Imaginary Schur-Weyl duality

2013/12/20 by Kleshchev, Alexander, Muth, Robert · 1 citation
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1312.6104

Abstract

We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules---one for each real positive root for the corresponding affine root system \tt X_l⁽¹⁾, as well as irreducible imaginary modules---one for each l-multipartition. We study imaginary modules by means of `imaginary Schur-Weyl duality'. We introduce an imaginary analogue of tensor space and the imaginary Schur algebra. We construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra. We construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.

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