Geodesic
CHARACTER LEVELS AND CHARACTER BOUNDS
Forum of Mathematics, Pi, Tome 8 (2020)

Voir la notice de l'article provenant de la source Cambridge University Press

We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods. 
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ROBERT M. GURALNICK; MICHAEL LARSEN; PHAM HUU TIEP. CHARACTER LEVELS AND CHARACTER BOUNDS. Forum of Mathematics, Pi, Tome 8 (2020). doi: 10.1017/fmp.2019.9

[AH] Arad, Z. and Herzog, M. (Eds.), Products of Conjugacy Classes in Groups, Springer Lecture Notes, 1112 (Springer, Berlin, 1985).Google Scholar | DOI

[A] Aschbacher, M., Finite Group Theory, 2nd edn, Cambridge Studies in Advanced Mathematics, 10 (Cambridge University Press, Cambridge, 2000).Google Scholar | DOI

[BLST] Bezrukavnikov, R., Liebeck, M. W., Shalev, A. and Tiep, P. H., ‘Character bounds for finite groups of Lie type’, Acta Math. 221 (2018), 1–57.Google Scholar | DOI

[BDK] Brundan, J., Dipper, R. and Kleshchev, A. S., ‘Quantum linear groups and representations of [[()[]mml:mo lspace="1em" rspace="0em"[]()]]GL[[()[]/mml:mo[]()]](𝔽)’, Mem. Amer. Math. Soc. 149(706) (2001).Google Scholar

[C] Carter, R., Finite Groups of Lie type: Conjugacy Classes and Complex Characters, (Wiley, Chichester, 1985).Google Scholar

[DS] Diaconis, P. and Shahshahani, M., ‘Generating a random permutation with random transpositions’, Z. Wahrsch. Verw. Gebiete 57 (1981), 159–179.Google Scholar | DOI

[DM1] Digne, F. and Michel, J., ‘Foncteurs de Lusztig et caractères des groupes linéaires et unitaires sur un corps fini’, J. Algebra 107 (1987), 217–255.Google Scholar | DOI

[DM2] Digne, F. and Michel, J., Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts, 21 (Cambridge University Press, Cambridge, 1991).Google Scholar | DOI

[E] Ennola, V., ‘On the characters of the finite unitary groups’, Ann. Acad. Sci. Fenn. Ser. A I(323) (1963), 35 pages.Google Scholar

[FS] Fong, P. and Srinivasan, B., ‘The blocks of finite general and unitary groups’, Invent. Math. 69 (1982), 109–153.Google Scholar | DOI

[FG1] Fulman, J. and Guralnick, R. M., ‘Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements’, Trans. Amer. Math. Soc. 364 (2012), 3023–3070.Google Scholar | DOI

[FG2] Fulman, J. and Guralnick, R. M., ‘Enumeration of commuting pairs in Lie algebras over finite fields’, Ann. Comb. 22 (2018), 295–316.Google Scholar | DOI

[Ga] Gantmacher, F. R., Matrix Theory, Vol. 2 (Chelsea, New York, 1977).Google Scholar

[Geck] Geck, M., ‘On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes’, Doc. Math. 1 (1996), 293–317.Google Scholar

[GM] Geck, M. and Malle, G., ‘On the existence of a unipotent support for the irreducible characters of a finite group of Lie type’, Trans. Amer. Math. Soc. 352 (2000), 429–456.Google Scholar | DOI

[GP] Geck, M. and Pfeiffer, G., Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras, London Mathematical Society Monographs (Oxford University Press, Oxford, 2000).Google Scholar

[Ge] Gérardin, P., ‘Weil representations associated to finite fields’, J. Algebra 46 (1977), 54–101.Google Scholar | DOI

[GKNT] Giannelli, E., Kleshchev, A. S., Navarro, G. and Tiep, P. H., ‘Restriction of odd degree characters and natural correspondences’, Int. Math. Res. Not. IMRN 2017(20) (2017), 6089–6118.Google Scholar

[Gr] Gross, B. H., ‘Group representations and lattices’, J. Amer. Math. Soc. 3 (1990), 929–960.Google Scholar | DOI

[Gu] Guralnick, R. M., ‘On the singular value decomposition over finite fields and orbits of ’, Preprint, 2018, .Google Scholar

[GLT] Guralnick, R. M., Larsen, M. and Tiep, P. H., ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, .Google Scholar

[GMST] Guralnick, R. M., Magaard, K., Saxl, J. and Tiep, P. H., ‘Cross characteristic representations of symplectic groups and unitary groups’, J. Algebra 257 (2002), 291–347. Addendum, J. Algebra (2006), 443–446.Google Scholar | DOI

[GT] Guralnick, R. M. and Tiep, P. H., ‘Cross characteristic representations of even characteristic symplectic groups’, Trans. Amer. Math. Soc. 356 (2004), 4969–5023.Google Scholar | DOI

[GH1] Gurevich, S. and Howe, R., ‘Small representations of finite classical groups’, inRepresentation Theory, Number Theory, and Invariant Theory, Progress in Mathematics, 323 (Birkhäuser/Springer, Cham, 2017), 209–234.Google Scholar | DOI

[GH2] Gurevich, S. and Howe, R., ‘Rank and duality in representation theory’, Preprint.Google Scholar

[Hw] Howe, R. E., ‘On the characters of Weil’s representations’, Trans. Amer. Math. Soc. 177 (1973), 287–298.Google Scholar

[Is] Isaacs, I. M., Character Theory of Finite Groups (AMS-Chelsea, Providence, 2006).Google Scholar

