Classification of Coxeter groups with finitely many elements of a-value 2
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 331-364.

We consider Lusztig’s a-function on Coxeter groups (in the equal parameter case) and classify all Coxeter groups with finitely many elements of a-value 2 in terms of Coxeter diagrams.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.95
Classification: 05E10,  20C08
Keywords: Coxeter groups, Hecke algebras, Lusztig’s a-function, fully commutative elements, heaps, star operations
Green, R. M. 1; Xu, Tianyuan 2

1 Department of Mathematics University of Colorado Boulder, Campus Box 395 Boulder, Colorado USA, 80309
2 Department of Mathematics and Statistics Queen’s University Kingston, Ontario Canada, K7L 3N6
@article{ALCO_2020__3_2_331_0,
     author = {Green, R. M. and Xu, Tianyuan},
     title = {Classification of {Coxeter} groups with finitely many elements of <strong>a</strong>-value 2},
     journal = {Algebraic Combinatorics},
     pages = {331--364},
     publisher = {MathOA foundation},
     volume = {3},
     number = {2},
     year = {2020},
     doi = {10.5802/alco.95},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.95/}
}
TY  - JOUR
TI  - Classification of Coxeter groups with finitely many elements of <strong>a</strong>-value 2
JO  - Algebraic Combinatorics
PY  - 2020
DA  - 2020///
SP  - 331
EP  - 364
VL  - 3
IS  - 2
PB  - MathOA foundation
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.95/
UR  - https://doi.org/10.5802/alco.95
DO  - 10.5802/alco.95
LA  - en
ID  - ALCO_2020__3_2_331_0
ER  - 
%0 Journal Article
%T Classification of Coxeter groups with finitely many elements of <strong>a</strong>-value 2
%J Algebraic Combinatorics
%D 2020
%P 331-364
%V 3
%N 2
%I MathOA foundation
%U https://doi.org/10.5802/alco.95
%R 10.5802/alco.95
%G en
%F ALCO_2020__3_2_331_0
Green, R. M.; Xu, Tianyuan. Classification of Coxeter groups with finitely many elements of a-value 2. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 331-364. doi : 10.5802/alco.95. https://alco.centre-mersenne.org/articles/10.5802/alco.95/

[1] Bezrukavnikov, Roman On tensor categories attached to cells in affine Weyl groups, Representation theory of algebraic groups and quantum groups (Adv. Stud. Pure Math.), Volume 40, Math. Soc. Japan, Tokyo, 2004, pp. 69-90 | MR: 2074589 | Zbl: 1078.20044

[2] Bezrukavnikov, Roman; Finkelberg, Michael; Ostrik, Victor On tensor categories attached to cells in affine Weyl groups. III, Isr. J. Math., Volume 170 (2009), pp. 207-234 | Article | MR: 2506324 | Zbl: 1210.20004

[3] Bezrukavnikov, Roman; Ostrik, Victor On tensor categories attached to cells in affine Weyl groups. II, Representation theory of algebraic groups and quantum groups (Adv. Stud. Pure Math.), Volume 40, Math. Soc. Japan, Tokyo, 2004, pp. 101-119 | Article | MR: 2074591 | Zbl: 1078.20045

[4] Biagioli, Riccardo; Jouhet, Frédéric; Nadeau, Philippe Fully commutative elements in finite and affine Coxeter groups, Monatsh. Math., Volume 178 (2015) no. 1, pp. 1-37 | Article | MR: 3384889 | Zbl: 1323.05136

[5] Billey, Sara C.; Jones, Brant C. Embedded factor patterns for Deodhar elements in Kazhdan–Lusztig theory, Ann. Comb., Volume 11 (2007) no. 3-4, pp. 285-333 | Article | MR: 2376108 | Zbl: 1189.20008

[6] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR: 2133266 (2006d:05001) | Zbl: 1110.05001

[7] Elias, Ben; Williamson, Geordie The Hodge theory of Soergel bimodules, Ann. Math. (2), Volume 180 (2014) no. 3, pp. 1089-1136 | Article | MR: 3245013 | Zbl: 1326.20005

[8] Ernst, Dana C. Diagram calculus for a type affine C Temperley-Lieb algebra, II, J. Pure Appl. Algebra, Volume 222 (2018) no. 12, pp. 3795-3830 | Article | MR: 3818281 | Zbl: 06902465

