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Bob Howlett: research

Representation theory of finite groups of Lie type

I began working in this area as a PhD student. My supervisor, R. J. Clarke, had shown that many finite symplectic groups do not have doubly transitive permutation representations; I set about attempting to extend this result to a larger class of groups. It was known that the classification of doubly transitive permutation representations of finite Chevalley groups could be completed if it could be shown that the nontrivial principal series characters all have degree divisible by the natural characteristic of the group. I succeeded in proving this in paper L1. In my thesis I extended the results to include generalized principal series characters, by describing the structure of relevant endomorphism algebras. These results are published in papers L3 and L4 (my co-author, R. W. Kilmoyer, having obtained similar results independently).

Following a suggestion of Gus Lehrer, I attempted to apply the ideas of my earlier papers to investigate endomorphism algebras of induced cuspidal representations, with a view to proving a conjecture of Springer. After I had developed an auxiliary theory on the structure of normalizers of parabolic subgroups of finite reflection groups (see paper R1), Lehrer and I were able to obtain presentations for the the relevant endomorphism algebras, and prove Springer's conjecture. This work appears in paper L5, and is also described in Chapter 10 of R. W. Carter's book Finite Groups of Lie Type: conjugacy classes and complex characters.

In paper R2 Lehrer and I describe a duality operation in the character ring of the normalizer of a parabolic subgroup of a finite reflection group, paralleling Curtis-Alvis duality for groups of Lie type, via the character correspondence derived from the theory dealt with in L5. Papers L6 and L7 (also joint with Lehrer) are devoted to proving naturality of this correspondence, and, in particular, that it preserves inner products. In papers L8 and L9, which are not closely related to the earlier Howlett-Lehrer papers, we describe natural embeddings of endomorphism algebras of certain permutation representations into the group algebra, without requiring that the characteristic of the ground ring be zero. Paper L10 is also concerned with representation theory over fields of nonzero characteristic, and extension of Harish-Chandra theory to such cases.

Paper L11, written jointly with Charles Zworestine, presents the very elegant proof of a theorem of Klyachko that forms the major part of Charles' PhD thesis. The theorem is that a certain sum of induced characters of GL(n,q) contains every irreducible character with multiplicity exactly 1. (The proof in the paper contains a significant shortening of the version that appeared in Charles' thesis.)

Finite Coxeter groups

My interest in these derived initially from their connections with Lie theory, but now extends beyond those connections. My work on normalizers of parabolic subgroups, in papers R1 and R2, although motivated originally by Lie theory, is of interest in its own right, and extends naturally to infinite Coxeter groups—cf. papers by V. Deodhar (The root system of a Coxeter group, Communications in Algebra, 10 (1982) 611–630), A. Cohen (Recent results on Coxeter groups, in Polytopes: abstract, convex and computational, Kluwer Acad. Publ. (Dordrecht, 1994)), B. Brink (Centralizers of reflections in Coxeter groups, Bull Lond. Math. Soc. 28 (1996) 465–470), and also a joint paper by B. Brink and myself: Normalizers of parabolic subgroups in Coxeter groups, Inventiones Mathematicae 136 (1999) 323–351.

In the paper R4 Gus Lehrer and I prove that the elements of minimal reflection length in certain cosets of parabolic subgroups form a single conjugacy class. (In the course of his investigations of representation theory of reductive groups Gus observed that this result had to be true; hence we were motivated to seek an elementary proof.)

The paper R3, with F. Bergeron, N. Bergeron and D. E. Taylor, investigates idempotents in the Solomon (or descent) algebras of finite Coxeter groups.

Infinite Coxeter groups

Paper C1 was concerned with Coxeter elements and their eigenvalues. In C2 I gave a unified description of the Schur multipliers of all Coxeter groups of finite rank; this extended the work of Ihara and Yokonuma (J. Fac. Sci. Univ. Tokyo Sect. I 11 (1965), 155–171) and Yokonuma (ibid. Sect. I 11 (1965), 173–186) on Euclidean and affine reflection groups, avoiding case by case arguments, and describing the answer in terms of invariants that can be readily found by an inspection of the Coxeter diagram.

