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Abstract
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We introduce the alternating Schur algebra
as the commutant of the action of the alternating group
on the
-fold tensor power
of an
-dimensional
-vector space. When
has characteristic
different from
,
we give a basis of
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
-graded algebra.
The algebra
is
-graded, having the
classical Schur algebra
as its even part. This leads to an approach to Koszul duality for
-modules
that is amenable to combinatorial methods. We characterize the category of
-modules in terms
of
-modules
and their Koszul duals. We use the graphical basis of
to study the dependence of the behavior of derived Koszul duality on
and
.
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Keywords
Schur algebra, Koszul duality, Schur–Weyl duality,
alternating group
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Mathematical Subject Classification 2010
Primary: 05E10, 20G05, 20G43
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Milestones
Received: 15 May 2019
Revised: 20 December 2019
Accepted: 2 January 2020
Published: 14 June 2020
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