Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-23T13:58:08.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Categorification via blocks of modular representations for $\mathfrak {sl}_n$

Part of: Special varieties Lie groups Algebraic groups

Published online by Cambridge University Press:  19 May 2020

Vinoth Nandakumar  and
Gufang Zhao
Vinoth Nandakumar*
Affiliation:
Max Planck Institute of Mathematics (Bonn) Vivatsgasse 7, 53111Bonn, Germany and School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia
Gufang Zhao
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA01003, USA and The University of Melbourne, School of Mathematics and Statistics, Parkville, VIC3010, Australia e-mail: gufangz@unimelb.edu.au
*
e-mail: vinoth.nandakumar@sydney.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

Keywords

Categorificationmodular representationFourier–Mukai transformlocalization

MSC classification

Primary: 22E47: Representations of Lie and real algebraic groups
Secondary: 14M15: Grassmannians, Schubert varieties, flag manifolds 14L35: Classical groups (geometric aspects)
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 4 , August 2021 , pp. 1095 - 1123
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

Dedicated to our friend, Dmitry Vaintrob

References

Anno, R. and Nandakumar, V., Exotic t-structures for two-block Springer fibers. Preprint, 2016. arXiv:1602.00768Google Scholar
Ariki, S., On the decomposition numbers of the Hecke algebra of G(m, 1, n). J. Math. Kyoto Univ. 36(1996), no. 4, 789–808.Google Scholar
Beilinson, A. A., Lusztig, G., and MacPherson, R., A geometric setting for the quantum deformation of GLn. Duke Math. J. 61(1990), no. 2, 655–677.CrossRefGoogle Scholar
Bernstein, J., Frenkel, I., and Khovanov, M., A categorification of the Temperley-Lieb algebra and Schur quotients of ${U}_q\left({\mathfrak{sl}}_2\right)$ via projective and Zuckerman functors. Selecta Math. 5(1999), no. 2, 199–241.CrossRefGoogle Scholar
Bezrukavnikov, R. and Mirković, I., Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution with an Appendix by E. Sommers. Ann. Math. 178(2013), 835–919.CrossRefGoogle Scholar
Bezrukavnikov, R., Mirkovic, I., and Rumynin, D., Singular localization and intertwining functors for reductive lie algebras in prime characteristic. Nagoya J. Math. 184(2006), 1–55.CrossRefGoogle Scholar
Bezrukavnikov, R., Mirkovic, I., and Rumynin, D., Localization of modules for a semisimple Lie algebra in prime characteristic. Ann. Math. 167(2008), no. 3, 945–991.CrossRefGoogle Scholar
Bezrukavnikov, R. and Riche, S., Affine braid group actions on derived categories of Springer resolutions. Ann. Sci. Éc. Norm. Supér. 45(2012), no. 4, 535–599.CrossRefGoogle Scholar
Brundan, J., On the definition of Kac-Moody 2-category. Math. Ann. 364(2016), 353–372.CrossRefGoogle Scholar
Cautis, S., Dodd, C., and Kamnitzer, J., Associated graded of Hodge modules and categorical ${\mathfrak{sl}}_2$ actions. Preprint, 2016. arXiv:1603.07402 Google Scholar
Cautis, S. and Kamnitzer, J., Braiding via geometric Lie algebra actions. Compos. Math. 148(2012), no. 2, 464–506.CrossRefGoogle Scholar
Cautis, S. and Kamnitzer, J., Categorical geometric symmetric Howe duality. Preprint, 2016. arXiv:1611.02210CrossRefGoogle Scholar
Cautis, S., Kamnitzer, J., and Licata, A., Coherent sheaves and categorical ${\mathfrak{sl}}_2$ actions. Duke Math. J. 154(2010), no. 1, 135–179.CrossRefGoogle Scholar
Cautis, S., Kamnitzer, J., and Licata, A., Derived equivalences for cotangent bundles of Grassmannians via categorical ${\mathfrak{sl}}_2$ actions. J. Reine Angew. Math. 675(2013), 53–99.Google Scholar
Cautis, S., Kamnitzer, J., and Licata, A., Coherent sheaves on quiver varieties and categorification. Math. Ann. 357(2013), no. 3, 805–854.CrossRefGoogle Scholar
Cautis, S. and Koppensteiner, C., Exotic t-structures and actions of quantum affine algebras. Preprint, 2016. arXiv:1611.02777Google Scholar
Cautis, S. and Lauda, A., Implicit structure in 2-representations of quantum groups. Select. Math. 21(2015), 201–244.CrossRefGoogle Scholar
Chriss, N. and Ginzburg, V., Representation theory and complex geometry. Birkhäuser, Boston, MA and Basel, Berlin, Germany, 1997.Google Scholar
