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

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Schur algebras for the alternating group and Koszul duality

### Thangavelu Geetha, Amritanshu Prasad and Shraddha Srivastava

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 $d$.

##### 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
##### Authors
 Thangavelu Geetha School of Mathematics Indian Institute of Science Education and Research Thiruvananthapuram India Amritanshu Prasad Institute of Mathematical Sciences (HBNI) Chennai India Shraddha Srivastava Institute of Mathematical Sciences (HBNI) Chennai India