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

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.