Indecomposable Soergel bimodules for universal Coxeter groups
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- by Ben Elias and Nicolas Libedinsky; with an appendix by Ben Webster PDF
- Trans. Amer. Math. Soc. 369 (2017), 3883-3910 Request permission
Abstract:
We produce an explicit recursive formula which computes the idempotent projecting to any indecomposable Soergel bimodule for a universal Coxeter system. This gives the exact set of primes for which the positive characteristic analogue of Soergel’s conjecture holds. Along the way, we introduce the multicolored Temperley-Lieb algebra.References
- Alexandre BeÄlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris SĂ©r. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
- Jie Du, Brian Parshall, and Leonard Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Phys. 195 (1998), no. 2, 321–352. MR 1637785, DOI 10.1007/s002200050392
- Matthew Dyer, On some generalisations of the Kazhdan-Lusztig polynomials for “universal” Coxeter systems, J. Algebra 116 (1988), no. 2, 353–371. MR 953157, DOI 10.1016/0021-8693(88)90223-2
- Ben Elias, Dihedral cathedral, Preprint, arXiv:1308.6611.
- Ben Elias, Quantum algebraic geometric satake, Preprint, arXiv:1403.5570.
- Ben Elias and Aaron D. Lauda, Trace decategorification of the Hecke category, J. Algebra 449 (2016), 615–634. MR 3448186, DOI 10.1016/j.jalgebra.2015.11.028
- 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
- Ben Elias and Geordie Williamson, Soergel calculus, Represent. Theory 20 (2016), 295–374. MR 3555156, DOI 10.1090/ert/481
- Peter Fiebig, The combinatorics of Coxeter categories, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4211–4233. MR 2395170, DOI 10.1090/S0002-9947-08-04376-6
- Frederick M Goodman and Hans Wenzl, Ideals in the Temperley Lieb category, 2002, arXiv:math/0206301.
- 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
- Daniel Juteau, Carl Mautner, and Geordie Williamson, Perverse sheaves and modular representation theory, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, Soc. Math. France, Paris, 2012, pp. 315–352 (English, with English and French summaries). MR 3203032
- 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
- Aaron D. Lauda, An introduction to diagrammatic algebra and categorified quantum $\mathfrak {sl}_2$, Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 2, 165–270. MR 3024893
- Helmut Lenzing, http://webusers.imj-prg.fr/~bernhard.keller/ictp2006/lecturenotes/lenzing1. pdf.
- Nicolas Libedinsky, Light leaves and Lusztig’s conjecture, Adv. Math. 280 (2015), 772–807. MR 3350234, DOI 10.1016/j.aim.2015.04.022
- 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
- Scott Morrison, A formula for the Jones-Wenzl projections, Preprint, available at http://tqft.net/math/JonesWenzlProjections.pdf.
- Wolfgang Soergel, The combinatorics of Harish-Chandra bimodules, J. Reine Angew. Math. 429 (1992), 49–74. MR 1173115, DOI 10.1515/crll.1992.429.49
- 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
- H. N. V. Temperley and E. H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A 322 (1971), no. 1549, 251–280. MR 498284, DOI 10.1098/rspa.1971.0067
- Hans Wenzl, On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), no. 1, 5–9. MR 873400
- B. W. Westbury, The representation theory of the Temperley-Lieb algebras, Math. Z. 219 (1995), no. 4, 539–565. MR 1343661, DOI 10.1007/BF02572380
- 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
- Geordie Williamson, Schubert calculus and torsion, Preprint, arXiv:1309.5055.
- Geordie Williamson, Singular Soergel bimodules, Int. Math. Res. Not. IMRN 20 (2011), 4555–4632. MR 2844932, DOI 10.1093/imrn/rnq263
Additional Information
- Ben Elias
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
- MR Author ID: 896756
- Email: belias@uoregon.edu
- Nicolas Libedinsky
- Affiliation: Department of Mathematics, Universidad de Chile, Santiago, Chile
- Email: nlibedinsky@gmail.com
- Ben Webster
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 794563
- Received by editor(s): October 6, 2014
- Received by editor(s) in revised form: April 2, 2015, and May 22, 2015
- Published electronically: December 7, 2016
- Additional Notes: The first author was supported by NSF grant DMS-1103862
The second author was supported by Fondecyt iniciacion 11121118 - © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3883-3910
- MSC (2010): Primary 20C20, 20G40; Secondary 20C33, 20C08
- DOI: https://doi.org/10.1090/tran/6754
- MathSciNet review: 3624396