# Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in $\mathbf{a}(2)$-finite Coxeter systems

@inproceedings{Green2021KazhdanLusztigCO, title={Kazhdan--Lusztig cells of \$\mathbf\{a\}\$-value 2 in \$\mathbf\{a\}(2)\$-finite Coxeter systems}, author={R. M. Green and Tianyuan Xu}, year={2021} }

A Coxeter group is said to be a(2)-finite if it has finitely many elements of a-value 2 in the sense of Lusztig. In this paper, we give explicit combinatorial descriptions of the left, right, and two-sided Kazhdan–Lusztig cells of a-value 2 in an irreducible a(2)-finite Coxeter group. In particular, we introduce elements we call stubs to parameterize the one-sided cells and we characterize the one-sided cells via both star operations and weak Bruhat orders. We also compute the cardinalities of… Expand

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SHOWING 1-10 OF 48 REFERENCES

STAR REDUCIBLE COXETER GROUPS

- Mathematics
- Glasgow Mathematical Journal
- 2006

We define “star reducible” Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a… Expand

On the Kazhdan-Lusztig cells in type E8

- Computer Science, Mathematics
- Math. Comput.
- 2015

Algorithmic aspects of this theory for finite W which are important in applications are addressed, e.g., run explicitly through all left cells, determine the values of Lusztig’s a-function, identify the characters of left cell representations. Expand

On the Subregular J-Rings of Coxeter Systems

- Mathematics
- Algebras and Representation Theory
- 2018

We recall Lusztig’s construction of the asymptotic Hecke algebra J of a Coxeter system (W,S) via the Kazhdan–Lusztig basis of the corresponding Hecke algebra. The algebra J has a direct summand JE… Expand

Kazhdan-Lusztig cells in the affine Weyl groups of rank 2

- Mathematics
- 2009

In this paper we determine the partition into Kazhdan-Lusztig cells of the affine Weyl groups of type $\tB_{2}$ and $\tG_{2}$ for any choice of parameters. Using these partitions we show that the… Expand

Simple transitive $2$-representations of small quotients of Soergel bimodules

- Mathematics
- Transactions of the American Mathematical Society
- 2018

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the… Expand

Hecke Algebras With Unequal Parameters

- Mathematics
- 2003

Introduction Coxeter groups Partial order on $W$ The algebra ${\mathcal H}$ The bar operator The elements $c_w$ Left or right multiplication by $c_s$ Dihedral groups Cells Cosets of parabolic… Expand

The Kazhdan-Lusztig cells in certain affine Weyl groups

- Mathematics
- 1986

Coxeter groups, Hecke algebras and their representations.- Applications of Kazhdan-Lusztig theory.- Geometric interpretations of the Kazhdan-Lusztig polynomials.- The algebraic descriptions of the… Expand

Fully commutative elements in finite and affine Coxeter groups

- Mathematics
- 2014

An element of a Coxeter group $$W$$W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were… Expand

Fully commutative elements in the Weyl and affine Weyl groups

- Mathematics
- 2005

Abstract Let W be a Weyl or an affine Weyl group and let W c be the set of fully commutative elements in W. We associate each w ∈ W c to a digraph G ( w ) . By using G ( w ) , we give a… Expand

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

- Mathematics
- 2009

In this paper, we present an infinite dimensional associative diagram algebra that satisfies the relations of the generalized Temperley--Lieb algebra having a basis indexed by the fully commutative… Expand