Gr\"obner-Shirshov bases for Temperley-Lieb algebras of the complex reflection group of type $G(d,1,n)$

TL;DR

This paper constructs a Gr"obner-Shirshov basis for the Temperley-Lieb algebra associated with the complex reflection group G(d,1,n), providing a combinatorial interpretation and a dimension formula.

Contribution

It generalizes previous results for type B Coxeter groups to complex reflection groups G(d,1,n) by constructing a basis and interpreting standard monomials combinatorially.

Findings

Established a Gr"obner-Shirshov basis for the algebra

Provided a combinatorial interpretation of monomials

Derived the dimension formula for the algebra

Abstract

We construct a Gr\"obner-Shirshov basis of the Temperley-Lieb algebra $\mathfrak{T}(d,n)$ of the complex reflection group $G(d,1,n)$, inducing the standard monomials expressed by the generators $\{E_i\}$ of $\mathfrak{T}(d,n)$. This result generalizes the one for the Coxeter group of type $B_n$ in \cite{KimSSLeeDI}. We also give a combinatorial interpretation of the standard monomials of $\mathfrak{T}(d,n)$, relating to the fully commutative elements of the complex reflection group $G(d,1,n)$. In this way, we obtain the dimension formula of $\mathfrak{T}(d,n)$.