On the Structure of nil-Temperley-Lieb Algebras of type A

TL;DR

This paper explores the structure of nil-Temperley-Lieb algebras of type A, revealing their dimensions match Catalan numbers and establishing a combinatorial correspondence with Dyck paths.

Contribution

It provides a detailed description of monomials in these algebras and connects their structure to well-known combinatorial objects, offering new insights into their properties.

Findings

Algebras' dimensions are Catalan numbers.

Monomials correspond bijectively to Dyck paths.

Distribution of monomials by degree matches Dyck path statistics.

Abstract

We investigate nil-Temperley-Lieb algebras of type A. We give a general description of the structure of monomials formed by the generators. We also show that the dimensions of these algebras are the famous Catalan numbers by providing a bijection between the monomials and Dyck paths. We show that the distribution of these monomials by degree is the same as the distribution of Dyck paths by the sum of the heights of the peaks minus the number of peaks.