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6 - An Introduction to Diagrammatic Soergel Bimodules

Published online by Cambridge University Press:  25 November 2023

Edited by
David Jordan ,
Nadia Mazza  and
Sibylle Schroll
David Jordan
Affiliation:
University of Edinburgh
Nadia Mazza
Affiliation:
Lancaster University
Sibylle Schroll
Affiliation:
Universität zu Köln
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Summary

This chapter is a survey of the category of diagrammatic Soergel bimodules and the main results about these representations.

Keywords

Soergel bimodulediagrammatic presentation of categories
Type
Chapter
Information
Modern Trends in Algebra and Representation Theory , pp. 212 - 226
Publisher: Cambridge University Press
Print publication year: 2023

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References

[1]Achar, Pramod N., Makisumi, Shotaro, Riche, Simon, and Williamson, Geordie. 2017 (Mar.). Free-monodromic mixed tilting sheaves on flag varieties. arXiv: 1703.05843.Google Scholar
[2]Be˘ilinson, A., and Bernstein, J. 1981. Localisation de g-modules. C. R. Acad. Sci. Paris Se´r. I Math., 292(1), 15–18.Google Scholar
[3]Bowman, Chris, Cox, Anton, and Hazi, Amit. 2020 (May). Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras. arXiv:2005.02825.Google Scholar
[4]Brylinski, J.-L. and Kashiwara, M. 1981. Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math., 64(3), 387–410.Google Scholar
[5]Elias, Ben. 2016. The two-color Soergel calculus. Compos. Math., 152(2), 327–398.Google Scholar
[6]Elias, Ben, and Khovanov, Mikhail. 2010. Diagrammatics for Soergel categories. Int. J. Math. Math. Sci., Art. ID 978635, 58.Google Scholar
[7]Elias, Ben, and Williamson, Geordie. 2013 (Mar.). Soergel bimodules and Kazhdan-Lusztig conjectures. www.maths.usyd.edu.au/u/geordie/aarhus/.Google Scholar
[8]Elias, Ben, and Williamson, Geordie. 2014. The Hodge theory of Soergel bimodules. Ann. of Math. (2), 180(3), 1089–1136.Google Scholar
[9]Elias, Ben, and Williamson, Geordie. 2016. Soergel calculus. Represent. Theory, 20, 295–374.Google Scholar
[10]Elias, Ben, Makisumi, Shotaro, Thiel, Ulrich, and Williamson, Geordie. 2020. Introduction to Soergel bimodules. RSME Springer Series, vol. 5. Springer, Cham.Google Scholar
[11]Humphreys, James E. 2008. Representations of semisimple Lie algebras in the BGG category O. Graduate Studies in Mathematics, vol. 94. American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
[12]Jensen, Lars Thorge, and Williamson, Geordie. 2017. The p-canonical basis for Hecke algebras. Pages 333–361 of: Categorification and higher representation theory. Contemp. Math., vol. 683. Amer. Math. Soc., Providence, RI.CrossRefGoogle Scholar
[13]Libedinsky, Nicolas. 2008. Sur la cate´gorie des bimodules de Soergel. J. Algebra, 320(7), 2675–2694.Google Scholar
[14]Riche, Simon, and Williamson, Geordie. 2018. Tilting modules and the p-canonical basis. Aste´risque, ix+184.Google Scholar
[15]Soergel, Wolfgang. 1990. Kategorie O, perverse Garben und Moduln u¨ber den Koinvarianten zur Weylgruppe. J. Amer. Math. Soc., 3(2), 421–445.Google Scholar
[16]Soergel, Wolfgang. 2000. On the relation between intersection cohomology and representation theory in positive characteristic. J. Pure Appl. Algebra, 152(1–3), 311–335. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998).CrossRefGoogle Scholar
[17]Soergel, Wolfgang. 2007. Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln u¨ber Polynomringen. J. Inst. Math. Jussieu, 6(3), 501–525.Google Scholar
[18]Williamson, Geordie. 2012 (Dec.). Soergel bimodules and representation theory. www.maths.usyd.edu.au/u/geordie/sydney/.Google Scholar
[19]Williamson, Geordie. 2017. Schubert calculus and torsion explosion. J. Amer. Math. Soc., 30(4), 1023–1046. With a joint appendix with Alex Kontorovich and Peter J. McNamara.Google Scholar

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