Schur Algebras for the Alternating Group and Koszul Duality

Abstract

We introduce the alternating Schur algebra $A{S}_F(n,d)$ 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 $A{S}_F(n,d)$ 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 $A{S}_F(n,d)$ 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 $A{S}_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $A{S}_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$.

27 pages, 1 figure, to appear in Pacific Journal of Mathematics