My former PhD student Brigitte Brink and I proved in paper C3 that Coxeter groups are automatic. The key idea in this work was the introduction of a partial order, which we call dominance, on the root system of the group, and the main theorem is that the set of elementary roots (positive roots which do not dominate other positive roots) is finite. (Confession: the proof given in C3 of Proposition 1.3 of that paper is flawed, although the result is correct and proved in the exercises of Bourbaki. The error is corrected in the version you can download.) Dominance is used by D. Krammer in his PhD thesis (Utrecht, 1994), and also by Peter Rowley, Don Taylor and myself in C4. In this latter paper we show that if there are no infinite bonds in the Coxeter diagram of a finitely generated Coxeter group W then the outer automorphism group of W is finite. We also show that if W is irreducible then the bases of the root system form a single orbit under the action of the group generated by W and -1. (J. Y. Hée has also proved this.) The set of elementary roots is completely described in all cases by Brink in her PhD thesis (University of Sydney, 1994) and in J. Algebra 206 (1998), no. 2, 371–412. (A preliminary version of this paper, which in my opinion is in some ways better than the published version is available as a University of Sydney algebra preprint entitled The set of dominance minimal roots.)

In C7 Brigitte Brink and I describe how to obtain a presentation for the normalizer of a parabolic subgroup of an arbitrary Coxeter group. This extends the work of Deodhar (Communications in Algebra, 10 (1982) 611–630) in which generators were given but not defining relations.

In C8 Bill Franzsen and I show that the automorphism group of an infinite Coxeter group of rank 3 whose Coxeter diagram has no infinite bonds is generated by inner and graph automorphisms. That is, there are never any interesting outer automorphisms. Bill has developed this further in his PhD thesis, extending also the work of Howlett, Rowley and Taylor in C4.

I gave some introductory lectures on Coxeter groups at workshops for postgraduate students which were held at the Australian National University in 1993 and 1996. I can supply copies of the texts of these lectures (C5 and C6).

Representation theory of finite solvable groups

This is a topic in which I had some interest in from the start of my career. In S1 I gave a simplified proof of a theorem of E. C. Dade (Characters of groups with normal extra special subgroups Math. Zeit. 152 (1976), 1–31) giving a criterion for a character of certain normal subgroups to be extendible to the whole group. This result has close connections with the theory of Weil representations of symplectic groups, a topic to which I returned in paper S4, in collaboration with S. P. Glasby. Here our main contribution was the discovery of a new formula for the sign of the character value at an arbitrary element. We used the theory of Weil representations to construct some interesting solvable groups.

My main work in this general area, done in collaboration with I. M. Isaacs, consisted in proving that groups which possess an irreducible complex character of degree equal to the square root of the index of the centre are necessarily solvable (paper S3), which had been a conjecture of long standing.

Paper S2 can perhaps also be classified as concerned with representation theory of finite solvable groups. The problem, which derived from the work of my co-authors on analysis on compact groups, was this: is it true that for every finite group G there exists a function from G to the unit circle which is orthogonal to all its translates? We proved that this is true for finite solvable groups; the general question remains unanswered.

Paper S5 deals with the following computational problem: given matrices which generate an absolutely irreducible subgroup of the general linear group of degree n over a finite field F, find a similarity transformation which simultaneously carries these to matrices whose entries lie in a subfield of F which is as small as possible. This was applied to give an algorithm for the construction of the absolutely irreducible characteristic p representations of a finite soluble group, each written over its minimal field.

Miscellaneous topics

In M3 Richard Levingston and I found a simple base for the laws of a certain variety of metabelian groups, L. G. Kovács and I M4 obtained an inequality relating the dimensions of two cohomology groups, and D. F. Holt and I M1 gave a bound on the exponent of a group that factorizes as a product of two abelian groups with given exponents. I subsequently improved this bound to the best possible in M2.

The paper M5 (mainly the work of my co-author Jian-yi Shi) is devoted to showing that a certain property of the root system characterizes the finite real reflection groups amongst the finite unitary relection groups.

PhD Students

Clive Saunders, PhD awarded 1989, thesis The reflection representation of certain Chevalley groups. Clive, who now works in the financial sector, gave an explicit construction of the principal series representation that corresponds to the natural representation of the Weyl group, for Chevalley groups of types A, D and E. Reprints of his paper, which appeared in J. Algebra 137 1991, 145–165, can be obtained from me.

