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

Part of: Representation theory of groups Representation theory of rings and algebras Linear algebraic groups and related topics

Published online by Cambridge University Press:  08 April 2025

GUS LEHRER  and
MENGFAN LYU
GUS LEHRER
Affiliation:
School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia e-mail: gustav.lehrer@sydney.edu.au
MENGFAN LYU
Affiliation:
School of Computer, Data and Mathematical Sciences, Western Sydney University, Locked Bag 1797, Penrith, N.S.W. 2751, Australia e-mail: mengfan.lyu@westernsydney.edu.au
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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 $\sigma$. We prove the cellularity of $TL_{r, p, n}$ by proving that $\sigma$ 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.

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets 20C08: Hecke algebras and their representations
Secondary: 20G42: Quantum groups (quantized function algebras) and their representations
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 178 , Issue 2 , March 2025 , pp. 193 - 227
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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