# memo_has_moved_text();Imaginary Schur-Weyl duality

Alexander Kleshchev and Robert Muth

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

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Chapters

• Chapter 1. Introduction
• Chapter 2. Preliminaries
• Chapter 3. Khovanov-Lauda-Rouquier algebras
• Chapter 4. Imaginary Schur-Weyl duality
• Chapter 5. Imaginary Howe duality
• Chapter 6. Morita equaivalence
• Chapter 7. On formal characters of imaginary modules
• Chapter 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 modulesâĂŤone for each real positive root for the corresponding affine root system , as well as irreducible imaginary modulesâĂŤone for each -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.