# Collaborations

Although more than 20 items in Scott's bibliography are singly-authored, he often has preferred to work in collaboration. His principal collaborators are listed alphabetically below, followed by synopses of the collaborations, indexed to his bibliography. (You can access that bibliography using the button at the bottom of this page. Then, you can return to this page using a browser "back" button.) Some newer collaborations may not yet be listed.

- Michael Aschbacher, Professor of Mathematics, California Institute of Technology (NAS)
- George Avrunin, Professor of Mathematics, University of Massachusetts
- Christopher Bendel, Ph.D and Mathematics Department Chair, University of Wisconsin-Stout
- Edward Cline, Professor of Mathematics Emeritus (deceased), University of Oklahoma
- Charles Curtis, Professor of Mathematics Emeritus, University of Oregon
- Jie Du, Associate Professor of Mathematics, University of New South Wales, Sydney
- Vlastimil Dlab, Distinguished Professor Emeritus, Carleton University, Ottawa (FRSC)
- Charles Dunkl, Professor of Mathematics Emeritus, The University of Virginia
- Walter Feit, Professor Emeritus (deceased), Yale University (NAS)
- Wilburd van der Kallen, Department of Mathematics, University of Utrecht
- David Goldschmidt, Director of the Center for Communications Research, Princeton,

formerly Professor of Mathematics, University of California at Berkeley - Peter Hoefsmit, Industrial mathematician, Airborne Express Company
- I. Martin Isaacs, Professor of Mathematics, University Wisconsin
- Roger Lyndon, Professor Emeritus (deceased), University of Michigan
- Gordon Keller, Professor of Mathematics (deceased), The University of Virginia
- Daniel Nakano, Disintinguished Research Professor of Mathematics, University of Georgia
- Peter Neumann, O.B.E, Emeritus Fellow of Queen's College, and former Assessor, Oxford University
- Don Newhart, Mathematician, National Security Agency
- Michael O'Nan, Professor of Mathematics, Rutgers University
- Brian Parshall, Whyburn Professor of Mathematics, The University of Virginia
- Cornelius Pillen, Professor of Mathematics, University of South Alabama
- Frank Raymond, Professor of Mathematics Emeritus, University of Michigan
- Klaus Roggenkamp, Professor of Mathematics Emeritus, University of Stuttgart
- David Stewart, G.H. Hardy Fellow in Mathematics, Oxford University
- Jian-pan Wang, Professor of Mathematics and former President, East China Normal University, Shanghai
- Nanhua Xi, Professor of Mathematics, Academy of Mathematics & Systems Science, Chinese Academy of Sciences, Beijing (CAS)

*Aschbacher-Scott [35]*

obtained a general theory of maximal subgroups of finite groups, reducing most questions to groups which were almost simple, and others to irreducible modules for the latter groups. Some of this had been proposed in the program Scott described in [29]. The latter paper adapted Dynkin's maximal subgroups program in Lie theory to finite group theory, and contained the often quoted O'Nan-Scott theorem, which described the types of maximal subgroups possible in the alternating groups. Actually, there were two theorems, and another gave an early version of the Aschbacher-Scott work. There was no collaboration between O'Nan and Scott, but both arrived at the 1979 Santa Cruz conference with similar results, and agreed to share credit. O'Nan never published separately. "Same damn theorem" was Michael O'Nan's comment after seeing the manuscript Scott had brought with him. In spite of this vote of confidence, that version turned out to have some inaccuracies, and [35] includes corrections. For a revisiting of the theory of maximal subgroups in the case of finite groups of Lie type, and its several connections with linear representation theory, see [74].*Bendel-Nakano-Pillen-Parshall-Scott-Stewart [99]*

is a six-author paper, but also can be viewed as a collaboration between the group Bendel-Nakano-Pillen, which has authored many papers together, with Parshall-Scott-Stewart, which has recently authored a separate paper [98]. In the current paper, bounds for any given Lie rank and cohomology degree n are given, for the dimension of the degree n cohomology group of a finite group of Lie type with the given Lie rank, with coefficiens in any irreducible module in the defining charactersitic. Similar results are given when the finite group of Lie type is replaced by a Frobenius kernel, or, more generally, the kernel of any of the endomorphims whose fixed points are used to define a finite group of Lie type. There are also some results for Ext groups. The paper both generalizes corresponding results in the cohomology degree 1 case by CPS[88] (and for Ext1 in PS[92]) as well as giving finite group and Frobenius kernel versions of degree n results of PS[92]. Both of these papers [88] and [92] had used methods of Bendel-Nakano-Pillen in the degree 1 case to go from the algebraic groups to the finite groups. But those methods had not advanced far enough to handle degree n, when the degee n algebraic groups results of [92] were obtained.. This was remedied in the collaboration under discussion, with both of the two groups making considerable progress toward the same goal before the collaboration took place, then deciding to join efforts.*Avrunin-Scott [33]*

