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Representation Theory
ISSN 1088-4165

     

Highest weight categories arising from Khovanov's diagram algebra III: category $ \mathcal{O}$

Author(s): Jonathan Brundan; Catharina Stroppel
Journal: Represent. Theory 15 (2011), 170-243.
MSC (2010): Primary 17B10, 16S37
Posted: March 7, 2011
MathSciNet review: 2781018
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We prove that integral blocks of parabolic category $ \mathcal{O}$ associated to the subalgebra $ \mathfrak{gl}_m(\mathbb{C}) \oplus \mathfrak{gl}_n(\mathbb{C})$ of $ \mathfrak{gl}_{m+n}(\mathbb{C})$ are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although this result is in principle known, the existing proof is quite indirect, going via perverse sheaves on Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl duality for higher levels. As a by-product we get a concrete combinatorial construction of $ 2$-Kac-Moody representations in the sense of Rouquier corresponding to level two weights in finite type $ A$.

References:

[AS]
T. Arakawa and T. Suzuki, Duality between $ \mathfrak{sl}_n(\mathbb{C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), 288-304. MR 1652134 (99h:17005)

[Ba]
E. Backelin, Koszul duality for parabolic and singular category $ \mathcal{O}$, Represent. Theory 3 (1999), 139-152. MR 1703324 (2001c:17034)

[BGS]
A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473-527. MR 1322847 (96k:17010)

[BFK]
J. Bernstein, I. Frenkel and M. Khovanov, A categorification of the Temperley-Lieb algebra and Schur quotients of $ U(\mathfrak{sl}_2)$ via projective and Zuckerman functors, Selecta Math. 5 (1999), 199-241. MR 1714141 (2000i:17009)

[BG]
J. Bernstein and S. Gelfand, Tensor products of finite and infinite representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245-285. MR 581584 (82c:17003)

[BN]
B. Boe and D. Nakano, Representation type of the blocks of category $ \mathcal{O}_S$, Adv. Math. 196 (2005), 193-256. MR 2159299 (2006f:17006)

[Bra]
T. Braden, Perverse sheaves on Grassmannians, Canad. J. Math. 54 (2002), 493-532. MR 1900761 (2003e:32053)

[Bre]
F. Brenti, Kazhdan-Lusztig and $ R$-polynomials, Young's lattice, and Dyck partitions, Pacific J. Math. 207 (2002), 257-286. MR 1972246 (2004e:20008)

[B1]
J. Brundan, Dual canonical bases and Kazhdan-Lusztig polynomials, J. Algebra 306 (2006), 17-46. MR 2271570 (2007m:05229)

[B2]
J. Brundan, Centers of degenerate cyclotomic Hecke algebras and parabolic category $ \mathcal{O}$, Represent. Theory 12 (2008), 236-259. MR 2424964 (2010d:20008)

[BK1]
J. Brundan and A. Kleshchev, Representations of shifted Yangians and finite $ W$-algebras, Mem. Amer. Math. Soc. 196 (2008), no. 918, 107 pp.. MR 2456464 (2009i:17020)

[BK2]
J. Brundan and A. Kleshchev, Schur-Weyl duality for higher levels, Selecta Math. 14 (2008), 1-57. MR 2480709

[BK3]
J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), 451-484. MR 2551762

[BK4]
J. Brundan and A. Kleshchev, The degenerate analogue of Ariki's categorification theorem, Math. Z. 266 (2010), 877-919. MR 2729296

[BK5]
J. Brundan and A. Kleshchev, Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math. 222 (2009), 1883-1942. MR 2562768

[BKW]
J. Brundan, A. Kleshchev and W. Wang, Graded Specht modules, to appear in J. Reine Angew. Math.; arXiv:0901.0218.

[BS1]
J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov's diagram algebra I: cellularity; arXiv:0806.1532.

[BS2]
J. Brundan and C. Stroppel, Highest weight categories arising from Khovanov's diagram algebra II: Koszulity, Transform. Groups 15 (2010), 1-45. MR 2600694

[C]
Y. Chen, Categorification of level two representations of quantum $ \mathfrak{sl}_n$ via generalized arc rings; arXiv:math/0611012.

[CK]
Y. Chen and M. Khovanov, An invariant of tangle cobordisms via subquotients of arc rings; arXiv:math/0610054.

[CWZ]
S.-J. Cheng, W. Wang and R.B. Zhang, Super duality and Kazhdan-Lusztig polynomials, Trans. Amer. Math. Soc. 360 (2008), 5883-5924. MR 2425696 (2009e:17008)

[CR]
J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and $ \mathfrak{sl}_2$-categorification, Ann. of Math. 167 (2008), 245-298. MR 2373155 (2008m:20011)

[CPS]
E. Cline, B. Parshall and L. Scott, Finite dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85-99. MR 961165 (90d:18005)

[D]
V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Func. Anal. Appl. 20 (1986), 56-58. MR 831053 (87m:22044)

[Du]
J. Du, IC bases and quantum linear groups, Proc. Symp. Pure Math. 56 (1994), Part 2, 135-148. MR 1278732 (95d:17010)

