised: 20 December 2019

Accepted: 2 January 2020

Published: 14 June 2020