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 $\sigma$. 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.