Chuang, J. and Rouquier, R., Derived equivalences for symmetric groups and ${\mathfrak{sl}}_2$ -categorification. Ann. Math. 167(2008), no. 1, 245–298.CrossRefGoogle Scholar
Chuang, J. and Rouquier, R., Perverse equivalences. Preprint, 2017. www.math.ucla.edu/~rouquier/papers/perverse.pdf Google Scholar
Elias, B. and Williamson, G., The Hodge theory of Soergel bimodules. Ann. Math. 180(2014), 1089–1136.CrossRefGoogle Scholar
Fiebig, P., Lusztig's conjecture as a moment graph problem. Bull. Lond. Math. Soc. 42(2010), 957–972.CrossRefGoogle Scholar
Frenkel, I., Khovanov, M., and Stroppel, C., A categorification of finite-dimensional irreducible representations of quantum $\mathfrak{sl}_{2}$ and their tensor products . Select. Math. 12(2006), nos. 3–4, 379–431.CrossRefGoogle Scholar
Frenkel, I. B., Khovanov, M. G., and Kirillov, A. A. Jr., Kazhdan-Lusztig polynomials and canonical basis . Transform. Groups 3(1998), no. 4, 321–336.CrossRefGoogle Scholar
Grojnowski, I., Affine ${\mathfrak{sl}}_p$ controls the representation theory of the symmetric group and related Hecke algebras. Preprint, 1999. arXiv:9907129Google Scholar
Hartshorne, R., Algebraic geometry. Graduate Texts in Mathematics, 52, Springer-Verlag, New York, NY, and Heidelberg, Germany, 1977.Google Scholar
Humphreys, J. E., Projective modules for $SL(2,q)$ . J. Algebra 25(1973), 513–518.CrossRefGoogle Scholar
Humphreys, J. E., Modular representations of finite groups of Lie type. London Mathematical Society Lecture Note Series, 326, Cambridge University Press, Cambridge, UK, 2006.Google Scholar
Humphreys, J. E., Notes on Weyl modules for semisimple algebraic groups. Unpublished notes, 2014. http://people.math.umass.edu/jeh/pub/weyl.pdf Google Scholar
Jantzen, J. C., Representations of Lie algebras in positive characteristic. Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., 131, Academic Press, Boston, MA, 1987.Google Scholar
Jantzen, J. C., Representations of algebraic groups. 2nd ed., Mathematical Surveys and Monographs, 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
Jensen, L. T. and Williamson, G., The $p$ -canonical basis for Hecke algebras. Preprint, 2015. arXiv:1510.01556 Google Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups I. Representation theory. Am. Math. Soc. 13(2009), no. 14, 309–347.Google Scholar
Khovanov, M. and Lauda, A., A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363(2011), no. 5, 2685–2700.CrossRefGoogle Scholar
Lin, Z., Representations of Chevalley groups arising from admissible lattices. Proc. AMS. 114(1992), 651–660.CrossRefGoogle Scholar
Nakano, D. K., Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and connections. In: Hu, N., Shu, B., and Wang, J.P. (eds.), Lie and representation theory, Surv. Modern Math. II, International Press, Boston, MA, 2012, 151–176.Google Scholar
Nandakumar, V. and Yang, D., Modular representations in type A with a two-row nilpotent p-character. Preprint, 2017. arXiv:1710.08754.Google Scholar
Nandakumar, V. and Zhao, G., Categorification via blocks of modular representations II. Preprint, 2017. arXiv:2005.08248 http://sites.google.com/view/vinothmn/ Google Scholar
Riche, S., Koszul duality and modular representations of semi-simple Lie algebras. Duke Math. J. 154(2010), no. 1, 31–134.CrossRefGoogle Scholar
Riche, S., Koszul duality and Frobenius structure for restricted enveloping algebras. Preprint, 2010. arXiv:1010.0495 Google Scholar
Riche, S. and Williamson, G., Tilting modules and the p-canonical basis. Ast risque, to appear. arXiv:1512.08296 Google Scholar
Rouquier, R., 2-Kac-Moody Lie algebras. Preprint, 2008. arXiv:0812.5023 Google Scholar
Sussan, J., Category $\mathbf{\mathcal{O}}$ and ${\mathfrak{sl}}_k$ link invariants. Ph.D. thesis, Yale University, New Haven, 2007. arXiv:0701045 Google Scholar
Webster, B., Tensor product algebras, Grassmannians and Khovanov homology. Preprint, 2013. arXiv:1312.7357 Google Scholar
Webster, B., Knot invariants and higher representation theory . Mem. Am. Math. Soc. 250(2017), no. 1191, 133.Google Scholar
Weyman, J. and Zhao, G., Noncommutative desingularization of orbit closures for some representations of ${GL}_n$ . Preprint, 2012. arXiv:1204.0488 Google Scholar