REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE By HUNG NGOC NGUYEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 1 °c 2008 Hung Ngoc Nguyen 2 ACKNOWLEDGMENTS I am indebted to my advisor, Prof. Pham Huu Tiep, who has given me constant encouragement and devoted guidance over the last four years. I would like to thank Prof. Alexander Dranishnikov, Prof. Peter Sin, Prof. Meera Sitharam and Prof. Alexandre Turull, for serving on my supervisory committee and giving me valuable suggestions. I am also pleased to thank the Department of Mathematics at the University of Florida and the College of Liberal Arts and Sciences for honoring me with the 4-year Alumni Fellowship and the CLAS Dissertation Fellowship. Finally, I thank my parents, my sister, and my wife for all their love and support. 3 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................. 3 LIST OF TABLES ..................................... 6 LIST OF SYMBOLS .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1 INTRODUCTION .................................. 9 1.1 Overview and Motivation ............................ 9 1.2 Results ...................................... 11 2 IRREDUCIBLE RESTRICTIONS FOR G2(q) ................... 15 2.1 Preliminaries .................................. 16 2.2 Degrees of Irreducible Brauer Characters of G2(q) .............. 19 2.3 Proofs ...................................... 21 2.4 Small Groups G2(3) and G2(4) ......................... 28 3 IRREDUCIBLE RESTRICTIONS FOR SUZUKI AND REE GROUPS ..... 56 3.1 Suzuki Groups .................................. 56 3.2 Ree Groups ................................... 57 3 4 IRREDUCIBLE RESTRICTIONS FOR D4(q) .................. 60 4.1 Basic Reduction ................................. 60 3 p 4.2 Restrictions to G2(q) and to D4( q) ..................... 62 4.3 Restrictions to Maximal Parabolic Subgroups ................ 64 5 LOW-DIMENSIONAL CHARACTERS OF THE SYMPLECTIC GROUPS .. 67 5.1 Preliminaries .................................. 67 5.1.1 Strategy of the Proofs .......................... 67 5.1.2 Centralizers of Semi-simple Elements ................. 68 5.2 Unipotent Characters .............................. 75 5.2.1 Unipotent Characters of SO2n+1(q) and P CSp2n(q) ......... 75 ¡ 0 5.2.2 Unipotent Characters of P (CO2n(q) ) ................. 83 + 0 5.2.3 Unipotent Characters of P (CO2n(q) ) ................. 88 5.3 Non-unipotent Characters ........................... 93 6 LOW-DIMENSIONAL CHARACTERS OF THE ORTHOGONAL GROUPS . 106 6.1 Odd Characteristic Orthogonal Groups in Odd Dimension ......... 106 4 6.2 Even Characteristic Orthogonal Groups in Even Dimension ......... 109 6.3 Odd Characteristic Orthogonal Groups in Even Dimension ......... 111 § 6.4 Groups Spin12(3) ................................ 115 REFERENCES ....................................... 118 BIOGRAPHICAL SKETCH ................................ 123 5 LIST OF TABLES Table page 2-1 Degrees of irreducible complex characters of G2(q) ................. 48 2-2 Degrees of 2-Brauer characters of G2(q), q odd ................... 49 2-3 Degrees of 3-Brauer characters of G2(q), 3 - q ................... 50 2-4 Degrees of `-Brauer characters of G2(q), ` ¸ 5 and ` - q .............. 51 2-5 Degrees of `-Brauer characters of G2(q), ` ¸ 5 and ` - q .............. 52 2-6 Fusion of conjugacy classes of P in G2(3) ...................... 53 2-7 Fusion of conjugacy classes of P in G2(4) ...................... 54 2-8 Fusion of conjugacy classes of Q in G2(4) ...................... 55 5-1 Low-dimensional unipotent characters of Sp2n(q), n ¸ 6, q odd, I ........ 103 5-2 Low-dimensional irreducible characters of Sp2n(q), n ¸ 6, q odd, II ....... 104 5-3 Low-dimensional irreducible characters of Sp2n(q), n ¸ 6, q odd, III ....... 105 6 LIST OF SYMBOLS, NOMENCLATURE, OR ABBREVIATIONS Let G be a ¯nite group, ` be a prime number, and F be an algebraically closed ¯eld of characteristic `. We write Z(G): for the center of G O`(G): for the maximal normal `-subgroup of G Irr(G): for the set of irreducible complex characters of G, or the set of irreducible CG-representation IBr`(G): for the set of irreducible `-Brauer characters of G, or the set of absolutely irreducible G-representation in characteristic ` d`(G): for the smallest degree of irreducible FG-modules of dimension > 1 m`(G): for the largest degree of irreducible FG-modules d2;`(G): for the second smallest degree of irreducible FG-modules of dimension > 1 dC(G); mC(G); d2;C(G): for d0(G), m0(G), d2;0(G), respectively Âb : for the restriction of a character  2 Irr(G) to `-regular elements of G Furthermore, we write + GLn (q): for GLn(q) ¡ GLn (q): for GUn(q) CSp2n(q): see the discussion before Lemma 5.1.4 P CSp2n(q): for the quotient of CSp2n(q) by its center § § 0 CO2n(q);CO2n(q) : see the discussion before Lemma 5.1.5 § 0 § 0 P (CO2n(q) ): for the quotient of CO2n(q) by its