Gröbner-Shirshov Bases for Temperley-Lieb Algebras of Complex Reflection Groups

Jeong-Yup Lee,Dong-Il Lee,Sungsoon Kim

doi:10.3390/sym10100438

Abstract

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra T ( d , n ) of the complex reflection group G ( d , 1 , n ) , inducing the standard monomials expressed by the generators { E i } of T ( d , n ) . This result generalizes the one for the Coxeter group of type B n in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of T ( d , n ) , relating to the fully commutative elements of the complex reflection group G ( d , 1 , n ) . More generally, the Temperley-Lieb algebra T ( d , r , n ) of the complex reflection group G ( d , r , n ) is defined and its dimension is computed.

Highlights

The Temperley-Lieb algebra appears originally in the context of statistical mechanics [1], and later its structure has been studied in connection with knot theory, where it is known to be a quotient of theHecke algebra of type A in [2].Our approach to understanding the structure of Temperley-Lieb algebras is from the noncommutative Gröbner basis theory, or the Gröbner-Shirshov basis theory more precisely, which provides a powerful tool for understanding the structure ofassociative algebras and their representations, especially in computational aspects
The main idea of the Composition-Diamond Lemma is to establish an algorithm for constructing standard monomials of a quotient algebra by a two-sided ideal generated by a set of relations called a Gröbner-Shirshov basis
Our set of standard monomials in this algorithm is a minimal set of monomials which are indivisible by any leading monomial of the polynomials in the Gröbner-Shirshov basis

Summary

Introduction

The Temperley-Lieb algebra appears originally in the context of statistical mechanics [1], and later its structure has been studied in connection with knot theory, where it is known to be a quotient of the. In the first part of this paper, extending the result for type Bn in [16], we construct a Gröbner-Shirshov basis for the Temperley-Lieb algebra T (d, n) of the complex reflection group of type G (d, 1, n) and compute the dimension of T (d, n), by enumerating the standard monomials which are in bijection with the fully commutative elements. Fan [21] and Stembridge [22] studied the fully commutative elements for Coxeter groups and proved that these elements parametrize the bases of the corresponding Temperley-Lieb algebras. Gröbner-Shirshov basis, the combinatorial aspects in Section is that there is a bijection between the standard monomials of having the same number of elements

Preliminaries

Temperley-Lieb Algebra of Type Bn

The generalized