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STABILITY PATTERNS IN REPRESENTATION THEORY

Part of: Algebraic combinatorics General commutative ring theory Categories with structure Linear algebraic groups and related topics

Published online by Cambridge University Press:  15 June 2015

STEVEN V SAM  and
ANDREW SNOWDEN
STEVEN V SAM
Affiliation:
Department of Mathematics, University of California, Berkeley, CA, USA; svs@math.berkeley.edu
ANDREW SNOWDEN
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, USA; asnowden@umich.edu

Abstract

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We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

MSC classification

Primary: 05E05: Symmetric functions and generalizations 20G05: Representation theory
Secondary: 13A50: Actions of groups on commutative rings; invariant theory 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e11
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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/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Benkart, G., Chakrabarti, M., Halverson, T., Leduc, R., Lee, C. and Stroomer, J., ‘Tensor product representations of general linear groups and their connections with Brauer algebras’, J. Algebra 166(3) (1994), 529–567.Google Scholar
Berele, A. and Regev, A., ‘Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras’, Adv. Math. 64(2) (1987), 118–175.Google Scholar
Boij, M. and Söderberg, J., ‘Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture’, J. Lond. Math. Soc. (2) 78(1) (2008), 85–106; arXiv:math/0611081v2.Google Scholar
Brauer, R., ‘On algebras which are connected with the semisimple continuous groups’, Ann. of Math. (2) 38(4) (1937), 857–872.Google Scholar
Brundan, J., ‘Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n)’, J. Amer. Math. Soc. 16(1) (2003), 185–231; arXiv:math/0203011v3.Google Scholar
Brundan, J. and Stroppel, C., ‘Highest weight categories arising from Khovanov’s diagram algebra III: category 𝓞’, Represent. Theory 15 (2011), 170–243; arXiv:0812.1090v3.CrossRefGoogle Scholar
Brundan, J. and Stroppel, C., ‘Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup’, J. Eur. Math. Soc. 14 (2012), 373–419; arXiv:0907.2543v2.Google Scholar
Brundan, J. and Stroppel, C., ‘Gradings on walled Brauer algebras and Khovanov’s arc algebra’, Adv. Math. 231 (2012), 709–773; arXiv:1107.0999v1.Google Scholar
Cheng, S.-J. and Lam, N., ‘Irreducible characters of general linear superalgebra and super duality’, Comm. Math. Phys. 298(3) (2010), 645–672; arXiv:0905.0332v2.Google Scholar
Cheng, S.-J., Lam, N. and Wang, W., ‘Super duality and irreducible characters of ortho-symplectic Lie superalgebras’, Invent. Math. 183(1) (2011), 189–224; arXiv:0911.0129v2.CrossRefGoogle Scholar
Church, T., Ellenberg, J. and Farb, B., ‘FI-modules and stability for representations of symmetric groups’, Duke Math. J. to appear, arXiv:1204.4533v3.Google Scholar
Church, T. and Farb, B., ‘Representation theory and homological stability’, Adv. Math. 245 (2013), 250–314; arXiv:1008.1368v3.CrossRefGoogle Scholar
Comes, J. and Wilson, B., ‘Deligne’s category Rep(GL 𝛿) and representations of general linear supergroups’, Represent. Theory 16 (2012), 568–609; arXiv:1108.0652v1.Google Scholar
Dan-Cohen, E., Penkov, I. and Serganova, V., ‘A Koszul category of representations of finitary Lie algebras’, Preprint, 2011, arXiv:1105.3407v2.Google Scholar
Deligne, P., ‘Catégories tannakiennes’, in:The Grothendieck Festschrift, Vol. II, Progress in Mathematics 87 (Birkhäuser Boston, Boston, MA, 1990), 111–195.Google Scholar
Deligne, P., ‘La catégorie des représentations du groupe symétrique S t , lorsque t n’est pas un entier naturel’, in:Algebraic Groups and Homogeneous Spaces (Tata Inst. Fund. Res., Mumbai, 2007), 209–273.Google Scholar
Eisenbud, D., Fløystad, G. and Weyman, J., ‘The existence of equivariant pure free resolutions’, Ann. Inst. Fourier (Grenoble) 61(3) (2011), 905–926; arXiv:0709.1529v5.Google Scholar
Eisenbud, D. and Schreyer, F.-O., ‘Betti numbers of graded modules and cohomology of vector bundles’, J. Amer. Math. Soc. 22(3) (2009), 859–888; arXiv:0712.1843v3.Google Scholar
Fulton, W. and Harris, J., Representation Theory: A First Course, Graduate Texts in Mathematics, 129 (Springer, New York, 1991).Google Scholar
Goodman, R. and Wallach, N. R., Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, 255 (Springer, New York, 2009).Google Scholar
Halverson, T. and Ram, A., ‘Partition algebras’, European J. Combin. 26(6) (2005), 869–921; arXiv:math/0401314v2.Google Scholar
