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

Green, R. M.; Xu, Tianyuan

doi:10.5802/alco.275

Kazhdan–Lusztig cells of

a

-value 2 in

a (2)

-finite Coxeter systems

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

Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 727-772.

Abstract

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.

Received: 2022-03-01
Revised: 2022-09-22
Accepted: 2022-11-14
Published online: 2023-06-19

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

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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},
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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, Volume 6 (2023) no. 3, pp. 727-772. doi : 10.5802/alco.275. https://alco.centre-mersenne.org/articles/10.5802/alco.275/

References
Cited by

[1] Alvis, D. Subrings of the asymptotic Hecke algebra of type $H_{4}$ , Experiment. Math., Volume 17 (2008) no. 3, pp. 375-383 | DOI | MR | Zbl

[2] Ariki, S. Robinson–Schensted correspondence and left cells, Combinatorial methods in representation theory (Kyoto, 1998) (Adv. Stud. Pure Math.), Volume 28, Kinokuniya, Tokyo, 2000, pp. 1-20 | DOI | MR | Zbl

[3] Barbasch, D.; Vogan, D. Primitive ideals and orbital integrals in complex classical groups, Math. Ann., Volume 259 (1982) no. 2, pp. 153-199 | DOI | MR | Zbl

[4] Bezrukavnikov, R.; Ostrik, V. 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 | DOI | MR | Zbl

[5] Björner, A.; Brenti, F. Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR

[6] Bonnafé, C. On Kazhdan–Lusztig cells in type $B$ , J. Algebraic Combin., Volume 31 (2010) no. 1, pp. 53-82 | DOI | MR | Zbl

[7] Bonnafé, C. Kazhdan–Lusztig cells with unequal parameters, Algebra and Applications, 24, Springer, Cham, 2017, xxv+348 pages | DOI | MR

[8] Bremke, K.; Fan, C. K. Comparison of $a$ -functions, J. Algebra, Volume 203 (1998) no. 2, pp. 355-360 | DOI | MR | Zbl

[9] Chen, Y. Left cells in the Weyl group of type $E_{8}$ , J. Algebra, Volume 231 (2000) no. 2, pp. 805-830 | DOI | MR

[10] Chmutov, M.; Pylyavskyy, P.; Yudovina, E. Matrix-ball construction of affine Robinson–Schensted correspondence, Selecta Math. (N.S.), Volume 24 (2018) no. 2, pp. 667-750 | DOI | MR | Zbl

[11] Ernst, D. C. Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II, J. Pure Appl. Algebra, Volume 222 (2018) no. 12, pp. 3795-3830 | DOI | MR | Zbl

[12] Fan, C. K. Structure of a Hecke algebra quotient, J. Amer. Math. Soc., Volume 10 (1997) no. 1, pp. 139-167 | DOI | MR | Zbl

[13] Garfinkle, D. On the classification of primitive ideals for complex classical Lie algebras. III, Compositio Math., Volume 88 (1993) no. 2, pp. 187-234 | Numdam | MR | Zbl

[14] Geck, M.; Halls, A. On the Kazhdan–Lusztig cells in type $E_{8}$ , Math. Comp., Volume 84 (2015) no. 296, pp. 3029-3049 | DOI | MR | Zbl

[15] Green, R. M. Decorated tangles and canonical bases, J. Algebra, Volume 246 (2001) no. 2, pp. 594-628 | DOI | MR | Zbl

[16] Green, R. M. Star reducible Coxeter groups, Glasg. Math. J., Volume 48 (2006) no. 3, pp. 583-609 | DOI | MR | Zbl

[17] Green, R. M.; Losonczy, J. Canonical bases for Hecke algebra quotients, Math. Res. Lett., Volume 6 (1999) no. 2, pp. 213-222 | DOI | MR | Zbl

[18] Green, R. M.; Losonczy, J. Fully commutative Kazhdan–Lusztig cells, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 4, pp. 1025-1045 | DOI | Numdam | MR | Zbl

[19] Green, R. M.; Xu, T. Classification of Coxeter groups with finitely many elements of $a$ -value 2, Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 331-364 | DOI | Numdam | MR | Zbl

