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Imaginary Schur-Weyl duality
About this Title
Alexander Kleshchev, Department of Mathematics, University of Oregon, Eugene, Oregon 97403 and Robert Muth, Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Publication: Memoirs of the American Mathematical Society
Publication Year:
2017; Volume 245, Number 1157
ISBNs: 978-1-4704-2249-3 (print); 978-1-4704-3603-2 (online)
DOI: https://doi.org/10.1090/memo/1157
Published electronically: July 15, 2016
MSC: Primary 20C08, 20C30, 05E10
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Khovanov-Lauda-Rouquier algebras
- 4. Imaginary Schur-Weyl duality
- 5. Imaginary Howe duality
- 6. Morita equaivalence
- 7. On formal characters of imaginary modules
- 8. Imaginary tensor space for non-simply-laced types
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 $\texttt {X}_l^{(1)}$, as well as irreducible imaginary modules—one for each $l$-multiplication. 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.- Pierre Baumann, Joel Kamnitzer, and Peter Tingley, Affine Mirković-Vilonen polytopes, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 113–205. MR 3270589, DOI 10.1007/s10240-013-0057-y
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