#### Vol. 306, No. 1, 2020

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Schur algebras for the alternating group and Koszul duality

Vol. 306 (2020), No. 1, 153–184
##### 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 $d$.

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