Geodesic
Kazhdan–Lusztig cells of a-value 2 in a(2)-finite Coxeter systems
Green, R. M.1 ; Xu, Tianyuan2
1 Department of Mathematics University of Colorado Boulder Boulder, CO, USA
2 Department of Mathematics and Statistics Haverford College Haverford, PA, USA
Algebraic Combinatorics, Tome 6 (2023) no. 3, pp. 727-772

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A Coxeter group is said to be a(2)-finite if it has finitely many elements of a-value (in the sense of Lusztig) equal to 2. 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 all the one-sided and two-sided cells of a-value 2 in irreducible a(2)-finite Coxeter groups.

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DOI : 10.5802/alco.275
Classification : 20F55, 20C08
Keywords: Kazhdan–Lusztig cells, Lusztig’s $\mathbf{a}$-function, Coxeter groups, fully commutative elements, heaps, star operations

Green, R. M. 1 ; Xu, Tianyuan 2

1 Department of Mathematics University of Colorado Boulder Boulder, CO, USA
2 Department of Mathematics and Statistics Haverford College Haverford, PA, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     author = {Green, R. M. and Xu, Tianyuan},
     title = {Kazhdan{\textendash}Lusztig cells of $\protect \mathbf{a}$-value 2 in $\protect \mathbf{a}(2)$-finite {Coxeter} systems},
     journal = {Algebraic Combinatorics},
     pages = {727--772},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {3},
     year = {2023},
     doi = {10.5802/alco.275},
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%J Algebraic Combinatorics
%D 2023
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Green, R. M.; Xu, Tianyuan. Kazhdan–Lusztig cells of $\protect \mathbf{a}$-value 2 in $\protect \mathbf{a}(2)$-finite Coxeter systems. Algebraic Combinatorics, Tome 6 (2023) no. 3, pp. 727-772. doi: 10.5802/alco.275

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