proved Carlson's conjecture that support and rank varieties defined the same sets, and established a Quillen stratification theory of cohomology of groups with nontrivial coefficients. Another main result was that the support variety of the tensor product of modules was the intersection of their respective support varieties. This is often inaccurately attributed to Carlson, who had only proved such a result for rank varieties for elementary groups. To work with cohomology varieties in the global case, Avrunin-Scott proved the (new) compatibility of support varieties with the inverse image of cohomological restriction. Another important method of proof in this paper was to regard the group algebra of an elementary abelian *p*-group as the restricted enveloping algebra of a commutative Lie *p*-algebra. The purpose was to get the *right* Hopf structure. This introduction of such algebras was the beginning of the support variety theory of Lie *p*-algebras later developed by Parshall-Friedlander.*Cline-Parshall-Scott (CPS) [9], [14], [17], [20], [22], [23], [24], [26], [34], [36], [38], [40], [41], [45], [49], [52], [55], [58], [59], [60], [66], [67], [75], [76], [78], [81], [88]*

Though multiply-authored papers are common in many scientific disciplines, most papers in mathematics have only one or two authors. CPS has collaborated as a group of three. The collaboration holds, by all accounts, the record for the longest continuously productive three-person collaboration in the history of mathematics. Their achievements include the proof [20] of Green's conjecture on exact induction and affine quotients, the demonstration [22] with van der Kallen of generic cohomology, the creation [45] of the theory of highest weight categories and quasi-hereditary algebras ([44], [47], [45]), the best purely algebraic reductions [55] of the Lusztig conjecture, and an expansion of many ideas of the defining characteristic theory of finite groups of Lie type into the nondefining arena, cf. [76]. The paper [78] intertwines quantum and algebraic group representation theory and finite groups of Lie type. A main theme of the paper is developing homological results dependent on the Lusztig conjecture “beyond the Jantzen region.”. The results are especially effective for 1-cohomoogy of finte groups of Lie type, where a bound for all primes, depending only on the rank, is established in the defining characteristic for irreducible coefficients. The CPS collaboration’s 1975 IHES paper [9] on 1-cohomology calculated many 1-cohomology results with irreducible coefficients, more extensively and more quickly than ever before, and is quoted in Andrew Wiles' proof of Fermat's Last Theorem (*Annals of Math.*, 1995).*Cline-Parshall-Scott-van der Kallen [22]*

See above.*Curtis-Scott*

produced manuscripts in 1974 with the first generic block theory for Hecke algebras (including a complete defect 0 theory) and finite groups of Lie type in nondefining characteristics. There work was not published, but influenced work of Curtis's student R. Boyce and work of Hoefsmit-Scott described below.*Du-Scott [54], [57], [69] (see also Du-Parshall-Scott below)*

developed in [57] some of the first *integral* theory of quantum enveloping algebras, and applied it to both the defining and nondefining characteristic representation theory of finite groups of Lie type. They discussed in the same framework conjectures of Lusztig and James (proving special cases), and introduced the notion of a generalized *q*-Schur algebra. Later, they introduced the *q*-Schur^{2} algebra, a decomposition-number preserving enlargement of an endomorphism algebra of type B important in the nondefining theory. This enlargement was quasihereditary in all cases, unlike the endomorphism algebra.*Du-Parshall-Scott [70], [71], [72]*

undertook a broader *enlargement* project (see above) for nondefining characteristic endomorphism algebras in all root system types, using Kazhdan-Lusztig cells and the new CPS notion of a stratified algebra. As a by-product, they obtained in [72] [the most complete version (the full integral quantum case) of the Schur-Weyl double centralizer theorem].*Dlab-Scott [61]*

edited a volume of invited papers from the 1993 Canadian Mathemtics Society conference on Representations of Algebras, especially devoted to finite dimensional algebras and Lie theory, and marking a new level of interaction between these areas.*Dunkl-Scott*

is not so well known as a collaboration, but Dunkl and Scott were jointly responsible, cf. Scott's abstract [8], for the Krein condition, or rather, the realization that there was a positivity condition in finite permutation groups and combinatorics, perfectly analogous to a condition obtained in harmonic analysis by the Soviet mathematician M. G. Krein. An especially elementary proof, using observations by D. Higman on work of Schur, was presented by Scott in [12], who also presented further results. The impact of the Krein condition in combinatorics was substantial, and many (former) Soviets, especially the associates of Krein, were very grateful for the choice of name.*Feit-Lyndon-Scott [15]*

gave a one-page elementary combinatorial proof of a result on permutations that had previously been obtained only through the theory of Riemann surfaces. Later, in his 1979 Annals paper, Scott [21] generalized the theory from permutations to matrices, incorporating the Riemann surface geometry into some inequalities based on algebraic 1-cohomology. This led, in the same paper of Scott, to a generalization, now widely applied, of a famous method of Brauer (the *Brauer trick*) for constructing nonobvious subgroups of finite groups.*Goldschmidt-Scott [25]*

