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Balanced semisimple filtrations for tilting modules


Author: Amit Hazi
Journal: Represent. Theory 21 (2017), 4-19
MSC (2010): Primary 20G42
DOI: https://doi.org/10.1090/ert/495
Published electronically: March 8, 2017
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Abstract: Let $ U_l$ be a quantum group at an $ l$th root of unity, obtained via Lusztig's divided powers construction. Many indecomposable tilting modules for $ U_l$ have been shown to have what we call a balanced semisimple filtration, or a Loewy series whose semisimple layers are symmetric about some middle layer. The existence of such filtrations suggests a remarkably straightforward algorithm for calculating these characters if the irreducible characters are already known. We first show that the results of this algorithm agree with Soergel's character formula for the regular indecomposable tilting modules. We then show that these balanced semisimple filtrations really do exist for these tilting modules.


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  • [1] Henning Haahr Andersen, An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 60 (1986), no. 2, 125-153. MR 840301, https://doi.org/10.1016/S0001-8708(86)80008-1
  • [2] Henning Haahr Andersen and Masaharu Kaneda, Loewy series of modules for the first Frobenius kernel in a reductive algebraic group, Proc. London Math. Soc. (3) 59 (1989), no. 1, 74-98. MR 997252, https://doi.org/10.1112/plms/s3-59.1.74
  • [3] Henning Haahr Andersen and Masaharu Kaneda, Rigidity of tilting modules, Mosc. Math. J. 11 (2011), no. 1, 1-39, 181 (English, with English and Russian summaries). MR 2808210
  • [4] Henning Haahr Andersen, Patrick Polo, and Ke Xin Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1-59. MR 1094046, https://doi.org/10.1007/BF01245066
  • [5] Henning Haahr Andersen and Daniel Tubbenhauer, Diagram categories for $ u_q$-tilting modules at roots of unity, Transformation Groups (2016), 1-61.
  • [6] C. Bowman, S. R. Doty, and S. Martin, Decomposition of tensor products of modular irreducible representations for $ {\rm SL}\sb 3$ (with an appendix by C. M. Ringel), Int. Electron. J. Algebra 9 (2011), 177-219. MR 2753767 (2012c:20119)
  • [7] C. Bowman, S. R. Doty, and S. Martin, Decomposition of tensor products of modular irreducible representations for $ {\rm SL}_3$: the $ p\geq 5$ case, Int. Electron. J. Algebra 17 (2015), 105-138. MR 3310689
  • [8] Vinay V. Deodhar, Duality in parabolic set up for questions in Kazhdan-Lusztig theory, J. Algebra 142 (1991), no. 1, 201-209. MR 1125213, https://doi.org/10.1016/0021-8693(91)90225-W
  • [9] Amit Hazi, Radically filtered quasi-hereditary algebras and rigidity of tilting modules, Math. Proc. Camb. Phil. Soc., in press. DOI 10.1017/S0305004116001006
  • [10] Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
  • [11] Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level, Duke Math. J. 77 (1995), no. 1, 21-62. MR 1317626, https://doi.org/10.1215/S0012-7094-95-07702-3
  • [12] Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case, Duke Math. J. 84 (1996), no. 3, 771-813. MR 1408544, https://doi.org/10.1215/S0012-7094-96-08424-0
  • [13] Masaki Kashiwara and Toshiyuki Tanisaki, Parabolic Kazhdan-Lusztig polynomials and Schubert varieties, J. Algebra 249 (2002), no. 2, 306-325. MR 1901161, https://doi.org/10.1006/jabr.2000.8690
  • [14] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. I, II, J. Amer. Math. Soc. 6 (1993), no. 4, 905-947, 949-1011. MR 1186962, https://doi.org/10.2307/2152745
  • [15] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. III, J. Amer. Math. Soc. 7 (1994), no. 2, 335-381. MR 1239506, https://doi.org/10.2307/2152762
  • [16] D. Kazhdan and G. Lusztig, Tensor structures arising from affine Lie algebras. IV, J. Amer. Math. Soc. 7 (1994), no. 2, 383-453. MR 1239507, https://doi.org/10.2307/2152763
  • [17] G. Lusztig, Modular representations and quantum groups, Classical groups and related topics (Beijing, 1987) Contemp. Math., vol. 82, Amer. Math. Soc., Providence, RI, 1989, pp. 59-77. MR 982278, https://doi.org/10.1090/conm/082/982278
  • [18] George Lusztig, Some problems in the representation theory of finite Chevalley groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 313-317. MR 604598
  • [19] George Lusztig, Monodromic systems on affine flag manifolds, Proc. Roy. Soc. London Ser. A 445 (1994), no. 1923, 231-246. MR 1276910, https://doi.org/10.1098/rspa.1994.0058
  • [20] Simon Riche and Geordie Williamson, Tilting modules and the p-canonical basis, January 2016, arXiv:math.RT/1512.08296v2.
  • [21] Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83-114. MR 1444322, https://doi.org/10.1090/S1088-4165-97-00021-6
  • [22] Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-448. MR 1663141, https://doi.org/10.1090/S1088-4165-98-00057-0

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Additional Information

Amit Hazi
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Email: A.Hazi@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/ert/495
Received by editor(s): October 11, 2016
Received by editor(s) in revised form: February 15, 2017
Published electronically: March 8, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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