Super duality and Kazhdan-Lusztig polynomials

Cheng, Shun-Jen; Wang, Weiqiang; Zhang, R.

doi:10.1090/S0002-9947-08-04447-4

Super duality and Kazhdan-Lusztig polynomials
HTML articles powered by AMS MathViewer

by Shun-Jen Cheng, Weiqiang Wang and R. B. Zhang PDF

Trans. Amer. Math. Soc. 360 (2008), 5883-5924 Request permission

Abstract:

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type $A$) via Fock space reformulations of their Kazhdan-Lusztig theories. As a consequence, the characters of finite-dimensional irreducible modules of the general linear Lie superalgebra are computed by the usual parabolic Kazhdan-Lusztig polynomials of type $A$. In addition, we establish closed formulas for canonical and dual canonical bases for the tensor product of any two fundamental representations of type $A$.

References

Alexandre Beĭlinson and Joseph Bernstein, Localisation de $g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137
Alexander Beilinson, Victor Ginzburg, and Wolfgang Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), no. 2, 473–527. MR 1322847, DOI 10.1090/S0894-0347-96-00192-0
J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
I. N. Bernšteĭn and D. A. Leĭtes, A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series $\textrm {Gl}$ and $\textrm {sl}$, C. R. Acad. Bulgare Sci. 33 (1980), no. 8, 1049–1051 (Russian). MR 620836
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. MR 884183, DOI 10.1016/0001-8708(87)90007-7
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, Tilting modules for Lie superalgebras, Comm. Algebra 32 (2004), no. 6, 2251–2268. MR 2100468, DOI 10.1081/AGB-120037218
Luis G. Casian and David H. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), no. 4, 581–600. MR 900346, DOI 10.1007/BF01166705
David H. Collingwood and Ronald S. Irving, A decomposition theorem for certain self-dual modules in the category ${\scr O}$, Duke Math. J. 58 (1989), no. 1, 89–102. MR 1016415, DOI 10.1215/S0012-7094-89-05806-7
Edward Cline, Brian Parshall, and Leonard Scott, Abstract Kazhdan-Lusztig theories, Tohoku Math. J. (2) 45 (1993), no. 4, 511–534. MR 1245719, DOI 10.2748/tmj/1178225846
Shun-Jen Cheng and Weiqiang Wang, Howe duality for Lie superalgebras, Compositio Math. 128 (2001), no. 1, 55–94. MR 1847665, DOI 10.1023/A:1017594504827
Shun-Jen Cheng and R. B. Zhang, Analogue of Kostant’s $\mathfrak {u}$-cohomology formula for the general linear superalgebra, Int. Math. Res. Not. 1 (2004), 31–53. MR 2036954, DOI 10.1155/S1073792804131437
Vinay V. Deodhar, On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials, J. Algebra 111 (1987), no. 2, 483–506. MR 916182, DOI 10.1016/0021-8693(87)90232-8
Stephen Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), no. 1, 39–60. MR 1200163, DOI 10.1007/BF02571640
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
Jie Du, $\textrm {IC}$ bases and quantum linear groups, Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 135–148. MR 1278732
Thomas J. Enright and Brad Shelton, Categories of highest weight modules: applications to classical Hermitian symmetric pairs, Mem. Amer. Math. Soc. 67 (1987), no. 367, iv+94. MR 888703, DOI 10.1090/memo/0367
I. B. Frenkel, M. G. Khovanov, and A. A. Kirillov Jr., Kazhdan-Lusztig polynomials and canonical basis, Transform. Groups 3 (1998), no. 4, 321–336. MR 1657524, DOI 10.1007/BF01234531
Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
Ronald S. Irving, Singular blocks of the category $\scr O$, Math. Z. 204 (1990), no. 2, 209–224. MR 1055986, DOI 10.1007/BF02570868
Jens C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), no. 1, 53–65. MR 439902, DOI 10.1007/BF01391218
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
Michio Jimbo, Quantum $R$ matrix for the generalized Toda system, Comm. Math. Phys. 102 (1986), no. 4, 537–547. MR 824090
J. Van der Jeugt, J. W. B. Hughes, R. C. King, and J. Thierry-Mieg, A character formula for singly atypical modules of the Lie superalgebra $\textrm {sl}(m/n)$, Comm. Algebra 18 (1990), no. 10, 3453–3480. MR 1063989, DOI 10.1080/00927879008824086
Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
J. Van der Jeugt and R. B. Zhang, Characters and composition factor multiplicities for the Lie superalgebra $\mathfrak {g}\mathfrak {l}(m/n)$, Lett. Math. Phys. 47 (1999), no. 1, 49–61. MR 1669394, DOI 10.1023/A:1007590920834
V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2
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
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
M. Kashiwara, T. Miwa, and E. Stern, Decomposition of $q$-deformed Fock spaces, Selecta Math. (N.S.) 1 (1995), no. 4, 787–805. MR 1383585, DOI 10.1007/BF01587910
S. M. Khoroshkin and V. N. Tolstoy, Universal $R$-matrix for quantized (super)algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617. MR 1134942
A. N. Kirillov and N. Reshetikhin, $q$-Weyl group and a multiplicative formula for universal $R$-matrices, Comm. Math. Phys. 134 (1990), no. 2, 421–431. MR 1081014
Jonathan Kujawa, Crystal structures arising from representations of $\textrm {GL}(m\vert n)$, Represent. Theory 10 (2006), 49–85. MR 2209849, DOI 10.1090/S1088-4165-06-00219-6
Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205–263. MR 1410572
Alain Lascoux and Marcel-Paul Schützenberger, Polynômes de Kazhdan & Lusztig pour les grassmanniennes, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266 (French). MR 646823
B. Leclerc, A Littlewood-Richardson rule for evaluation representations of $U_q(\widehat {\mathfrak {sl}}_n)$, Séminaire Lotharingien de Combinatoire, B50e, www.emis.ams.org/journals/SLC/.
Bernard Leclerc and Hyohe Miyachi, Constructible characters and canonical bases, J. Algebra 277 (2004), no. 1, 298–317. MR 2059632, DOI 10.1016/j.jalgebra.2004.02.023
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
Kailash Misra and Tetsuji Miwa, Crystal base for the basic representation of $U_q(\mathfrak {s}\mathfrak {l}(n))$, Comm. Math. Phys. 134 (1990), no. 1, 79–88. MR 1079801
Ivan Penkov and Vera Serganova, Representations of classical Lie superalgebras of type $\textrm {I}$, Indag. Math. (N.S.) 3 (1992), no. 4, 419–466. MR 1201236, DOI 10.1016/0019-3577(92)90020-L
Vera Serganova, Kazhdan-Lusztig polynomials for Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 151–165. MR 1237837
Vera Serganova, Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra ${\mathfrak {g}}{\mathfrak {l}}(m|n)$, Selecta Math. (N.S.) 2 (1996), no. 4, 607–651. MR 1443186, DOI 10.1007/PL00001385
W. Soergel, $\mathfrak {n}$-cohomology of simple highest weight modules on walls and purity, Invent. Math. 98 (1989), no. 3, 565–580. MR 1022307, DOI 10.1007/BF01393837
Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83–114. MR 1444322, DOI 10.1090/S1088-4165-97-00021-6
Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432–448. MR 1663141, DOI 10.1090/S1088-4165-98-00057-0
Wolfgang Soergel, Character formulas for tilting modules over quantum groups at roots of one, Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 161–172. MR 1698856
Eugene Stern, Semi-infinite wedges and vertex operators, Internat. Math. Res. Notices 4 (1995), 201–219. MR 1326065, DOI 10.1155/S107379289500016X
A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras $\textrm {Gl}(n,\,m)$ and $Q(n)$, Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430 (Russian). MR 735715
David A. Vogan Jr., Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), no. 4, 805–859. MR 552528
Minoru Wakimoto, Infinite-dimensional Lie algebras, Translations of Mathematical Monographs, vol. 195, American Mathematical Society, Providence, RI, 2001. Translated from the 1999 Japanese original by Kenji Iohara; Iwanami Series in Modern Mathematics. MR 1793723, DOI 10.1142/9789812810700
Yi Ming Zou, Categories of finite-dimensional weight modules over type I classical Lie superalgebras, J. Algebra 180 (1996), no. 2, 459–482. MR 1378540, DOI 10.1006/jabr.1996.0077