Schubert calculus and torsion explosion

Williamson, Geordie¹

¹ Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

Journal of the American Mathematical Society, Tome 30 (2017) no. 4, pp. 1023-1046

Résumé

The author observes that certain numbers occurring in Schubert calculus for $\text {SL}_n$ also occur as entries in intersection forms controlling decompositions of Soergel bimodules in higher rank. These numbers grow exponentially. This observation gives many counter-examples to the expected bounds in Lusztig’s conjecture on the characters of simple rational modules for $\text {SL}_n$ over fields of positive characteristic. The examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.

Détail

DOI : 10.1090/jams/868

Bibliographie

[1] Arkhipov, Sergey, Bezrukavnikov, Roman, Ginzburg, Victor Quantum groups, the loop Grassmannian, and the Springer resolution J. Amer. Math. Soc. 2004 595 678

[2] Andersen, H. H., Jantzen, J. C., Soergel, W. Representations of quantum groups at a 𝑝th root of unity and of semisimple groups in characteristic 𝑝: independence of 𝑝 Astérisque 1994 321

[3] Beilinson, A., Bezrukavnikov, R., Mirković, I. Tilting exercises Mosc. Math. J. 2004

[4] Bourgain, Jean, Gamburd, Alex, Sarnak, Peter Sieving and expanders C. R. Math. Acad. Sci. Paris 2006 155 159

[5] Bourgain, Jean, Gamburd, Alex, Sarnak, Peter Affine linear sieve, expanders, and sum-product Invent. Math. 2010 559 644

[6] Bourgain, Jean, Kontorovich, Alex On Zaremba’s conjecture Ann. of Math. (2) 2014 137 196

[7] Bezrukavnikov, Roman, Mirković, Ivan Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution Ann. of Math. (2) 2013 835 919

[8] Bezrukavnikov, Roman, Mirković, Ivan, Rumynin, Dmitriy Localization of modules for a semisimple Lie algebra in prime characteristic Ann. of Math. (2) 2008 945 991

[9] Carmichael, R. D. On the numerical factors of the arithmetic forms 𝛼ⁿ±𝛽ⁿ Ann. of Math. (2) 1913/14 49 70

[10] De Cataldo, Mark Andrea A., Migliorini, Luca The hard Lefschetz theorem and the topology of semismall maps Ann. Sci. École Norm. Sup. (4) 2002 759 772

[11] De Cataldo, Mark Andrea A., Migliorini, Luca The Hodge theory of algebraic maps Ann. Sci. École Norm. Sup. (4) 2005 693 750

[12] Donkin, Stephen On tilting modules for algebraic groups Math. Z. 1993 39 60

[13] Donkin, S. The 𝑞-Schur algebra 1998

[14] Elias, Ben, Khovanov, Mikhail Diagrammatics for Soergel categories Int. J. Math. Math. Sci. 2010

[15] Elias, Ben The two-color Soergel calculus Compos. Math. 2016 327 398

[16] Elias, Ben, Williamson, Geordie The Hodge theory of Soergel bimodules Ann. of Math. (2) 2014 1089 1136

[17] Fiebig, Peter The combinatorics of Coxeter categories Trans. Amer. Math. Soc. 2008 4211 4233

[18] Fiebig, Peter The multiplicity one case of Lusztig’s conjecture Duke Math. J. 2010 551 571

[19] Fiebig, Peter Sheaves on affine Schubert varieties, modular representations, and Lusztig’s conjecture J. Amer. Math. Soc. 2011 133 181

[20] Fiebig, Peter An upper bound on the exceptional characteristics for Lusztig’s character formula J. Reine Angew. Math. 2012 1 31

[21] Frolenkov, Dmitrii A., Kan, Igor D. A strengthening of a theorem of Bourgain-Kontorovich II Mosc. J. Comb. Number Theory 2014 78 117

[22] Fiebig, Peter, Williamson, Geordie Parity sheaves, moment graphs and the 𝑝-smooth locus of Schubert varieties Ann. Inst. Fourier (Grenoble) 2014 489 536

[23] Green, James A. Polynomial representations of 𝐺𝐿_{𝑛} 1981 124 140

[24] Hensley, Douglas A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets J. Number Theory 1996 9 45

[25] Huang, Shinnyih An improvement to Zaremba’s conjecture Geom. Funct. Anal. 2015 860 914

[26] James, Gordon The decomposition matrices of 𝐺𝐿_{𝑛}(𝑞) for 𝑛≤10 Proc. London Math. Soc. (3) 1990 225 265

[27] Jantzen, Jens Carsten Moduln mit einem höchsten Gewicht 1979

[28] Jantzen, Jens Carsten Representations of algebraic groups 2003

[29] Jantzen, Jens Carsten Character formulae from Hermann Weyl to the present 2008 232 270

[30] Juteau, Daniel, Mautner, Carl, Williamson, Geordie Parity sheaves J. Amer. Math. Soc. 2014 1169 1212

[31] Kato, Shin-Ichi On the Kazhdan-Lusztig polynomials for affine Weyl groups Adv. in Math. 1985 103 130

[32] Kazhdan, D., Lusztig, G. Tensor structures arising from affine Lie algebras. I, II J. Amer. Math. Soc. 1993

[33] Kazhdan, D., Lusztig, G. Tensor structures arising from affine Lie algebras. III J. Amer. Math. Soc. 1994 335 381

[34] Kazhdan, D., Lusztig, G. Tensor structures arising from affine Lie algebras. IV J. Amer. Math. Soc. 1994 383 453

[35] Kontorovich, Alex From Apollonius to Zaremba: local-global phenomena in thin orbits Bull. Amer. Math. Soc. (N.S.) 2013 187 228

[36] Kontorovich, Alex Levels of distribution and the affine sieve Ann. Fac. Sci. Toulouse Math. (6) 2014 933 966

[37] Kashiwara, Masaki, Tanisaki, Toshiyuki Kazhdan-Lusztig conjecture for affine Lie algebras with negative level Duke Math. J. 1995 21 62

[38] Kashiwara, Masaki, Tanisaki, Toshiyuki Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. II. Nonintegral case Duke Math. J. 1996 771 813

[39] Libedinsky, Nicolas Presentation of right-angled Soergel categories by generators and relations J. Pure Appl. Algebra 2010 2265 2278

[40] Libedinsky, Nicolas Light leaves and Lusztig’s conjecture Adv. Math. 2015 772 807

[41] Lusztig, George Some problems in the representation theory of finite Chevalley groups 1980 313 317

[42] Lusztig, George Monodromic systems on affine flag manifolds Proc. Roy. Soc. London Ser. A 1994 231 246

[43] Salehi Golsefidy, Alireza, Sarnak, Peter The affine sieve J. Amer. Math. Soc. 2013 1085 1105

[44] Soergel, Wolfgang Kategorie 𝒪, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe J. Amer. Math. Soc. 1990 421 445

[45] Soergel, Wolfgang The combinatorics of Harish-Chandra bimodules J. Reine Angew. Math. 1992 49 74

[46] Soergel, Wolfgang Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules Represent. Theory 1997 83 114

[47] Soergel, Wolfgang On the relation between intersection cohomology and representation theory in positive characteristic J. Pure Appl. Algebra 2000 311 335

[48] Soergel, Wolfgang Langlands’ philosophy and Koszul duality 2001 379 414

[49] Soergel, Wolfgang Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen J. Inst. Math. Jussieu 2007 501 525

[50] Williamson, Geordie, Braden, Tom Modular intersection cohomology complexes on flag varieties Math. Z. 2012 697 727