Abstract
We prove the linkage principle and describe blocks of the general linear supergroups GL(m|n) over the ground field K of characteristic p ≠ 2.
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H. H. Andersen, The strong linkage principle, J. reine angew. Math. 315 (1980), 53–59.
J. Brundan, J. Kujawa, A new proof of the Mullineux conjecture, J. Algebraic Comb. 18 (2003), 13–39.
S. J. Cheng, W.Wang, Dualities and Representations of Lie Superalgebras, Graduate Studies in Mathematics, Vol. 144, American Mathematical Society, Providence, RI, 2012.
M. Demazure, A very simple proof of Bott's theorem, Invent. Math. 33 (1976), no. 3, 271–272.
S. Donkin, The blocks of a semisimple algebraic group, J. Algebra 67 (1980), no. 1, 36–53.
S. Doty, The strong linkage principle, Amer. J. Math. 111 (1989), no. 1, 135–141.
A. N. Grishkov, A. N. Zubkov, Solvable, reductive and quasireductive supergroups, J. Algebra 452 (2016), 448–473.
C. Gruson, V. Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 3, 852–892.
J. Germoni, Indecomposable representations of special linear Lie superalgebras, J. Algebra 209 (1998), no. 2, 367–401.
J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.
J. C. Jantzen, Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren, Math. Z. 140 (1974), 127–149.
J. C. Jantzen, Darstellungen halbeinfacher Gruppen und kontravariante Formen, J. reine angew. Math. 290 (1977), 117–141.
J. Kujawa, The Steinberg tensor product theorem for GL(m|n), in: Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemp. Math., Vol. 413, American Mathematical Society, Providence, RI, 2006, pp. 123–132.
J. Kujawa, Crystal structures arising from representations of GL(m|n), Represent. Theory 10 (2006), 49–85.
F. Marko, Primitive vectors in induced supermodules for general linear supergroups, J. Pure Appl. Algebra 219 (2015), no. 4, 978–1007.
F. Marko, A. N. Zubkov, Pseudocompact algebras and highest weight categories, Algebr. Represent. Theory 16 (2013), no. 3, 689–728.
И. Б. Пeнькoв, Teopия Бopeля-Beйля-Бoттa для клaccичecкиx cупepгpупп Ли. Итoги нaуки и тexн., cep. Coвpeм. пpoбл. мaт. Hoв. дocтиж., тoм 32, VINITI, M., 1988, str. 71–124. Engl. transl.: I. B. Penkov, Borel-Weil-Bott theory for classical Lie supergroups. J. Soviet Math. 51 (1990), no. 1, 2108–2140.
I. Penkov, I. Skornyakov, Cohomologie des D-modules tordus typiques sur les super-varits de drapeaux, C. R. Acad. Sci. Paris Sr. I Math. 299 (1984), no. 20, 1005–1008.
B. Shu, W.Wang, Modular representations of the ortho-symplectic supergroups, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 251–271.
D. N. Verma, Rôle of affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras, in: Lie Groups and Their Representations, I.M. Gelfand ed., Proc. Budapest 1971, London, 1975, pp. 653–705.
A. N. Zubkov, Some homological properties of GL(m|n) in arbitrary characteristic, J. Algebra Appl. 15 (2016), no. 7, 1650119, 26 pp.
A. N. Zubkov, GL(m|n)-supermodules with good and Weyl filtrations, J. Pure Appl. Algebra 219 (2015), no. 12, 5259–5279.
A. N. Zubkov, F. Marko, The center of Dist(GL(m|n)) in positive characteristic, Algebr. Represent. Theory 19 (2016), no. 3, 613–639.
A. N. Zubkov, Affine quotients of supergroups, Transform. Groups 14 (2009), no. 3, 713–745.
A. H. Зубкoв, O нeкoтopыx cвoйcтвax oбщиx линeйныx cупepгpупп и cупepaлгeбp Шуpa, Aлгeбpa и лoгикa 45 (2006), no. 3, 257–299. Engl. transl.: A. N. Zubkov, Some properties of general linear supergroups and of Schur superalgebras, Algebra Logic 45 (2006), no. 3, 147–171.
A. N. Zubkov, On quotients of affine superschemes over finite supergroups, J. Algebra Appl. 10 (2011), no. 3, 391–408.
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MARKO, F., ZUBKOV, A.N. BLOCKS FOR THE GENERAL LINEAR SUPERGROUP GL(m|n). Transformation Groups 23, 185–215 (2018). https://doi.org/10.1007/s00031-017-9429-6
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DOI: https://doi.org/10.1007/s00031-017-9429-6