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Cohomological Tensor Functors on Representations of the General Linear Supergroup
About this Title
Th. Heidersdorf and R Weissauer
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 270, Number 1320
ISBNs: 978-1-4704-4714-4 (print); 978-1-4704-6528-5 (online)
DOI: https://doi.org/10.1090/memo/1320
Published electronically: June 23, 2021
Keywords: Supergroups,
tensor functor,
representations of the general linear supergroup,
Duflo-Serganova functor
Table of Contents
Chapters
- 1. Introduction
- 2. Cohomological Tensor Functors
- 3. The Main Theorem and its Proof
- 4. Consequences of the Main Theorem
Abstract
We define and study cohomological tensor functors from the category $T_n$ of finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$ for $0 <r \leq n$. In the case $DS: T_n \to T_{n-1}$ we prove a formula $DS(L) = \bigoplus \Pi ^{n_i} L_i$ for the image of an arbitrary irreducible representation. In particular $DS(L)$ is semisimple and multiplicity free. We derive a few applications of this theorem such as the degeneration of certain spectral sequences and a formula for the modified superdimension of an irreducible representation.- Brian D. Boe, Jonathan R. Kujawa, and Daniel K. Nakano, Complexity and module varieties for classical Lie superalgebras, Int. Math. Res. Not. IMRN 3 (2011), 696–724. MR 2764876, DOI 10.1093/imrn/rnq090
- Brian D. Boe, Jonathan R. Kujawa, and Daniel K. Nakano, Cohomology and support varieties for Lie superalgebras. II, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 19–44. MR 2472160, DOI 10.1112/plms/pdn019
- Brian D. Boe, Jonathan R. Kujawa, and Daniel K. Nakano, Cohomology and support varieties for Lie superalgebras, Trans. Amer. Math. Soc. 362 (2010), no. 12, 6551–6590. MR 2678986, DOI 10.1090/S0002-9947-2010-05096-2
- Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m|n)$, J. Amer. Math. Soc. 16 (2003), no. 1, 185–231. MR 1937204, DOI 10.1090/S0894-0347-02-00408-3
- Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra. II. Koszulity, Transform. Groups 15 (2010), no. 1, 1–45. MR 2600694, DOI 10.1007/s00031-010-9079-4
- Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra I: cellularity, Mosc. Math. J. 11 (2011), no. 4, 685–722, 821–822 (English, with English and Russian summaries). MR 2918294, DOI 10.17323/1609-4514-2011-11-4-685-722
- Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 373–419. MR 2881300, DOI 10.4171/JEMS/306
- Jonathan Brundan and Catharina Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709–773. MR 2955190, DOI 10.1016/j.aim.2012.05.016
- Daniel Bump and Anne Schilling, Crystal bases, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Representations and combinatorics. MR 3642318, DOI 10.1142/9876
- Jonathan Comes and Benjamin Wilson, Deligne’s category $\underline {\rm {Rep}}(GL_\delta )$ and representations of general linear supergroups, Represent. Theory 16 (2012), 568–609. MR 2998810, DOI 10.1090/S1088-4165-2012-00425-3
- P. Deligne, La catégorie des représentations du groupe symétrique $S_t$, lorsque $t$ n’est pas un entier naturel, Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 209–273 (French, with English and French summaries). MR 2348906
- Fr. Drouot, Quelques proprietes des representations de la super-algebre der Lie $\mathfrak {gl}(m,n)$, PhD thesis, (2009)
- M. Duflo and V. Serganova, On associated variety for Lie superalgebras, arXiv:math/0507198v1, (2005)
- Jérôme Germoni, Indecomposable representations of special linear Lie superalgebras, J. Algebra 209 (1998), no. 2, 367–401. MR 1659915, DOI 10.1006/jabr.1998.7520
- Nathan Geer, Jonathan Kujawa, and Bertrand Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, Selecta Math. (N.S.) 17 (2011), no. 2, 453–504. MR 2803849, DOI 10.1007/s00029-010-0046-7
- Gerhard Götz, Thomas Quella, and Volker Schomerus, Representation theory of $\mathfrak {sl}(2|1)$, J. Algebra 312 (2007), no. 2, 829–848. MR 2333186, DOI 10.1016/j.jalgebra.2007.03.012
- Thorsten Heidersdorf, Mixed tensors of the general linear supergroup, J. Algebra 491 (2017), 402–446. MR 3699103, DOI 10.1016/j.jalgebra.2017.08.012
- Thorsten Heidersdorf, On supergroups and their semisimplified representation categories, Algebr. Represent. Theory 22 (2019), no. 4, 937–959. MR 3985146, DOI 10.1007/s10468-018-9806-4
- Thorsten Heidersdorf and Rainer Weissauer, Pieri type rules and $GL(2|2)$ tensor products, Algebr. Represent. Theory 24 (2021), no. 2, 425–451. MR 4233502, DOI 10.1007/s10468-020-09954-0
- Thorsten Heidersdorf, Mixed tensors of the general linear supergroup, J. Algebra 491 (2017), 402–446. MR 3699103, DOI 10.1016/j.jalgebra.2017.08.012
- G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603. MR 54581, DOI 10.2307/1969740
- V. Kac, Representations of classical Lie superalgebras, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977) Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 597–626. MR 519631
- Jonathan Kujawa, The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra $\mathfrak {osp}(m|2n)$, Recent developments in Lie algebras, groups and representation theory, Proc. Sympos. Pure Math., vol. 86, Amer. Math. Soc., Providence, RI, 2012, pp. 201–215. MR 2977005, DOI 10.1090/pspum/086/1419
- Eckhard Meinrenken, Clifford algebras and Lie theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 58, Springer, Heidelberg, 2013. MR 3052646, DOI 10.1007/978-3-642-36216-3
- Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
- Manfred Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. An introduction. MR 537441
- V. Serganova Blocks in the category of finite-dimensional representations of $gl(m,n)$, 2006
- Caroline Gruson and Vera Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892. MR 2734963, DOI 10.1112/plms/pdq014
- Vera Serganova, On the superdimension of an irreducible representation of a basic classical Lie superalgebra, Supersymmetry in mathematics and physics, Lecture Notes in Math., vol. 2027, Springer, Heidelberg, 2011, pp. 253–273. MR 2906346, DOI 10.1007/978-3-642-21744-9_{1}2
- R. Weissauer, Monoidal model structures, categorial quotients and representations of super commutative Hopf algebras II: The case $Gl(m,n)$, arXiv e-prints, 1010.3217 (2010)