Bases and BGG Resolutions of Simple Modules of Temperley–Lieb Algebras of Type B

Dimitris Michailidis

doi:10.1093/imrn/rnab003

Abstract

Abstract We construct explicit bases of simple modules and Bernstein–Gelfand–Gelfand (BGG) resolutions of all simple modules of the (graded) Temperley–Lieb algebra of type B over a field of characteristic zero.

Highlights

Inspired by the study of certain models in physics, Martin and Saleur introduced the main hero of this paper, the Temperley-Lieb algebra of type B or blob algebra, as the diagrammatic two parameter generalisation of the Temperley-Lieb algebra of type A [20]
The blob algebra controls a portion of the representation theory of the Kac-Moody quantum algebra Uq(sl2) as Martin and Ryom-Hansen [19] established via Ringel duality
The blob algebra is quasi-hereditary and in that setting Hazi, Martin and Parker [13] determined the structure of the indecomposable tilting modules using the graded structure

Summary

Introduction

Inspired by the study of certain models in physics, Martin and Saleur introduced the main hero of this paper, the Temperley-Lieb algebra of type B or blob algebra, as the diagrammatic two parameter generalisation of the Temperley-Lieb algebra of type A [20]. The blob algebra is quasi-hereditary and in that setting Hazi, Martin and Parker [13] determined the structure of the indecomposable tilting modules using the graded structure Having their origins in Lie theory, alcove geometries play an important role in the understanding of the representation theory of Hecke and KLR algebras. In the context of modular representation theory of the symmetric group and Hecke algebras, BGG resolutions were first used by Bowman, Norton and Simental [3] They utilised resolutions of Specht modules in order to provide homological construction of unitary simple modules of Cherednik and Hecke algebras of type A. Simple modules indexed by onecolumn bipartitions which belong to hyperplanes have much easier BGG resolutions and they are used in the proof of the second main theorem of this paper which is the following. This construction is over a field of characteristic zero

Combinatorics of tableaux and paths

The Blob Algebra

Homomorphisms between cell modules

Bases of simple modules