eBook ISBN:  9781470436032 
Product Code:  MEMO/245/1157.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 
eBook ISBN:  9781470436032 
Product Code:  MEMO/245/1157.E 
List Price:  $75.00 
MAA Member Price:  $67.50 
AMS Member Price:  $45.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 245; 2016; 83 ppMSC: Primary 20; 05
The authors study imaginary representations of the KhovanovLaudaRouquier 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 \(l\)multiplication. The authors study imaginary modules by means of “imaginary SchurWeyl duality” and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of GelfandGraev representations, Ringel duality and the JacobiTrudy formula.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. KhovanovLaudaRouquier algebras

4. Imaginary SchurWeyl duality

5. Imaginary Howe duality

6. Morita equaivalence

7. On formal characters of imaginary modules

8. Imaginary tensor space for nonsimplylaced types


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The authors study imaginary representations of the KhovanovLaudaRouquier 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 \(l\)multiplication. The authors study imaginary modules by means of “imaginary SchurWeyl duality” and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of GelfandGraev representations, Ringel duality and the JacobiTrudy formula.

Chapters

1. Introduction

2. Preliminaries

3. KhovanovLaudaRouquier algebras

4. Imaginary SchurWeyl duality

5. Imaginary Howe duality

6. Morita equaivalence

7. On formal characters of imaginary modules

8. Imaginary tensor space for nonsimplylaced types