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Two-color Soergel Calculus and Simple Transitive 2-representations

Part of: Special aspects of infinite or finite groups Categories with structure Representation theory of groups Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 January 2019

Marco Mackaaij  and
Daniel Tubbenhauer
Marco Mackaaij
Affiliation:
Center for Mathematical Analysis, Geometry, and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal Email: mmackaay@ualg.pt
Daniel Tubbenhauer
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasses 190, Campus Irchel, Office Y27J32, CH-8057 Zürich, Switzerland Email: daniel.tubbenhauer@math.uzh.ch URL: www.dttubbenhauer.com
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Abstract

In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations.

Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.

Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.

Keywords

2-representation theorycategorificationSoergel bimoduleKazhdan–Lusztig theoryHecke algebras for dihedral groupszigzag algebra

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 17B10: Representations, algebraic theory (weights) 18D05: Double categories, $2$-categories, bicategories and generalizations 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 20F55: Reflection and Coxeter groups
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 6 , December 2019 , pp. 1523 - 1566
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

Author M. M. is partially supported by FCT/Portugal through the project UID/MAT/04459/2013

References

Andersen, H. H., The strong linkage principle for quantum groups at roots of 1 . J. Algebra 260(2003), 2–15. Special issue celebrating the 80th birthday of Robert Steinberg. https://doi.org/10.1016/S0021-8693(02)00618-X.Google Scholar
Andersen, H. H. and Tubbenhauer, D., Diagram categories for U q-tilting modules at roots of unity . Transform. Groups 22(2017), 29–89 https://doi.org/10.1007/s00031-016-9363-z.Google Scholar
Bénabou, J., Introduction to bicategories. In: Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77.Google Scholar
Brouwer, A. E. and Haemers, W. H., Spectra of graphs. Universitext, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-1939-6.Google Scholar
Borceux, F., Handbook of categorical algebra. 1. Encyclopedia of Mathematics and its Applications, 50, Cambridge University Press, Cambridge, 1994.Google Scholar
Elias, B., The two-color Soergel calculus Compos. Math. 152(2016), 327–398. https://doi.org/10.1112/S0010437X15007587.Google Scholar
Elias, B., Quantum Satake in type A. Part I . J. Comb. Algebra 1(2017), 63–125. https://doi.org/10.4171/JCA/1-1-4.Google Scholar
Gelfand, S. I. and Manin, Y. I., Methods of homological algebra. Second ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. https://doi.org/10.1007/978-3-662-12492-5.Google Scholar
Gadbled, A., Thiel, A.-L., and Wagner, E., Categorical action of the extended braid group of affine type A . Commun. Contemp. Math. 19(2017),. 1650024, 39. https://doi.org/10.1142/S0219199716500243.Google Scholar
Huerfano, R. S. and Khovanov, M., A category for the adjoint representation . J. Algebra 246(2001), 514–542. https://doi.org/10.1006/jabr.2001.8962.Google Scholar
Khovanov, M., Categorifications of the colored Jones polynomial . J. Knot Theory Ramifications 14(2005), 111–130. https://doi.org/10.1142/S0218216505003750.Google Scholar
Kirillov, A. Jr. and Ostrik, V., On a q-analogue of the McKay correspondence and the ADE classification of sl 2 conformal field theories . Adv. Math. 171(2002), 183–227. https://doi.org/10.1006/aima.2002.2072.Google Scholar
Kildetoft, T., Mackaay, M., Mazorchuk, V., and Zimmermann, J., Simple transitive 2-representations for some 2-categories of Soergel bimodules . J. Pure Appl. Algebra 221(2017), no. 3, 565–587. https://doi.org/10.1016/j.jpaa.2016.07.006.Google Scholar
Khovanov, M. and Seidel, P., Quivers, Floer cohomology, and braid group actions . J. Amer. Math. Soc. 15(2002), 203–271. https://doi.org/10.1090/S0894-0347-01-00374-5.Google Scholar
Leinster, T., Basic bicategories. 1998. arxiv:math/9810017.Google Scholar
Lusztig, G., Some examples of square integrable representations of semisimple p-adic groups . Trans. Amer. Math. Soc. 277(1983), 623–653. https://doi.org/10.2307/1999228.Google Scholar
Mac Lane, S., Categories for the working mathematician. Graduate Texts in Mathematics, 5, Springer-Verlag, New York, 1998.Google Scholar
Mazorchuk, V. and Miemietz, V., Cell 2-representations of finitary 2-categories . Compos. Math. 147(2011), 1519–1545. https://doi.org/10.1112/S0010437X11005586.Google Scholar
Mazorchuk, V. and Miemietz, V., Endomorphisms of cell 2-representations . Int. Math. Res. Not. IMRN 24(2016), 7471–7498. https://doi.org/10.1093/imrn/rnw025.Google Scholar
Mazorchuk, V. and Miemietz, V., Transitive 2-representations of finitary 2-categories . Trans. Amer. Math. Soc. 368(2016), 7623–7644. https://doi.org/10.1090/tran/6583.Google Scholar
Mackaay, M. and Mazorchuk, V., Simple transitive 2-representations for some 2-subcategories of Soergel bimodules . J. Pure Appl. Algebra 221(2017), 565–587. https://doi.org/10.1016/j.jpaa.2016.07.006.Google Scholar
Mackaay, M., Mazorchuk, V., Miemietz, V., and Tubbenhauer, D., Simple transitive 2-representations via (co)algebra 1-morphisms. Indiana Univ. Math. J., to appear, 2016. arxiv:1612.06325.Google Scholar
Mackaay, M., Mazorchuk, V., Miemietz, V., and Tubbenhauer, D., Trihedral Soergel bimodules. 2018. arxiv:1804.08920.Google Scholar
Ostrik, V., Module categories, weak Hopf algebras and modular invariants . Transform. Groups 8(2003), 177–206. https://doi.org/10.1007/s00031-003-0515-6.Google Scholar
Power, A. J., A general coherence result . J. Pure Appl. Algebra 57(1989), 165–173. https://doi.org/10.1016/0022-4049(89)90113-8.Google Scholar
Sawin, S. F., Quantum groups at roots of unity and modularity . J. Knot Theory Ramifications 15(2006), 1245–1277. https://doi.org/10.1142/S0218216506005160.Google Scholar
Schwenk, A. J., Almost all trees are cospectral. In: New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), Academic Press, New York, 1973, pp. 275–307.Google Scholar
Smith, J. H., Some properties of the spectrum of a graph. In: Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969), Gordon and Breach, New York, 1970, pp. 403–406.Google Scholar
Weibel, C. A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. https://doi.org/10.1017/CBO9781139644136.Google Scholar
Wenzl, H., On sequences of projections . C. R. Math. Rep. Acad. Sci. Canada 9(1987), 5–9.Google Scholar
Zimmermann, J., Simple transitive 2-representations of Soergel bimodules in type B 2 . J. Pure Appl. Algebra 221(2017), 666–690. https://doi.org/10.1016/j.jpaa.2016.07.011.Google Scholar