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On the subregular \(J\)-rings of Coxeter systems. (English) Zbl 1439.20041
Summary: 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 \(J_E\) for each two-sided Kazhdan-Lusztig cell of \(W\), and we study the summand \(J_C\) corresponding to a particular cell \(C\) called the subregular cell. We develop a combinatorial method involving truncated Clebsch-Gordan rules to compute \(J_C\) without using the Kazhdan-Lusztig basis. As applications, we deduce some connections between \(J_C\) and the Coxeter diagram of \(W\), and we show that for certain Coxeter systems \(J_C\) contains subalgebras that are free fusion rings in the sense of T. Banica and R. Vergnioux [J. Noncommut. Geom. 3, No. 3, 327–359 (2009; Zbl 1203.46048)], thereby connecting the subalgebras to compact quantum groups arising from operator algebra theory.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
20C08 Hecke algebras and their representations
GitHub; SageMath
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[1] Aho, A.V., Hopcroft, J.E.: The Design and Analysis of Computer Algorithms, 1st edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1974) · Zbl 0326.68005
[2] Bakalov, B., Kirillov, A. Jr.: Lectures on Tensor Categories and Modular Functors, University Lecture Series, vol. 21. American Mathematical Society, Providence (2001) · Zbl 0965.18002
[3] Banica, T.; Speicher, R., Liberation of orthogonal Lie groups, Adv. Math., 222, 1461-1501 (2009) · Zbl 1247.46064
[4] Banica, T.; Vergnioux, R., Fusion rules for quantum reflection groups, J. Noncommut. Geom., 3, 327-359 (2009) · Zbl 1203.46048
[5] Bezrukavnikov, R.: On tensor categories attached to cells in affine Weyl groups. In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, pp 69-90. Math. Soc., Japan (2004) · Zbl 1078.20044
[6] Bezrukavnikov, R.; Finkelberg, M.; Ostrik, V., On tensor categories attached to cells in affine Weyl groups. III, Israel J. Math., 170, 207-234 (2009) · Zbl 1210.20004
[7] Bezrukavnikov, R., Ostrik, V.: On tensor categories attached to cells in affine Weyl groups. II. In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, pp 101-1-19. Math. Soc., Japan (2004) · Zbl 1078.20045
[8] Björner, A., Brenti, F.: Combinatorics of Coxeter groups Graduate Texts in Mathematics, vol. 231. Springer, New York (2005) · Zbl 1110.05001
[9] Curtis, C.: The Hecke algebra of a finite Coxeter group. In: The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., vol. 47, pp 51-60. Amer. Math. Soc., Providence (1987)
[10] Curtis, CW; Iwahori, N.; Kilmoyer, R., Hecke algebras and characters of parabolic type of finite groups with (B, N)-pairs, Inst. Hautes Études Sci. Publ. Math., 40, 81-116 (1971) · Zbl 0254.20004
[11] Developers, T.S.: SageMath, the Sage Mathematics Software System (Version 7.2.). http://www.sagemath.org (2016)
[12] Dipper, R.; James, G., Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3), 52, 20-52 (1986) · Zbl 0587.20007
[13] Elias, B.; Williamson, G., The Hodge theory of Soergel bimodules, Ann. Math. (2), 180, 1089-1136 (2014) · Zbl 1326.20005
[14] Etingof, P., Gelaki, S., Nikshych, D., Ostrik, V.: Tensor Categories Mathematical Surveys and Monographs, vol. 205. American Mathematical Society, Providence (2015) · Zbl 1365.18001
[15] Etingof, P.; Khovanov, M., Representations of tensor categories and Dynkin diagrams, Internat. Math. Res. Notices, 5, 235-247 (1995) · Zbl 0854.17016
[16] Freslon, A., Fusion (semi)rings arising from quantum groups, J. Algebra, 417, 161-197 (2014) · Zbl 1354.46066
[17] Geck, M., Kazhdan-Lusztig cells and decomposition numbers, Represent. Theory, 2, 264-277 (1998) · Zbl 0901.20004
[18] Geck, M., Pfeiffer, G.: Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras, London Mathematical Society Monographs New Series, vol. 21. The Clarendon Press, Oxford University Press, New York (2000) · Zbl 0996.20004
[19] Kassel, C.: Quantum Groups Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
[20] Kazhdan, D.; Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184 (1979) · Zbl 0499.20035
[21] Lusztig, G., Some examples of square integrable representations of semisimple p-adic groups, Trans. Amer. Math. Soc., 277, 623-653 (1983) · Zbl 0526.22015
[22] Lusztig, G.: Characters of Reductive Groups over a Finite Field, Annals of Mathematics Studies, vol. 107. Princeton University Press, Princeton (1984) · Zbl 0556.20033
[23] Lusztig, G., Cells in affine Weyl groups. II, J. Algebra, 109, 536-548 (1987) · Zbl 0625.20032
[24] Lusztig, G., Cells in affine Weyl groups. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34, 223-243 (1987) · Zbl 0631.20028
[25] Lusztig, G.: Leading coefficients of character values of Hecke algebras The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., vol. 47, pp 235-262. Amer. Math. Soc., Providence (1987)
[26] Lusztig, G., Cells in affine Weyl groups. IV, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36, 297-328 (1989) · Zbl 0688.20020
[27] Lusztig, G., Cells in affine Weyl groups and tensor categories, Adv. Math., 129, 85-98 (1997) · Zbl 0884.20026
[28] Lusztig, G.: Hecke algebras with unequal parameters. arXiv:math/0208154 (2014) · Zbl 1051.20003
[29] Moore, G.; Seiberg, N., Classical and quantum conformal field theory, Comm. Math. Phys., 123, 177-254 (1989) · Zbl 0694.53074
[30] Raum, S., Isomorphisms and fusion rules of orthogonal free quantum groups and their free complexifications, Proc. Amer. Math. Soc., 140, 3207-3218 (2012) · Zbl 1279.46052
[31] Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds, de Gruyter Studies in Mathematics, revised, vol. 18. Walter de Gruyter & Co., Berlin (2010) · Zbl 1213.57002
[32] Xu, T.: Sage code for Hecke and asymptotic Hecke algebras. https://github.com/TianyuanXu/mysagecode (2017)
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