Charles Zworestine, PhD awarded 1993, thesis Bilinear forms and the representation theory of general linear groups. Charles gave an elegant alternative account of the model for the characters of finite general linear groups discovered by A. A. Klyachko (Models for the complex representations of the groups GL(n,q), Math. USSR Sbornik 48 (1984), 365–379), and discussed by Inglis and Saxl (An explicit model for the the complex representations of the finite general linear groups, Archiv der Math. 57 (1991), 424–431). We eventually completed a joint paper on this topic (L11), published in the Proceedings of the International Conference on Representation Theory that was held in Shanghai in July 1998.

Brigitte Brink, PhD awarded 1994, thesis On root systems and automaticity of Coxeter groups. Brigitte currently has a position at the Justus Liebig University, Giessen, Germany. See the sections on infinite Coxeter groups and finite Coxeter groups above for a brief description of some of the results contained in her thesis.

Ilknur Tulunay, PhD awarded 2001, thesis Cuspidal modules of finite general linear groups. Ilknur's aim was to provide an elementary and explicit construction of cuspidal modules of finite general linear groups. Her principal results are a new formula for certain so-called "Bessel functions" (matrix coefficients) for cuspidal representations, and a description of the irreducible constituents of the restriction of a cuspidal representation to the subgroup stabilizing a hyperplane and a point not on that hyperplane. (This restriction turns out to be multiplicity-free.) She is preparing two papers on these topics.

Bill Franzsen, PhD awarded 2001, thesis Automorphisms of Coxeter groups. Bill's main focus was on infinite Coxeter groups of finite rank such that all rank two parabolic subgroups are finite. We are yet to find such a group with an outer automorphism that is not a graph automorphism. (A graph automorphism is an automorphism derived from a symmetry of the Coxeter diagram: in other words, it is an automorphism that permutes a set of simple reflections.) Bill proved that Coxeter groups that are "nearly finite", in the sense that they have a finite maximal parabolic subgroup, and which have the property that all rank two parabolic subgroups are finite, do indeed have no outer automorphisms other than graph automorphisms. He also proved a number of other results in a similar vein. Along the way he gave what is in my view the definitive account of the automorphism groups of the finite Coxeter groups.

I currently have three PhD students: Yunchuan Yin, who is working on the construction of explicit representations of Hecke Algebras, Shona Yu, who is studying representations of algebras such as the Birman-Murakami-Wenzl algebra, and Mauro Grassi, who is going to study Coxeter groups from a geometrical perspective (and then explain it to me).

Publications

Representation theory of finite groups of Lie type

L1 On the Degrees of Steinberg Characters of Chevalley Groups, Mathematische Zeitschrift 135 (1974) 125–135. MR 50 #13228.
L2 Some irreducible characters of groups with BN pairs, PhD thesis, University of Adelaide (1976).
L3 Some irreducible characters of groups with BN pairs, Journal of Algebra 39 (1976) 571–592. MR 54 #388.
L4 Principal series representations of finite groups with split BN pairs, Communications in Algebra 8 (1980) 543–583. (Co-author R. W. Kilmoyer.) MR 81d 20009.
L5 Induced cuspidal representations and generalised Hecke rings, Inventiones Mathematicae 58 (1980) 37–64. (Co-author G. I. Lehrer.) MR 81j 20017.
L6 A comparison theorem and other formulae in the character ring of a finite group of Lie type, Contemporary Mathematics 9 (1982) 285–288. (Co-author G. I. Lehrer.) MR 84b 20055.
L7 Representations of generic algebras and finite groups of Lie type, Transactions of the American Mathematical Society 280 (1983) 753–779. (Co-author G. I. Lehrer.) MR 85i 20044.
L8 Embeddings of Hecke Algebras in group algebras, Journal of Algebra 105 (1987) 159–174. (Coauthor G. I. Lehrer.) MR 88c 20019.
L9 On the integral group algebra of a finite algebraic group, Astérisque 168 (1988) 141–155. (Co-author G. I. Lehrer.) (Preprint version on line as a dvi file.) MR 91b 20063.
L10 On Harish-Chandra induction and restriction for modules of Levi subgroup Journal of Algebra 165 (1994) 172–183. (Co-author G. I. Lehrer.) (Preprint version on line as a dvi file.) MR 95d 20025.
L11 Klyachko's model for the representations of finite general linear groups, in Representations and Quantizations (CHEP Beijing and Springer-Verlag Berlin Heidelberg, 2000): proceedings of the International Conference on Representation Theory, held in Shanghai in 1998. (Co-author C. Zworestine.) (Preprint version on line as a pdf file.)
L12 Matrix generators for exceptional groups of Lie type. Journal of Symbolic Computation 31 (2001), no. 4, 429–445. (Co-authors L. J. Rylands and D. E. Taylor.) (Preprint version on line as a pdf file.)

