Diagram calculus for a type affine C Temperley–Lieb algebra, II

Abstract

In a previous paper, we presented an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley-Lieb algebra having a basis indexed by the fully commutative elements of the Coxeter group of type affine C. We also provided an explicit description of a basis for the diagram algebra. In this paper, we show that the diagrammatic representation is faithful and establish a correspondence between the basis diagrams and the socalled monomial basis of the Temperley-Lieb algebra of type affine C.

Type $\tilde{C}$ Temperley-Lieb algebra quotients and Catalan combinatorics

“…Remark 2.5. In the literature (for example [Ern12]) we see that T L Cn (q, Q) was defined in the equal parameters case, i.e., when q = Q. Sometimes it was defined with a weight function (for example [Gra95]).…”

Classification of Coxeter groups with finitely many elements of $\protect \mathbf{a}$-value 2

“…Indeed, thanks to a result of Stembridge's in [20], W contains finitely many fully commutative elements if G is any other graph from Figure 1, so W must be a(2)-finite in these cases. It will be easy to show that W is a(2)-finite when G = I 2 (∞), and the case G =C n will also be easy thanks to a result of Ernst from [7] on the Temperley-Lieb algebra of typeC n , therefore the only case requiring more work is G = E q,r where min(q, r) ≥ 3. We will prove W is a(2)-finite in this case via a series of lemmas in Section 4.2, using arguments that involve heaps.…”

Exotic Springer fibers for orbits corresponding to one-row bipartitions

“…Two-boundary cup diagrams appear naturally in the representation theory of two-boundary Temperley-Lieb algebras, [dGN09]. These algebras admit a neat diagrammatic description and can be realized as quotients of multi-parameter affine Hecke algebras of type C, see [Ern12], [Ern18]. In light of [Kat09], the latter fact explains the appearance of the two-boundary cup diagrams when studying exotic Springer fibers.…”