[ES]
T. Enright and B. Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 367 (1987), 1-94. MR 888703 (88f:22052)

[FK]
I. Frenkel and M. Khovanov, Canonical bases in tensor products and graphical calculus for $ U_q(\mathfrak{sl}_2)$, Duke Math. J. 87 (1997), 409-480. MR 1446615 (99a:17019)

[FKS]
I. Frenkel, M. Khovanov and C. Stroppel, A categorification of finite-dimensional irreducible representations of quantum $ \mathfrak{sl}_2$ and their tensor products, Selecta Math. 12 (2006), 379-431. MR 2305608 (2008a:17014)

[G]
P. Gabriel, Des catégories Abéliennes, Bull. Soc. Math. France 90 (1962), 323-448. MR 0232821 (38:1144)

[GL]
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1-34. MR 1376244 (97h:20016)

[HM]
J. Hu and A. Mathas, Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A, Adv. Math. 225 (2010), 598-642. MR 2671176

[HK]
R. Huerfano and M. Khovanov, Categorification of some level two representations of $ \mathfrak{sl}_n$, J. Knot Theory Ramifications 15 (2006), 695-713. MR 2253831 (2008f:17025)

[H]
J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category $ \mathcal{O}$, Graduate Studies in Mathematics 94, AMS, 2008. MR 2428237 (2009f:17013)

[I]
R. Irving, Projective modules in the category $ \mathcal{O}_S$: self-duality, Trans. Amer. Math. Soc. 291 (1985), 701-732. MR 800259 (87i:17005)

[KL]
D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)

[K1]
M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000), 359-426. MR 1740682 (2002j:57025)

[K2]
M. Khovanov, A functor-valued invariant of tangles, Alg. Geom. Topology 2 (2002), 665-741. MR 1928174 (2004d:57016)

[KLa]
M. Khovanov and A. Lauda, A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309-347. MR 2525917

[K]
A. Kleshchev, Linear and Projective Representations of Symmetric Groups, Cambridge University Press, Cambridge, 2005. MR 2165457 (2007b:20022)

[LS]
A. Lascoux and M.-P. Schützenberger, Polynômes de Kazhdan et Lusztig pour les Grassmanniennes, Astérisque 87-88 (1981), 249-266. MR 646823 (83i:14045)

[LM]
B. Lerclerc and H. Miyachi, Constructible characters and canonical bases, J. Algebra 277 (2004), 298-317. MR 2059632 (2005g:17033)

[L]
G. Lusztig, Introduction to Quantum Groups, Birkhäuser, 1993. MR 1227098 (94m:17016)

[M]
S. Maclane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer, 1971. MR 1712872 (2001j:18001)

[MS]
V. Mazorchuk and C. Stroppel, A combinatorial approach to functorial quantum $ \mathfrak{sl}_k$ knot invariants, Amer. J. Math. 131 (2009), 1679-1713. MR 2567504

[MOY]
H. Murakami, I. Ohtsuki and S. Yamada, Homfly polynomial via an invariant of colored plane graphs, Enseign. Math. 44 (1998), 325-360. MR 1659228 (2000a:57023)

[NM]
J. Nagel and M. Moshinsky, Operators that lower or raise the irreducible vector spaces of $ {\rm U}_{n-1}$ contained in an irreducible vector space of $ {\rm U}_{n}$, J. Math. Phys. 6 (1965), 682-694. MR 0186188 (32:3648)

[R]
R. Rouquier, $ 2$-Kac-Moody algebras; arXiv:0812.5023.

[S1]
C. Stroppel, Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors, Duke Math. J. 126 (2005), 547-596. MR 2120117 (2005i:17011)

[S2]
C. Stroppel, TQFT with corners and tilting functors in the Kac-Moody case; arXiv:math/0605103.

[S3]
C. Stroppel, Parabolic category $ \mathcal{O}$, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology, Compositio Math. 145 (2009), 954-992. MR 2521250

[SW]
C. Stroppel and B. Webster, 2-block Springer fibers: convolution algebras, coherent sheaves and embedded TQFT, to appear in Comm. Math. Helv.; arXiv:0802.1943.

[Su]
J. Sussan, Category $ \mathcal{O}$ and $ \mathfrak{sl}(k)$ link invariants; arXiv:math/0701045.

[VV]
M. Varagnolo and E. Vasserot, Canonical bases and Khovanov-Lauda algebras; arXiv:0901.3992.

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Additional Information:

Jonathan Brundan
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: brundan@uoregon.edu

Catharina Stroppel
Affiliation: Department of Mathematics, University of Bonn, 53115 Bonn, Germany
Email: stroppel@math.uni-bonn.de

DOI: 10.1090/S1088-4165-2011-00389-7
PII: S 1088-4165(2011)00389-7
Received by editor(s): July 15, 2009
Received by editor(s) in revised form: June 22, 2010 and June 26, 2010
Posted: March 7, 2011
Additional Notes: The first author was supported in part by NSF grant no. DMS-0654147
The second author was supported by the NSF and the Minerva Research Foundation DMS-0635607.
Copyright of article: Copyright 2011, American Mathematical Society




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