The center of the affine nilTemperley–Lieb algebra

Abstract

We give a description of the center of the affine nilTemperley-Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley-Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley-Lieb algebra on N generators into the affine nilTemperley-Lieb algebra on N + 1 generators.

On the length of fully commutative elements

“…Now let e be an edge of Γ with m e < +∞. We distinguish the same sub-cases as we did in Case (2) in the definition of δ:…”

“…By hypothesis, we have an injective morphism from Γ 1 to Γ 2 so we can safely assume that S 1 ⊆ S 2 and that the morphism is the identity on Γ 1 . Notice that Γ 1 ≤ Γ 2 then means precisely that m (1) st ≤ m (2) st for all s, t ∈ S 1 . Let w be any element in W F C 1,l .…”

A Plactic Algebra Action on Bosonic Particle Configurations

“…This action is known to factor through the nilTemperley-Lieb algebra which acts faithfully on the linear span of fermionic particle configurations [11,Proposition 9.1], [5,Example 2.4], [1, Proposition 2.4.1], [2]. In [3] the case of affine fermionic particle configurations was studied, including a description of a normal form for monomials in the affine nilTemperley-Lieb algebra and its center. Motivated by these results we study the representation of the plactic algebra and the local plactic algebra on bosonic particle configurations more closely in this paper.…”

With Wronskian through the Looking Glass

“…The reader should compare this definition with operators defining a representation of the nil-Temperley-Lieb algebra from [2, equation (2.4.6)], cf. also[1] and references therein.…”

“…, and the matrix of W N with respect to the above two lexicographic coordinate systems is triangular with 1's on the diagonal.Corollary 3.10. The map W (1.1) is an embedding.Indeed, X is a union of open Schubert cells, the map W is GL n -equivariant, and each open cell may be transfered to N using an appropriate g ∈ GL n 1. …”