Jones--Wenzl projections and Dyck tilings: type $A$ and $B$

Abstract

The Jones--Wenzl projections are a special class of elements of the Temperley--Lieb algebra. We prove that the coefficient appearing in the Jones--Wenzl projection is given by a generating function of combinatorial objects, called Dyck tilings. We also show that this correspondence holds for type $B$ case.

Figures (5)

• Figure 2.1: A Temperley--Lieb diagram with six strands
• Figure 3.1: Cover-inclusive Dyck tilings
• Figure 3.5: Bijection between a Temperley--Lieb diagram and a Dyck path
• Figure 3.7: An example of the enlargement and addition of unit boxes.
• Figure 4.5: Diagrams $g_{4,i}$, $0\le i\le 4$, for type $B$.

Theorems & Definitions (28)

• Example 2.2
• Proposition 2.3: Wen87
• Definition 2.4
• Lemma 2.5: Proposition 3.3 in Mor17
• Proposition 2.6: Proposition 4.1 in Mor17
• Example 2.7
• Definition 3.2
• Example 3.3
• Example 3.4
• Theorem 3.6