[20] Guilhot, J. Kazhdan–Lusztig cells in affine Weyl groups of rank 2, Int. Math. Res. Not. IMRN (2010) no. 17, pp. 3422-3462 | DOI | MR | Zbl

[21] Hart, S. How many elements of a Coxeter group have a unique reduced expression?, J. Group Theory, Volume 20 (2017) no. 5, pp. 903-910 | DOI | MR | Zbl

[22] Kazhdan, D.; Lusztig, G. Representations of Coxeter groups and Hecke algebras, Invent. Math., Volume 53 (1979) no. 2, pp. 165-184 | DOI | MR | Zbl

[23] Kildetoft, T.; Mackaay, M.; Mazorchuk, V.; Zimmermann, J. Simple transitive $2$ -representations of small quotients of Soergel bimodules, Trans. Amer. Math. Soc., Volume 371 (2019) no. 8, pp. 5551-5590 | DOI | MR | Zbl

[24] Ko, H.; Mazorchuk, V. $2$ -representations of small quotients of Soergel bimodules in infinite types, Proc. Amer. Math. Soc., Volume 151 (2023) no. 6, pp. 2277-2290 | DOI | MR | Zbl

[25] Lusztig, G. Some examples of square integrable representations of semisimple $p$ -adic groups, Trans. Amer. Math. Soc., Volume 277 (1983) no. 2, pp. 623-653 | DOI | MR | Zbl

[26] Lusztig, G. Characters of reductive groups over a finite field, Annals of Mathematics Studies, 107, Princeton University Press, Princeton, NJ, 1984, xxi+384 pages | DOI | MR

[27] Lusztig, G. 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 | DOI | MR | Zbl

[28] Lusztig, G. Cells in affine Weyl groups. II, J. Algebra, Volume 109 (1987) no. 2, pp. 536-548 | DOI | MR | Zbl

[29] Lusztig, G. Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 36 (1989) no. 2, pp. 297-328 | MR | Zbl

[30] Lusztig, G. Hecke algebras with unequal parameters (2014) | arXiv

[31] Mackaay, M.; Mazorchuk, V.; Miemietz, V.; Tubbenhauer, D.; Zhang, X. Simple transitive 2-representations of Soergel bimodules for finite Coxeter types, Proc. Lond. Math. Soc. (2023) | DOI

[32] Shi, J. The Kazhdan–Lusztig cells in certain affine Weyl groups, Lecture Notes in Mathematics, 1179, Springer-Verlag, Berlin, 1986, x+307 pages | DOI | MR

[33] Shi, J. Left cells in the affine Weyl group of type ${\tilde{C}}_{4}$ , J. Algebra, Volume 202 (1998) no. 2, pp. 745-776 | DOI | MR

[34] Shi, J. Fully commutative elements in the Weyl and affine Weyl groups, J. Algebra, Volume 284 (2005) no. 1, pp. 13-36 | DOI | MR | Zbl

[35] Stembridge, J. R. On the fully commutative elements of Coxeter groups, J. Algebraic Combin., Volume 5 (1996) no. 4, pp. 353-385 | DOI | MR | Zbl

[36] Takahashi, K. The left cells and their $W$ -graphs of Weyl group of type $F_{4}$ , Tokyo J. Math., Volume 13 (1990) no. 2, pp. 327-340 | DOI | MR | Zbl

[37] Xi, N. Lusztig’s $a$ -function for Coxeter groups with complete graphs, Bull. Inst. Math. Acad. Sin. (N.S.), Volume 7 (2012) no. 1, pp. 71-90 | MR | Zbl

[38] Xie, X. Conjectures P1–P15 for Coxeter groups with complete graph, Adv. Math., Volume 379 (2021), p. Paper No. 107565, 62 | DOI | MR | Zbl

[39] Xu, T. On the subregular $J$ -rings of Coxeter systems, Algebr. Represent. Theory, Volume 22 (2019) no. 6, pp. 1479-1512 | DOI | MR | Zbl

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