Soergel calculus
HTML articles powered by AMS MathViewer
- by Ben Elias and Geordie Williamson PDF
- Represent. Theory 20 (2016), 295-374
Abstract:
The monoidal category of Soergel bimodules is an incarnation of the Hecke category, a fundamental object in representation theory. We present this category by generators and relations, using the language of planar diagrammatics. We show that Libedinsky’s light leaves give a basis for morphism spaces and give a new proof and a generalization of Soergel’s classification of the indecomposable Soergel bimodules.References
- Tom Braden and Robert MacPherson, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), no. 3, 533–551. MR 1871967, DOI 10.1007/s002080100232
- Vinay V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata 36 (1990), no. 1, 95–119. MR 1065215, DOI 10.1007/BF00181467
- Ben Elias and Mikhail Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. , posted on (2010), Art. ID 978635, 58. MR 3095655, DOI 10.1155/2010/978635
- B. Elias, A diagrammatic category for generalized Bott-Samelson bimodules and a diagrammatic categorification of induced trivial modules for Hecke algebras, Preprint, arXiv:1009.2120.
- B. Elias, Light ladders and clasp conjectures, Preprint, arXiv:1510.06840.
- B. Elias, Quantum satake in type A: part I, Preprint, arXiv:1403.5570.
- B. Elias, The two-color Soergel calculus, Preprint, arXiv:1308.6611.
- B. Elias and G. Williamson, Diagrammatics for Coxeter groups and their braid groups, Preprint, arXiv:1405.4928.
- Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089–1136. MR 3245013, DOI 10.4007/annals.2014.180.3.6
- Roger A. Fenn, Techniques of geometric topology, London Mathematical Society Lecture Note Series, vol. 57, Cambridge University Press, Cambridge, 1983. MR 787801
- Peter Fiebig, Sheaves on moment graphs and a localization of Verma flags, Adv. Math. 217 (2008), no. 2, 683–712. MR 2370278, DOI 10.1016/j.aim.2007.08.008
- Peter Fiebig, Lusztig’s conjecture as a moment graph problem, Bull. Lond. Math. Soc. 42 (2010), no. 6, 957–972. MR 2740015, DOI 10.1112/blms/bdq058
- Peter Fiebig, Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture, J. Amer. Math. Soc. 24 (2011), no. 1, 133–181. MR 2726602, DOI 10.1090/S0894-0347-2010-00679-0
- James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
- Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236 (1964). MR 165016
- Daniel Juteau, Carl Mautner, and Geordie Williamson, Parity sheaves, J. Amer. Math. Soc. 27 (2014), no. 4, 1169–1212. MR 3230821, DOI 10.1090/S0894-0347-2014-00804-3
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- Aaron D. Lauda, A categorification of quantum $\textrm {sl}(2)$, Adv. Math. 225 (2010), no. 6, 3327–3424. MR 2729010, DOI 10.1016/j.aim.2010.06.003
- Nicolas Libedinsky, Sur la catégorie des bimodules de Soergel, J. Algebra 320 (2008), no. 7, 2675–2694 (French, with French summary). MR 2441994, DOI 10.1016/j.jalgebra.2008.05.027
- Nicolas Libedinsky, Presentation of right-angled Soergel categories by generators and relations, J. Pure Appl. Algebra 214 (2010), no. 12, 2265–2278. MR 2660912, DOI 10.1016/j.jpaa.2010.02.026
- Nicolas Libedinsky, Light leaves and Lusztig’s conjecture, Adv. Math. 280 (2015), 772–807. MR 3350234, DOI 10.1016/j.aim.2015.04.022
- Yu. I. Manin and V. V. Schechtman, Arrangements of hyperplanes, higher braid groups and higher Bruhat orders, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 289–308. MR 1097620, DOI 10.2969/aspm/01710289
- Mark Ronan, Lectures on buildings, University of Chicago Press, Chicago, IL, 2009. Updated and revised. MR 2560094
- S. Riche and G. Williamson, Tilting modules and the $p$-canonical basis, Preprint, arXiv:1512.08296.
- Wolfgang Soergel, Kategorie $\scr O$, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), no. 2, 421–445 (German, with English summary). MR 1029692, DOI 10.1090/S0894-0347-1990-1029692-5
- Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, DOI 10.1090/S1088-4165-97-00021-6
- Wolfgang Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (2000), no. 1-3, 311–335. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1784005, DOI 10.1016/S0022-4049(99)00138-3
- Wolfgang Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen, J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525 (German, with English and German summaries). MR 2329762, DOI 10.1017/S1474748007000023
- T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4
- Bruce W. Westbury, Invariant tensors and cellular categories, J. Algebra 321 (2009), no. 11, 3563–3567. MR 2510062, DOI 10.1016/j.jalgebra.2008.07.004
- G. Williamson, Some examples of parity sheaves, Oberwolfach reports, 5 pages.
Additional Information
- Ben Elias
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 896756
- Email: belias@uoregon.edu
- Geordie Williamson
- Affiliation: Max-Planck-Institut für Mathematik, 53111 Bonn, Germany
- MR Author ID: 845262
- Email: geordie@mpim-bonn.mpg.de
- Received by editor(s): January 16, 2015
- Received by editor(s) in revised form: March 24, 2016
- Published electronically: October 7, 2016
- Additional Notes: The first-named author was supported by NSF Postdoctoral Fellowship DMS-1103862
- © Copyright 2016 by the authors
- Journal: Represent. Theory 20 (2016), 295-374
- MSC (2010): Primary 20C33, 20F55, 20G05; Secondary 22E46
- DOI: https://doi.org/10.1090/ert/481
- MathSciNet review: 3555156
Dedicated: To Mikhail Khovanov and Raphaël Rouquier, who taught us generators and relations