[Paper Review] A diagrammatic representation of an affine $C$ Temperley--Lieb algebra

Dana C. Ernst|ArXiv.org|May 27, 2009

Advanced Combinatorial Mathematics6 references6 citations

TL;DR

This paper constructs a diagrammatic representation of the affine $C$ Temperley–Lieb algebra as an infinite-rank associative algebra, using a basis indexed by fully commutative elements in the Coxeter group of type $\widetilde{C}_n$. By classifying weak star irreducible elements in $W(B_n)$ and $W(\widetilde{C}_n)$, the author establishes faithfulness of the diagrammatic homomorphism via inductive arguments, proving injectivity despite the absence of finite counting techniques applicable in finite-rank cases.

ABSTRACT

In this thesis, I present an associative diagram algebra that is a faithful representation of a particular Temperley--Lieb algebra of type affine $C$, which has a basis indexed by the fully commutative elements of the Coxeter group of the same type. The Coxeter group of type affine $C$ contains an infinite number of fully commutative elements, and so the corresponding Temperley--Lieb algebra is of infinite rank. With the exception of type affine $A$, all other generalized Temperley--Lieb algebras with known diagrammatic representations are of finite rank. In the finite rank case, counting arguments are employed to prove faithfulness, but these techniques are not available in the affine $C$ case. To prove faithfulness, I classify the fully commutative elements in Coxeter groups of types $B$ and affine $C$ that are irreducible under weak star reductions. The classification of these irreducible elements provides the groundwork for inductive arguments that are used to prove faithfulness. The classification of the weak star irreducible elements of type $B$ also verifies C.K. Fan's unproved claim about about the set of fully commutative elements in a Coxeter group of type $B$ having no generator appearing in the left or right descent set that can be left or right cancelled, respectively. The results of this thesis will be used to construct a trace on the Hecke algebra of type affine $C$, which will then be used to compute leading coefficients of certain Kazhdan--Lusztig polynomials in a non-recursive way.

Motivation & Objective

• To provide a diagrammatic realization of the affine $C$ Temperley–Lieb algebra, which is of infinite rank due to the infinite number of fully commutative elements in $W(\widetilde{C}_n)$.
• To overcome the limitation of standard counting arguments in proving faithfulness for infinite-rank algebras by developing a classification of weak star irreducible elements.
• To verify C.K. Fan’s unproved claim regarding fully commutative elements in $W(B_n)$ with no cancellable generators in left/right descent sets.
• To lay the foundation for constructing a trace on the Hecke algebra of type $\widetilde{C}_n$ to compute Kazhdan–Lusztig polynomials non-recursively.
• To establish that the diagram algebra $\mathbb{D}_n$ forms a basis indexed by admissible diagrams, ensuring a faithful representation of the Temperley–Lieb algebra.

Proposed method

• Classify weak star irreducible elements in $W(B_n)$ and $W(\widetilde{C}_n)$ using combinatorial reductions and heap representations.
• Define a diagram algebra $\mathbb{D}_n$ with decorated pseudo-diagrams, where diagrams are built from simple diagrams via multiplication rules.
• Construct a homomorphism $\theta: \mathrm{TL}(\widetilde{C}_n) \to \mathbb{D}_n$ mapping monomials to admissible diagrams.
• Use inductive arguments based on irreducible elements to prove that distinct monomials map to distinct diagrams, ensuring injectivity.
• Apply Bergman’s Diamond Lemma to verify that the set of admissible diagrams forms a basis for $\mathbb{D}_n$.
• Leverage properties of heap representations and convex subheaps to analyze edge structures and decoration placements in diagrams.

Experimental results

Research questions

• RQ1 How can a faithful diagrammatic representation be constructed for the infinite-rank affine $C$ Temperley–Lieb algebra?
• RQ2 What is the complete classification of weak star irreducible elements in $W(B_n)$ and $W(\widetilde{C}_n)$, and how does it support faithfulness?
• RQ3 Can the unproved claim by C.K. Fan about fully commutative elements in $W(B_n)$ be verified using this classification?
• RQ4 How does the diagram algebra $\mathbb{D}_n$ relate to the monomial basis of $\mathrm{TL}(\widetilde{C}_n)$, and what ensures its linear independence?
• RQ5 What structural properties of diagrams (e.g., edge connectivity, decoration placement) are necessary and sufficient for injectivity of $\theta$?

Key findings

• The diagram algebra $\mathbb{D}_n$ is shown to be a faithful representation of $\mathrm{TL}(\widetilde{C}_n)$, with the homomorphism $\theta$ injective.
• The classification of weak star irreducible elements in $W(B_n)$ confirms C.K. Fan’s claim about the absence of cancellable generators in left/right descent sets.
• The set of admissible diagrams in $\mathbb{D}_n$ forms a basis, and each monomial in $\mathrm{TL}(\widetilde{C}_n)$ maps to a unique admissible diagram.
• The injectivity of $\theta$ is established through inductive reduction arguments based on irreducible elements, avoiding finite counting techniques.
• Diagrams with $\mathbf{a}$-value 0 are undecorated, even if they contain loops, which is explicitly clarified in the revised version.
• The proof shows that no diagram with a single $\blacktriangle$ propagating edge from node 1 to $1'$ can arise from non-type I or II irreducible elements, leading to a contradiction if such a diagram were to appear.