On Kazhdan-Lusztig cells of a-value 2

Abstract: Lusztig's a-function on a Coxeter group $W$ is a function that takes

a constant nonnegative integer value on each Kazhdan--Lusztig cell of $W$. The function plays a key role in the study of the cells of $W$ and the cell representations of the Hecke algebra of $W$, but it is also very difficult to compute. For an element $w \in W$, it is known that $a(w)=0$ if and only if $W$ is the identity and $a(w)=1$ if and only if $w$ has a unique nonempty reduced word; moreover, the elements of $a$-values $0$ and $1$ each form a single two-sided cell. We report on some extensions of these results about the set $W_2$ of elements with $a$-value $2$. In particular, we classify Coxeter groups $W$ where $W_2$ is finite, count $W_2$ and describe the partition of $W_2$ into cells when $W_2$ is finite, and discuss a connection between cell representations associated to $W_2$ and the reflection representation of $W$ in types ADE. (Joint work with Richard Green.)

combinatoricscategory theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

( slides )

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Sherbrooke Meeting on Representation Theory of Algebras, Corona Edition (fully online)

Series comments: Please contact Thomas Brüstle or Juan Carlos Bustamante if you are interested to participate.

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