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The Modular Temperley-Lieb Algebra

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arxiv 2011.01328 v4 pith:LOUSINMO submitted 2020-11-02 math.RT

R. A. Spencer This is my paper · ORCID
classification math.RT
keywords modulesalgebratemperley-liebalgebrascharacteristicresultssimpletheory
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abstract

We investigate the representation theory of the Temperley-Lieb algebra, $TL_n(\delta)$, defined over a field of positive characteristic. The principle question we seek to answer is the multiplicity of simple modules in cell modules for $TL_n$ 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 $TL_n$ as endomorphism algebras of $U_q(\mathfrak{sl}_2)$ modules. Our results strictly generalise the known characteristic zero theory of the Temperley-Lieb algebras.

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  1. Representation gaps of rigid planar diagram monoids

    math.RT 2025-05 conditional novelty 6.0 of 10

    Rigid non-pivotal Temperley-Lieb, Motzkin, and planar rook monoids have smaller representation gaps than their pivotal counterparts, making them worse for cryptographic use.

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