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

We introduce the alternating Schur algebra ASF(n,d) as the commutant of the action of the alternating group Ad 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 ASF(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 Z2Z-graded algebra. The algebra ASF(n,d) is Z2Z-graded, having the classical Schur algebra SF(n,d) as its even part. This leads to an approach to Koszul duality for SF(n,d)-modules that is amenable to combinatorial methods. We characterize the category of ASF(n,d)-modules in terms of SF(n,d)-modules and their Koszul duals. We use the graphical basis of ASF(n,d) to study the dependence of the behavior of derived Koszul duality on n and d.

Schur algebra, Koszul duality, Schur–Weyl duality, alternating group
Mathematical Subject Classification 2010
Primary: 05E10, 20G05, 20G43
Received: 15 May 2019
Revised: 20 December 2019
Accepted: 2 January 2020
Published: 14 June 2020
Thangavelu Geetha
School of Mathematics
Indian Institute of Science Education and Research
Amritanshu Prasad
Institute of Mathematical Sciences (HBNI)
Shraddha Srivastava
Institute of Mathematical Sciences (HBNI)