[9] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor Tensor categories, Mathematical Surveys and Monographs, 205, American Mathematical Society, Providence, RI, 2015, xvi+343 pages | MR: 3242743 | Zbl: 1365.18001

[10] Fan, C. Kenneth A Hecke algebra quotient and some combinatorial applications, J. Algebr. Comb., Volume 5 (1996) no. 3, pp. 175-189 | Article | MR: 1394304 | Zbl: 0853.20028

[11] Geck, Meinolf Hecke algebras of finite type are cellular, Invent. Math., Volume 169 (2007) no. 3, pp. 501-517 | Article | MR: 2336039 | Zbl: 1130.20007

[12] Geck, Meinolf; Iancu, Lacrimioara Lusztig’s a-function in type B n in the asymptotic case, Nagoya Math. J., Volume 182 (2006), pp. 199-240 | Article | MR: 2235342 | Zbl: 1173.20301

[13] Green, Richard M. On rank functions for heaps, J. Comb. Theory, Ser. A, Volume 102 (2003) no. 2, pp. 411-424 | Article | MR: 1979543 | Zbl: 1027.06004

[14] Green, Richard M. Star reducible Coxeter groups, Glasg. Math. J., Volume 48 (2006) no. 3, pp. 583-609 | Article | MR: 2271387 | Zbl: 1149.20034

[15] Green, Richard M. Generalized Jones traces and Kazhdan–Lusztig bases, J. Pure Appl. Algebra, Volume 211 (2007) no. 3, pp. 744-772 | Article | MR: 2344227 | Zbl: 1172.20004

[16] Green, Richard M.; Losonczy, Jozsef Fully commutative Kazhdan–Lusztig cells, Ann. Inst. Fourier, Volume 51 (2001) no. 4, pp. 1025-1045 | Article | Numdam | MR: 1849213 | Zbl: 1008.20036

[17] Hart, Sarah How many elements of a Coxeter group have a unique reduced expression?, J. Group Theory, Volume 20 (2017) no. 5, pp. 903-910 | Article | MR: 3692054 | Zbl: 06786542

[18] Kazhdan, David; Lusztig, George Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979) no. 2, pp. 165-184 | Article | MR: 560412 | Zbl: 0499.20035

[19] Lusztig, George Some examples of square integrable representations of semisimple p-adic groups, Trans. Am. Math. Soc., Volume 277 (1983) no. 2, pp. 623-653 | Article | MR: 694380 (84j:22023) | Zbl: 0526.22015

[20] Lusztig, George Cells in affine Weyl groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983) (Adv. Stud. Pure Math.), Volume 6, North-Holland, Amsterdam, 1985, pp. 255-287 | Article | MR: 803338 | Zbl: 0569.20032

[21] Lusztig, George Cells in affine Weyl groups. II, J. Algebra, Volume 109 (1987) no. 2, pp. 536-548 | Article | MR: 902967 | Zbl: 0625.20032

[22] Lusztig, George Hecke algebras with unequal parameters, 2014 (https://arxiv.org/abs/math/0208154) | Zbl: 1051.20003

[23] Shi, Jian-yi Fully commutative elements in the Weyl and affine Weyl groups, J. Algebra, Volume 284 (2005) no. 1, pp. 13-36 | Article | MR: 2115002 | Zbl: 1079.20059

[24] Stembridge, John R. On the fully commutative elements of Coxeter groups, J. Algebr. Comb., Volume 5 (1996) no. 4, pp. 353-385 | Article | MR: 1406459 | Zbl: 0864.20025

[25] Stembridge, John R. The enumeration of fully commutative elements of Coxeter groups, J. Algebr. Comb., Volume 7 (1998) no. 3, pp. 291-320 | Article | MR: 1616016 | Zbl: 0897.05087

[26] Tits, Jacques Le problème des mots dans les groupes de Coxeter, Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, Academic Press, London, 1969, pp. 175-185 | MR: 0254129 | Zbl: 0206.03002

[27] Warrington, Gregory S. Equivalence classes for the μ-coefficient of Kazhdan–Lusztig polynomials in S n , Exp. Math., Volume 20 (2011) no. 4, pp. 457-466 | Article | MR: 2859901 | Zbl: 1263.05118

[28] Xu, Tianyuan On the Subregular J-Rings of Coxeter Systems, Algebr. Represent. Theory (2018), pp. 1-34 | Article | MR: 4034792 | Zbl: 07152729

Cited by Sources: