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

Dana C. Ernst

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 so-called monomial basis of the Temperley--Lieb algebra of type affine $C$.

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 . 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 so-called monomial basis of the Temperley--Lieb algebra of type affine .

Paper Structure

This paper contains 20 sections, 25 theorems, 39 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coxeter groups
  4. Fully commutative elements
  5. Heaps
  6. Hecke algebras
  7. Temperley--Lieb algebras
  8. Combinatorics in Coxeter groups of type affine $C$
  9. type I and type II elements
  10. Preparatory lemmas
  11. Weak star reductions and non-cancellable elements
  12. Diagram algebras
  13. Undecorated diagrams
  14. Decorated diagrams
  15. Admissible diagrams
  16. ...and 5 more sections

Key Result

Proposition 2.4

Let $(W,S)$ be a Coxeter system of type $\Gamma$ and let $w \in \mathop{\mathrm{FC}}\nolimits(\Gamma)$. Then $H'$ is a convex subheap of $H(w)$ if and only if $H'$ is the heap for some subword of some reduced expression for $w$. ∎

Figures (27)

  • Figure 1: Coxeter graphs corresponding to Coxeter systems of types $B_{n}$ and $\widetilde{C}_{n}$.
  • Figure 2: Labeled Hasse diagram and possible lattice point representation for the heap of an element from $\mathop{\mathrm{FC}}\nolimits(\widetilde{C}_5)$.
  • Figure 3: Various subheaps of the heap given in Figure \ref{['fig:first heap']}.
  • Figure 4: Impermissible convex subheaps for elements in $\mathop{\mathrm{FC}}\nolimits(\widetilde{C}_n)$.
  • Figure 5: Example of the heap for a type I element.
  • ...and 22 more figures

Theorems & Definitions (57)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Lemma 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Example 3.1
  • Lemma 3.2
  • ...and 47 more