[J1] James, G., ‘The irreducible representations of the finite general linear groups’, Proc. Lond. Math. Soc. (3) 52 (1986), 236–268.Google Scholar | DOI

[J2] James, G., ‘The decomposition matrices of [[()[]mml:mo lspace="1em" rspace="0em"[]()]]GL[[()[]/mml:mo[]()]](q) for n⩽10’, Proc. Lond. Math. Soc. (3) 60 (1990), 225–265.Google Scholar | DOI

[JK] James, G. and Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16 (Addison-Wesley Publishing Co., Reading, MA, 1981).Google Scholar

[K] Kawanaka, N., ‘Generalized Gelfand-Graev representations of exceptional simple algebraic groups over a finite field. I’, Invent. Math. 84 (1986), 575–616.Google Scholar | DOI

[KT1] Kleshchev, A. S. and Tiep, P. H., ‘Representations of finite special linear groups in non-defining characteristic’, Adv. Math. 220 (2009), 478–504.Google Scholar | DOI

[KT2] Kleshchev, A. S. and Tiep, P. H., ‘Representations of general linear groups which are irreducible over subgroups’, Amer. J. Math. 132 (2010), 425–473.Google Scholar | DOI

[Ku] Kupershmidt, B. A., ‘q-Newton binomial: from Euler to Gauss’, J. Nonlinear Math. Phys. 7 (2000), 244–262.Google Scholar | DOI

[LMT] Larsen, M., Malle, G. and Tiep, P. H., ‘The largest irreducible representations of simple groups’, Proc. Lond. Math. Soc. (3) 106 (2013), 65–96.Google Scholar | DOI

[LST] Larsen, M., Shalev, A. and Tiep, P. H., ‘Probabilistic Waring problems for finite simple groups’, Ann. of Math. (2) 190 (2019), 561–608.Google Scholar | DOI

[LBST1] Liebeck, M. W., O’Brien, E. A., Shalev, A. and Tiep, P. H., ‘The Ore conjecture’, J. Eur. Math. Soc. (JEMS) 12 (2010), 939–1008.Google Scholar | DOI

[LBST2] Liebeck, M. W., O’Brien, E. A., Shalev, A. and Tiep, P. H., ‘Products of squares in finite simple groups’, Proc. Amer. Math. Soc. 140 (2012), 21–33.Google Scholar | DOI

[LS] Liebeck, M. W. and Shalev, A., ‘Character degrees and random walks in finite groups of Lie type’, Proc. Lond. Math. Soc. (3) 90 (2005), 61–86.Google Scholar | DOI

[L] Lusztig, G., Characters of Reductive Groups over a Finite Field, Annals of Mathematics Studies, 107 (Princeton Univ. Press, Princeton, 1984).Google Scholar | DOI

[M] Macdonald, I. G., Symmetric Functions and Hall Polynomials, 2nd edn, Oxford Mathematical Monographs (Oxford University Press, New York, 1995), with contributions by A. Zelevinsky.Google Scholar

[Ma] Malle, G., ‘Unipotente Grade imprimitiver komplexer Spiegelungsgruppen’, J. Algebra 177 (1995), 768–826.Google Scholar | DOI

[Ol] Olsson, J. B., ‘Remarks on symbols, hooks and degrees of unipotent characters’, J. Combin. Theory Ser. A 42 (1986), 223–238.Google Scholar | DOI

[Sp] De Seguins Pazzis, C., ‘Invariance of simultaneous similarity and equivalence of matrices under extension of the ground field’, Linear Algebra Appl. 433 (2010), 618–624.Google Scholar | DOI

[Sh] Shalev, A., ‘Commutators, words, conjugacy classes and character methods’, Turk. J. Math. 31 (2007), 131–148.Google Scholar

[ST] Shalev, A. and Tiep, P. H., ‘Some conjectures on Frobenius’ character sum’, Bull. Lond. Math. Soc. 49 (2017), 895–902.Google Scholar | DOI

[S1] Srinivasan, B., ‘The modular representation ring of a cyclic p-group’, Proc. Lond. Math. Soc. (3) 14 (1974), 677–688.Google Scholar

[S2] Srinivasan, B., ‘Weil representations of finite classical groups’, Invent. Math. 51 (1979), 143–153.Google Scholar | DOI

[TaZ] Taussky, O. and Zassenhaus, H., ‘On the similarity transformation between a matrix and its transpose’, Pacific J. Math. 9 (1959), 893–896.Google Scholar | DOI

[TT] Taylor, J. and Tiep, P. H., ‘Lusztig induction, unipotent support, and character bounds’, Preprint, 2018, .Google Scholar

[ThV] Thiem, N. and Vinroot, C. R., ‘On the characteristic map of finite unitary groups’, Adv. Math. 210 (2007), 707–732.Google Scholar | DOI

[T1] Tiep, P. H., ‘Dual pairs of finite classical groups in cross characteristic’, Contemp. Math. 524 (2010), 161–179.Google Scholar | DOI

[T2] Tiep, P. H., ‘Weil representations of finite general linear groups and finite special linear groups’, Pacific J. Math. 279 (2015), 481–498.Google Scholar | DOI

[TZ] Tiep, P. H. and Zalesskii, A. E., ‘Some characterizations of the Weil representations of the symplectic and unitary groups’, J. Algebra 192 (1997), 130–165.Google Scholar | DOI

[W] Weil, A., ‘Sur certains groupes d’opérateurs unitaires’, Acta Math. 111 (1964), 143–211.Google Scholar | DOI

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