×

Deligne’s category \(\underline{\mathrm{Rep}}(\operatorname{GL}_{\delta})\) and representations of general linear supergroups. (English) Zbl 1302.17010

Summary: We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric Schur polynomials, and give a method for decomposing their tensor products. Along the way, we describe indecomposable objects in \( \underline{\mathrm{Rep}}(GL_\delta )\) and explain how to decompose their tensor products.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20G05 Representation theory for linear algebraic groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. · Zbl 0765.16001
[2] Georgia Benkart, Manish Chakrabarti, Thomas Halverson, Robert Leduc, Chanyoung Lee, and Jeffrey Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras, J. Algebra 166 (1994), no. 3, 529 – 567. · Zbl 0815.20028
[3] D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. · Zbl 0718.20001
[4] A. Berele and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math. 64 (1987), no. 2, 118 – 175. · Zbl 0617.17002
[5] Jonathan Brundan and Catharina Stroppel, Gradings on walled Brauer algebras and Khovanov’s arc algebra, Adv. Math. 231 (2012), no. 2, 709 – 773. · Zbl 1326.17006
[6] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra. II. Koszulity, Transform. Groups 15 (2010), no. 1, 1 – 45. · Zbl 1205.17010
[7] 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). · Zbl 1275.17012
[8] Jonathan Brundan and Catharina Stroppel, Highest weight categories arising from Khovanov’s diagram algebra III: category \?, Represent. Theory 15 (2011), 170 – 243. · Zbl 1261.17006
[9] 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. · Zbl 1243.17004
[10] Jonathan Comes and Victor Ostrik, On blocks of Deligne’s category \?\?\?(\?_{\?}), Adv. Math. 226 (2011), no. 2, 1331 – 1377. · Zbl 1225.18005
[11] Anton Cox and Maud De Visscher, Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra, J. Algebra 340 (2011), 151 – 181. · Zbl 1269.20037
[12] Anton Cox, Maud De Visscher, Stephen Doty, and Paul Martin, On the blocks of the walled Brauer algebra, J. Algebra 320 (2008), no. 1, 169 – 212. · Zbl 1196.20004
[13] C. J. Cummins and R. C. King, Composite Young diagrams, supercharacters of \?(\?/\?) and modification rules, J. Phys. A 20 (1987), no. 11, 3121 – 3133. · Zbl 0645.17002
[14] P. Deligne, La catégorie des représentations du groupe symétrique \?_{\?}, lorsque \? 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). · Zbl 1165.20300
[15] Pierre Deligne, La série exceptionnelle de groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 4, 321 – 326 (French, with English and French summaries). · Zbl 0910.22008
[16] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001
[17] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1 – 34. · Zbl 0853.20029
[18] Kazuhiko Koike, On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math. 74 (1989), no. 1, 57 – 86. · Zbl 0681.20030
[19] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. · Zbl 0899.05068
[20] Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. · Zbl 0906.18001
[21] E. M. Moens and J. van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, J. Algebraic Combin. 17 (2003), no. 3, 283 – 307. · Zbl 1020.05070
[22] E. M. Moens and J. Van der Jeugt, On characters and dimension formulas for representations of the Lie superalgebra \?\?(\?|\?), Lie theory and its applications in physics V, World Sci. Publ., River Edge, NJ, 2004, pp. 64 – 73. · Zbl 1229.17005
[23] P. Selinger, A survey of graphical languages for monoidal categories, New structures for physics, Lecture Notes in Phys., vol. 813, Springer, Heidelberg, 2011, pp. 289 – 355. · Zbl 1217.18002
[24] Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra \?\?(\?|\?), Selecta Math. (N.S.) 2 (1996), no. 4, 607 – 651. · Zbl 0881.17005
[25] A. N. Sergeev, Representations of the Lie superalgebras \?\?(\?,\?) and \?(\?) in a space of tensors, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 80 – 81 (Russian).
[26] John R. Stembridge, Rational tableaux and the tensor algebra of \?\?_{\?}, J. Combin. Theory Ser. A 46 (1987), no. 1, 79 – 120. · Zbl 0626.20030
[27] V. G. Turaev, Operator invariants of tangles, and \?-matrices, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 5, 1073 – 1107, 1135 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 2, 411 – 444.
[28] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.