Generalised Temperley-Lieb algebras of type G(r,p,n)

2022/11/17 by G Lehrer, Lehrer, Gus, Mengfan Lyu +1
Computer Science · Mathematics · #16G20 #20C08 (Primary) #20G42 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Coding theory and cryptography #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2211.09377

openalex publication_date 2022/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In an earlier work, we defined a ``generalised Temperley-Lieb algebra'' TL_r,1,n corresponding to the imprimitive reflection group G(r,1,n) as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley-Lieb algebra TL_r,p,n which corresponds to the complex reflection group G(r,p,n). Our definition identifies TL_r,p,n as the fixed-point subalgebra of TL_r,1,n under a certain automorphism σ. We prove the cellularity of TL_r,p,n by proving that σ induces a special shift automorphism with respect to the cellular structure of TL_r,1,n. We also give a description of the cell modules of TL_r,p,n and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

Related

• Generalised Temperley-Lieb algebras of type G(r,1,n)
• Representations of Temperley--Lieb Algebras
• A cellular quotient of the Temperley--Lieb algebra of type D
• Representation theory and an isomorphism theorem for the Framisation of the Temperley-Lieb algebra
• Diagrammatic representations of Generalized Temperley-Lieb algebras of affine type \widetildeB and \widetildeD