Finite Coxeter groups

R1 Normalizers of parabolic subgroups of reflection groups, Journal of the London Mathematical Society (2) 21 (1980) 62–80. MR 81g 20094.
R2 Duality in the normalizer of a parabolic subgroup of a finite Coxeter group, Bulletin of the London Mathematical Society 14 (1982) 133–136. (Co-author G. I. Lehrer.) MR 83e 20049.
R3 A decomposition of the descent algebra of a finite Coxeter group, Journal of Algebraic Combinatorics 1 (1992) 23–44. (Co-authors F. Bergeron, N. Bergeron and D. E. Taylor.) MR 93g 20079.
R4 On reflection length in reflection groups Archiv de Mathematik 73 (1999) 321–326. (Co-author G. I. Lehrer.) Preprint on line in pdf form. MR 2000g:20097.

Infinite Coxeter groups

C1 Coxeter groups and M-matrices, Bulletin of the London Mathematical Society 14 (1982) 137–141 MR 83g 20032.
C2 On the Schur multipliers of Coxeter groups, J. London Math. Soc. (2) 38 (1988) 263–276. (Preprint version on line as a dvi file.) MR 90e 20010.
C3 A finiteness property and an automatic structure for Coxeter groups, Mathematische Annalen 296 (1993) 179–190. (Co-author Brigitte Brink.) (Preprint version on line as a dvi file.) MR 94d 20045.
C4 On the outer automorphism groups of Coxeter groups, Manuscripta Mathematica 93 (1997) 499–513. (Co-authors P. J. Rowley and D. E. Taylor.) MR 98j 20056.
C5 Miscellaneous facts about Coxeter groups (Lectures delivered at the A.N.U. Group Actions Workshop 1993), University of Sydney Mathematics Research Report no. 93–38. (Preprint on line as a postscript file.)
C6 Introduction to Coxeter groups (Lectures delivered at the A.N.U. Geometric Group Theory Workshop, January/February 1996). (Preprint on line as a postscript file.)
C7 Normalizers of parabolic subgroups in Coxeter groups Inventiones Math. 136 (1999) 323–351. (Co-author B. Brink.) MR 2000b:20048.
C8 Automorphisms of Rank 3 Coxeter Groups, Proc. Amer. Math. Soc. 129 (2001) no. 9 2607–2616. (Co-author W. Franzsen.) Preprint version on line in pdf form.

Representation theory of finite solvable groups

S1 Extending characters from normal subgroups, Topics in Algebra (the proceedings of the 18th Summer Research Institute of the Australian Mathematical Society), Lecture Notes in Mathematics 697 (Springer-Verlag, 1978). MR 80b 20009.
S2 Extreme values for the Sidon constant, Proceedings of the American Mathematical Society 81 (1981) 531–537. (Co-authors D. I. Cartwright and J. R. McMullen.) MR 82c 43005.
S3 On Groups of Central Type, Mathematische Zeitschrift 179 (1982) 555–569. (Co-author I. M. Isaacs.) MR 83j 20020.
S4 Extraspecial towers and Weil representations, Journal of Algebra 151 (1992) 236–260. (Co-author S. P. Glasby.) MR 93j 20072.
S5 Writing representations over minimal fields, Communications in Algebra 25 (1997) 1703–1711. (Co-author S. P. Glasby.) MR 98c 20019

Miscellaneous

M1 On groups which are the product of two Abelian groups, Journal of the London Mathematical Society (2) 29 (1984) 453–461. (Co-author D. F. Holt.) MR 86f 20028.
M2 On the exponent of certain factorizable groups, Journal of the London Mathematical Society (2) 31 (1985) 265–271. MR 87b 20040.
M3 On the laws of certain varieties of groups, Bulletin of the Australian Mathematical Society 31 (1985) 145–154. (Co-author R. W. Levingston.) MR 86e 20030.
M4 On the first cohomology of a group with coefficients in a simple module, Journal of Algebra 99 (1985) 518–519 (Co-author L. G. Kovács.) MR 87h 20098.
M5 On regularity of finite reflection groups, Manuscripta Mathematica 102 (2000), no. 3, 325–333. (Co-author J-y. Shi.) (Preprint version on line in pdf form.) MR 2001f:20081.