On monoidal Koszul duality for the Hecke category

Shotaro Makisumi

doi:10.15446/recolma.v53nsupl.84084

Publicado

2019-12-11

On monoidal Koszul duality for the Hecke category

DOI:

https://doi.org/10.15446/recolma.v53nsupl.84084

Palabras clave:

Hecke algebra, Hecke category, Soergel bimodules, Koszul duality (en)
Álgebra de Hecke, categoría de Hecke, bimódulos de Soergel, dualidad de Koszul (es)

Descargas

PDF

Autores/as

Shotaro Makisumi Columbia University

Resumen (en)
Resumen (es)

We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work [1, 2] of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) SL₂ using Soergel bimodules, a concrete algebraic model of the Hecke category.

Procuramos dar una amable, si bien ahistórica, introducción a los fenómenos de dualidad de Koszul en la categoría de Hecke, con foco en la forma de esta dualidad estudiada en los trabajos [1, 2] en colaboración con Achar, Riche y Williamson. Ilustramos algunos fenómenos y construcciones claves en el ejemplo no trivial más simple, SL₂ finito, usando bimódulos de Soergel, un modelo algebraico concreto de la categoría de Hecke.

Referencias

Pramod N. Achar, Shotaro Makisumi, Simon Riche, and Geordie Williamson, Free-monodromic mixed tilting sheaves on flag varieties, preprint arXiv:1703.05843.

Pramod N. Achar, Shotaro Makisumi, Simon Riche, and Geordie Williamson, Koszul duality for Kac-Moody groups and characters of tilting modules, J. Amer. Math. Soc. 32 (2019), 261-310.

Pramod N. Achar and Simon Riche, Dualité de Koszul formelle et théorie des représentations des groupes algébriques réductifs en caractéristique positive, preprint arXiv:1703.05843.

Pramod N. Achar and Simon Riche, Koszul duality and semisimplicity of Frobenius, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 4, 1511-1612. MR 3137361

Pramod N. Achar and Simon Riche, Modular perverse sheaves on flag varieties, II: Koszul duality and formality, Duke Math. J. 165 (2016), no. 1, 161-215. MR 3450745

Pramod N. Achar and Simon Riche, Reductive groups, the loop Grassmannian, and the Springer resolution, Invent. Math. 214 (2018), no. 1, 289-436. MR 3858401

Pramod N. Achar, Simon Riche, and Cristian Vay, Mixed perverse sheaves on flag varieties of Coxeter groups, preprint arXiv:1802.07651, to appear in Canad. J. Math.

Pramod N. Achar and Laura Rider, The affine Grassmannian and the Springer resolution in positive characteristic, Compos. Math. 152 (2016), no. 12, 2627-2677. MR 3594290

Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473-527. MR 1322847 (96k:17010)

I. N. Bernstein, I. M. Gel0fand, and S. I. Gel0fand, Algebraic vector bundles on Pn and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66-67. MR 509387

Roman Bezrukavnikov and Zhiwei Yun, On Koszul duality for Kac-Moody groups, Represent. Theory 17 (2013), 1-98. MR 3003920

Ben Elias, The two-color Soergel calculus, Compos. Math. 152 (2016), no. 2, 327-398. MR 3462556

Ben Elias and Mikhail Khovanov, Diagrammatics for Soergel categories, Int. J. Math. Math. Sci. (2010), Art. ID 978635, 58. MR 3095655

Ben Elias and Geordie Williamson, The Hodge theory of Soergel bimodules, Ann. of Math. (2) 180 (2014), no. 3, 1089-1136. MR 3245013

Ben Elias and Geordie Williamson, Soergel calculus, Represent. Theory 20 (2016), 295-374. MR 3555156

Peter Fiebig, The combinatorics of Coxeter categories, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4211-4233. MR 2395170

Peter Fiebig, Sheaves on moment graphs and a localization of Verma flags, Adv. Math. 217 (2008), no. 2, 683-712. MR 2370278

Eugene Gorsky and Matthew Hogancamp, Hilbert schemes and y-ification of Khovanov-Rozansky homology, preprint arXiv:1712.03938.

