The Modular Temperley-Lieb Algebra

Abstract

We investigate the representation theory of the Temperley-Lieb algebra, $T{L}_n(\delta )$, defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for $T{L}_n$ over arbitrary rings. This provides us with the decomposition numbers for this algebra, as well as the dimensions of all simple modules. We obtain these results from diagrammatic principles, without appealing to realisations of $T{L}_n$ as endomorphism algebras of $U_q({\mathfrak{sl}}_2)$ modules. Our results strictly generalise the known characteristic zero theory of the Temperley-Lieb algebras.

33 pages, fix statement of Thm 4.6 and proof of Thm 8.3, add applications lower bound on irreducible dimensions and prospects

Cited by in corpus (7)

• SL2 tilting modules in the mixed case
• Growth rates of the number of indecomposable summands in tensor powers
• Monoidal categories, representation gap and cryptography
• The homology of the Temperley-Lieb algebras
• Sandwich cellularity and a version of cell theory
• Fractal behavior of tensor powers of the two dimensional space in prime characteristic
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors