Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

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)
Published electronically: July 15, 2016
MSC: Primary 20C08, 20C30, 05E10

View full volume PDF

View other years and numbers:

Table of Contents


  • 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


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.

References [Enhancements On Off] (What's this?)