Daniel Juteau, Carl Mautner, and Geordie Williamson, Parity sheaves, J. Amer. Math. Soc. 27 (2014), no. 4, 1169-1212. MR 3230821

David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412

David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185-203. MR 573434

Shotaro Makisumi, Mixed modular perverse sheaves on moment graphs, preprint arXiv:1703.01571.

Shotaro Makisumi, Modular Koszul duality for Soergel bimodules, preprint arXiv:1703.01576.

Carl Mautner and Simon Riche, Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 9, 2259-2332. MR 3836847

Simon Riche, Geometric Representation Theory in positive characteristic, (2016), Mémoire d'habilitation, Université Blaise Pascal (Clermont Ferrand 2), available from https://tel.archives-ouvertes.fr/tel-01431526.

Simon Riche and Geordie Williamson, Tilting modules and the p-canonical basis, Astérisque (2018), no. 397, ix+184. MR 3805034

Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83-114. MR 1444322

Wolfgang Soergel, Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen, J. Inst. Math. Jussieu 6 (2007), no. 3, 501-525. MR 2329762

Geordie Williamson, Parity sheaves and the Hecke category, Proceedings of the International Congress of Mathematicians 2018 (ICM 2018).

Geordie Williamson, Algebraic representations and constructible sheaves, Jpn. J. Math. 12 (2017), no. 2, 211-259. MR 3694932

Cómo citar

APA

Makisumi, S. (2019). On monoidal Koszul duality for the Hecke category. Revista Colombiana de Matemáticas, 53(supl), 195–222. https://doi.org/10.15446/recolma.v53nsupl.84084

ACM

[1]

Makisumi, S. 2019. On monoidal Koszul duality for the Hecke category. Revista Colombiana de Matemáticas. 53, supl (dic. 2019), 195–222. DOI:https://doi.org/10.15446/recolma.v53nsupl.84084.

ACS

(1)

Makisumi, S. On monoidal Koszul duality for the Hecke category. rev.colomb.mat 2019, 53, 195-222.

ABNT

MAKISUMI, S. On monoidal Koszul duality for the Hecke category. Revista Colombiana de Matemáticas, [S. l.], v. 53, n. supl, p. 195–222, 2019. DOI: 10.15446/recolma.v53nsupl.84084. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/84084. Acesso em: 19 abr. 2024.

Chicago

Makisumi, Shotaro. 2019. «On monoidal Koszul duality for the Hecke category». Revista Colombiana De Matemáticas 53 (supl):195-222. https://doi.org/10.15446/recolma.v53nsupl.84084.

Harvard

Makisumi, S. (2019) «On monoidal Koszul duality for the Hecke category», Revista Colombiana de Matemáticas, 53(supl), pp. 195–222. doi: 10.15446/recolma.v53nsupl.84084.

IEEE

[1]

S. Makisumi, «On monoidal Koszul duality for the Hecke category», rev.colomb.mat, vol. 53, n.º supl, pp. 195–222, dic. 2019.

MLA

Makisumi, S. «On monoidal Koszul duality for the Hecke category». Revista Colombiana de Matemáticas, vol. 53, n.º supl, diciembre de 2019, pp. 195-22, doi:10.15446/recolma.v53nsupl.84084.

Turabian

Makisumi, Shotaro. «On monoidal Koszul duality for the Hecke category». Revista Colombiana de Matemáticas 53, no. supl (diciembre 11, 2019): 195–222. Accedido abril 19, 2024. https://revistas.unal.edu.co/index.php/recolma/article/view/84084.

Vancouver

1.

Makisumi S. On monoidal Koszul duality for the Hecke category. rev.colomb.mat [Internet]. 11 de diciembre de 2019 [citado 19 de abril de 2024];53(supl):195-222. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/84084

Descargar cita

CrossRef Cited-by

2

1. Nicolas Libedinsky. (2022). IntroSurvey of Representation Theory. Journal of the Indian Institute of Science, 102(3), p.907. https://doi.org/10.1007/s41745-022-00301-4.

2. Ben Elias, Shotaro Makisumi, Ulrich Thiel, Geordie Williamson. (2020). Introduction to Soergel Bimodules. RSME Springer Series. 5, p.529. https://doi.org/10.1007/978-3-030-48826-0_26.