##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Tianyuan Xu

#### University of Colorado

## On elements of a-value 2 in Coxeter groups

##### Abstract:

The a-function on a Coxeter group W is a function a : W $\rightarrow$ N defined by Lusztig which is intimately related to the partition of W into Kazhdan--Lusztig cells and to the representation theory of the Hecke algebra of W. It is known that the identity element of W is the only element with a-value 0, while a non-identity element has a-value 1 if and only if it has a unique reduced word. However, as its definition relies on the Kazhdan--Lusztig basis of the Hecke algebra, thea-function is often difficult to compute for general elements. In this talk we will focus on elements of a-value 2, or a-2 elements. We show that a-2 elements arefully commutative in the sense of Stembridge, which allows us to associate to them certain posets called heaps and, in many cases, certain generalized Temperley--Lieb diagrams. Using heaps and Temperley--Lieb diagrams, we conjecture a combinatorial characterization of a-2 elements, classify all Coxeter groups with finitely many a-2 elements, and enumerate a-2 elements for all groups from the classification. Joint with Richard Green.

Host: Brendon Rhoades

### January 28, 2020

### 1:00 PM

### AP&M 7321

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