Hashimoto, M. and Hayashi, T., ‘Quantum multilinear algebra’, Tohoku Math. J. (2) 44(4) (1992), 471–521.Google Scholar
Howe, R., Tan, E.-C. and Willenbring, J. F., ‘Stable branching rules for classical symmetric pairs’, Trans. Amer. Math. Soc. 357(4) (2005), 1601–1626; arXiv:math/0311159v2.Google Scholar
Jones, V. F. R., ‘The Potts model and the symmetric group’, in:Subfactors (Kyuzeso, 1993) (World Scientific Publications, River Edge, NJ, 1994), 259–267.Google Scholar
King, R. C., ‘Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups’, J. Math. Phys. 12 (1971), 1588–1598.Google Scholar
Koike, K., ‘On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters’, Adv. Math. 74(1) (1989), 57–86.Google Scholar
Koike, K. and Terada, I., ‘Young-diagrammatic methods for the representation theory of the classical groups of type B n , C n , D n’, J. Algebra 107(2) (1987), 466–511.Google Scholar
Littlewood, D. E., The Theory of Group Characters and Matrix Representations of Groups reprint of the second (1950) edition, (AMS Chelsea Publishing, Providence, RI, 2006).Google Scholar
Littlewood, D. E., ‘Products and plethysms of characters with orthogonal, symplectic and symmetric groups’, Canad. J. Math. 10 (1958), 17–32.Google Scholar
Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd edition, (1995), With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York.Google Scholar
Martin, P., ‘Temperley–Lieb algebras for nonplanar statistical mechanics—the partition algebra construction’, J. Knot Theory Ramifications 3(1) (1994), 51–82.Google Scholar
Nagpal, R., V Sam, S. and Snowden, A., ‘Noetherianity of some degree two twisted commutative algebras’, Preprint, 2015, arXiv:1501.06925v1.Google Scholar
Okounkov, A. and Vershik, A., ‘A new approach to representation theory of symmetric groups’, Selecta Math. (N.S.) 2(4) (1996), 581–605; arXiv:math/0503040v3.Google Scholar
Ol’shanskiĭ, G. I., ‘Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians’, in:Topics in Representation Theory, Adv. Soviet Math., 2 (American Mathematical Society, Providence, RI, 1991), 1–66.Google Scholar
Ottaviani, G. and Rubei, E., ‘Quivers and the cohomology of homogeneous vector bundles’, Duke Math. J. 132(3) (2006), 459–508; arXiv:math/0403307v2.Google Scholar
Penkov, I. and Serganova, V., ‘Categories of integrable sl (∞)-, o (∞)-, sp (∞)-modules’, in:Representation Theory and Mathematical Physics, Contemporary Mathematics, 557 (American Mathematical Society, Providence, RI, 2011), 335–357. arXiv:1006.2749v1.Google Scholar
Penkov, I. and Styrkas, K., ‘Tensor representations of classical locally finite Lie algebras’, in:Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288 (Birkhäuser Boston, Inc, Boston, MA, 2011), 127–150. arXiv:0709.1525v1.Google Scholar
Proctor, R. A., ‘Odd symplectic groups’, Invent. Math. 92(2) (1988), 307–332.Google Scholar
V Sam, S. and Snowden, A., ‘GL-equivariant modules over polynomial rings in infinitely many variables’, Trans. Amer. Math. Soc. to appear, arXiv:1206.2233v2.Google Scholar
V Sam, S. and Snowden, A., ‘Introduction to twisted commutative algebras’, Preprint, 2012, arXiv:1209.5122v1.Google Scholar
V Sam, S. and Snowden, A., ‘GL-equivariant modules over polynomial rings in infinitely many variables II’, in preparation.Google Scholar
V Sam, S. and Snowden, A., ‘Infinite rank spinor and oscillator representations’, in preparation.Google Scholar
V Sam, S., Snowden, A. and Weyman, J., ‘Homology of Littlewood complexes’, Selecta Math. (N.S.) 19(3) (2013), 655–698; arXiv:1209.3509v2.CrossRefGoogle Scholar
Serganova, V., ‘Kazhdan–Lusztig polynomials and character formula for the Lie superalgebra gl(m|n)’, Selecta Math. (N.S.) 2(4) (1996), 607–651.Google Scholar
Serganova, V., ‘Classical Lie superalgebras at infinity’, in:Advances in Lie Superalgebras, Springer INdAM Ser., 7 (Springer, Cham, 2014), 181–201.CrossRefGoogle Scholar
Sergeev, A. N., ‘The tensor algebra of the identity representation as a module over the Lie superalgebras Gl(n, m) and Q (n)’, Math. USSR Sbornik 51 (1985), 419–427.Google Scholar
Snowden, A., ‘Syzygies of Segre embeddings and Δ-modules’, Duke Math. J. 162(2) (2013), 225–277; arXiv:1006.5248v4.Google Scholar
Stanley, R. P., Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, 62 (Cambridge University Press, Cambridge, 1999).Google Scholar
Wenzl, H., ‘On the structure of Brauer’s centralizer algebras’, Ann. of Math. (2) 128(1) (1988), 173–193.Google Scholar
Weyl, H., The Classical Groups: their Invariants and Representations, Fifteenth printing, Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 1997).Google Scholar