center § Spinn (q): for the Spin group. Modulo some exceptions, it is the universal cover of the § simple orthogonal group P ­n (q) semi-simple element: for an element which is diagonalizable unipotent character: see the discussion at the beginning of x5:1 7 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Ful¯llment of the Requirements for the Degree of Doctor of Philosophy REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE By Hung Ngoc Nguyen August 2008 Chair: Pham Huu Tiep Major: Mathematics Let G be a ¯nite quasi-simple group of Lie type. One of the important problems in modular representation theory is to determine when the restriction of an absolutely irreducible representation of G to its proper subgroups is still irreducible. The solution of this problem is a key step towards classifying all maximal subgroups of ¯nite classical groups. We solve this problem for the cases where G is a Lie group of the following types: 2 2 3 G2(q), B2(q), G2(q), and D4(q). One of the main tools to approach the above problem, and many others, is the classi¯cation of low-dimensional representations of ¯nite groups of Lie type. Low-dimensional complex representations were ¯rst studied in [70] for ¯nite classical groups and then in [50] for exceptional groups. We extend the results in [70] for symplectic and orthogonal groups to a larger bound. More explicitly, we classify irreducible complex characters of the § symplectic groups Sp2n(q) and orthogonal groups Spinn (q) of degrees up to the bound D, where D = (qn ¡ 1)q4n¡10=2 for symplectic groups, D = q4n¡8 for orthogonal groups in odd dimension, and D = q4n¡10 for orthogonal groups in even dimension. 8 CHAPTER 1 INTRODUCTION 1.1 Overview and Motivation Finite primitive permutation groups have been studied since the pioneering work of Galois and Jordan on group theory; they have had important applications in many di®erent areas of mathematics. If G is a primitive permutation group with a point stabilizer M then M is a maximal subgroup of G. Thanks to works of Aschbacher, O'Nan, Scott [3], and of Liebeck, Praeger, Saxl, and Seitz [47], [48], most problems involving such a G can be reduced to the case where G is a ¯nite classical group. If G is a ¯nite classical group, a fundamental theorem of Aschbacher [2] states that any maximal subgroup M of G is a member of either one of eight families Ci; 1 · i · 8, of naturally de¯ned subgroups of G or the collection S of 8 certain quasi-simple subgroups of G. Conversely, if M 2 [i=1Ci, then the maximality of M has been determined by Kleidman and Liebeck in [40]. It remains to determine which M 2 S are indeed maximal subgroups of G. This question leads to a number of important problems concerning modular representations of ¯nite quasi-simple groups. One of them is the irreducible restriction problem. Problem A. Let F be an algebraically closed ¯eld of characteristic `. Classify all triples (G; V; M) where G is a ¯nite quasi-simple group, V is an irreducible FG-module of dimension greater than one, and M is a proper subgroup of G such that the restriction V jM is irreducible. We recall that a ¯nite group G is called quasi-simple if G = [G; G] and G=Z(G) is simple. Moreover, any non-abelian ¯nite simple groups is one of alternating groups, 26 sporadic groups, or ¯nite simple groups of Lie type. The solution for Problem A when G=Z(G) is a sporadic group is largely computational. When G is a cover of a symmetric or alternating group, Problem A is quite complicated and almost done in [6], [41], [42], and [57]. Our main focus is on the case where G is a 9 ¯nite group of Lie type in characteristic di®erent from `. This has been solved recently in [43] when G is of type A. One of the important ingredients needed to solve Problem A is the classi¯cation of low-dimensional irreducible representations of ¯nite groups of Lie type. This is the reason why we have been working simultaneously on the irreducible restriction problem and the problem of determining low-dimensional representations. Suppose that G = G(q) is a ¯nite group of Lie type de¯ned over a ¯eld of q elements, where q = pn is a prime power. Lower bounds for the degrees of nontrivial irreducible representations of G in cross characteristic ` (i.e. ` 6= p) were found by Landazuri, Seitz and Zalesskii in [45], [59] and improved later by many people. These bounds have proved to be very useful in various applications. We are interested in not only the smallest representation, but more importantly, the low-dimensional representations. Low-dimensional complex representations were ¯rst studied by Tiep and Zalesskii [70] for ¯nite classical groups and then by LÄubeck [50] for exceptional groups. For representations over ¯elds of cross characteristics, this problem has been studied recently in [5], [23] for SLn(q); [25], [32] for SUn(q); [25] for Sp2n(q) with q odd; and [24] for Spn(q) with q even.