gave a counterexample to an open question of W. A. Manning, the leading expert in his day on permutation groups, of more than fifty years standing. The coauthors had exchanged ideas on previous occasions as well, and were office mates at Yale in 1970-71. Also, Goldschmidt had earlier been a graduate student while Scott was an instructor at the University of Chicago.*Hoefsmit-Scott*

produced only a manuscript (1975), but it influenced later work of Scott's Ph.D. student, Leonard Jones, and Scott's postdoctoral student Jie Du. The manuscript contained, with some inaccuracies, the first effort at proving the Nakayama conjecture for Hecke algebras, as well as results on vertex and source theory, and a Mackey induction theory in the Hecke algebra case. Before he decided on a career in industry, Hoefsmit was Scott's postdoctoral student, following a well-known but never published Ph.D. dissertation at UBC, Vancouver.*Issacs-Scott [5]*

proved some concrete results about blocks and restriction of characters. They also exchanged many ideas on permutation groups and subnormal subgroups as postdoctoral office mates at the University of Chicago in 1968-70.*Keller-Scott*

produced only an abstract [2] from their common interests in permutation groups and representation, yet the collaboration was the major catalyst in Scott's decision to move from Yale to Virginia in 1971. Ed Cline (the 'C' in CPS above) had earlier been Keller's collaborator and colleague at the University of Minnesota, and Keller and Scott jointly sponsored Cline's visit in 1973-75 to the University of Virginia Mathematics Department.*Neumann-Scott [11], [18]*

found several applications of character theory, especially modular theory, to finite permutation groups. They are unpublished, except for the abstract [18] by Scott. Nevertheless, the collaboration began Scott's long association with Oxford University, occasioning two visits there instigated by Neumann. In turn, Neumann met with Scott on two visits to the US, once as a short term visitor to the University Virginia. In addition, Neumann sent his Ph.D. student, Cheryl Praeger, to study as a postdoctoral student with Keller and Scott at Virginia, and his Ph.D. student, Jan Saxl, for a shorter visit.*Newhart-Scott [30]*

are coauthors of a virtually complete handwritten manuscript applying the representation theory of permutation module endomorphism rings to combinatorial issues of interest in coding theory. Newhardt was the Ph.D. student of Scott's colleague, H. N. Ward, at the University of Virginia, and is currently employed by the National Security Agency.*O'Nan-Scott*

See the discussion of Aschbacher-Scott above.*Parshall-Scott [28], [47], [63], [82], [85], [89], [90], [92], [93], [94], [95], [96], [97] (see also Cline-Parshall-Scott and Du-Parshall-Scott above, as well as Parshall-Scott-Stewart and Parshall-Scott-Wang below)*

in their lectures (detailed, with additions, in the *Moose* notes [47]) at the 1989 Ottawa-Moosonee conference introduced, with considerable effect, the finite dimensional algebras community to the new notions of quasi-hereditary algebras and highest weight categories, born in the CPS research on algebraic group representations, cf. [41], [44], and [45]. In [63], combining methods of classical algebra and of etale cohomology, they produced a new proof (and, actually, the first into print) of the very strong *Koszulity* property for categories of *perverse sheaves* important in Lie representaion theory. A recent advance has been the notion of a "forced grading" on a (generally) finite dimenisonal algebra found in Lie theory. This gets past an obstacle noted in the CPS paper [66] that an abstract finite dimensional algebra A can satisfy a huge number of favorable Lie theoretic properities, but still fail to be Koszul, or even have a "tight" grading (making the algebra gr A constructed from the radical filtration isomorphic to A). The remedy has been to work with gr A. This is much more difficult than might be realized, and it is quite hard to transfer many nice properties that A might have to gr A. Nevertheless, we have been successful with it in important cases [95],[96],[97] and plan on going much further with this theme.*Parshall-Scott-Stewart [98]*

revisted the theory of generic cohomology [22]. In the case of irreducible coefficients (or Ext with each coefficient group irreducible) they are able to make consierable improvements. A key ingredient is an earlier conjecture made by David Stewart about bounding, in terms of the Ext degree and the root system rank, the "digits" of difference that can occur two dominant weights, when a degree n Ext group for irreducible modules with these high weights is nonzero. This conjecture is proved in this paper in the cohomology case in complete generality. As a consequence of the paper, it is shown that, for a given Lie rank and cohomology degree, all cohomology groups dimenisons with irreducible coefficients, for finite Chevelley groups, can be computed, with only finitely many exceptions, in terms of cohomology of an explicitly described related irreducible module for the ambient algebraic group. One nice thing here is that we stay within the world of irreducible modules.*Parshall-Scott-Wang [77]*

presented a complete revision of the theory of Borel subalgebras, and many new results, especially in the infinitesimal and quantum cases. In the process they gave replacements for an incorrect result in Scott's appendix [65], and alternative connections to Kazhdan-Lusztig theory, in the spirit of proposed results of S. Koenig.*Raymond-Scott [19]*

using 3-cohomology, answered negatively (for higher dimensions) the question as to whether or not a well-known result of Nielsen in differential topology (on homotopy-periodic diffeomorphisms) generalized beyond 2-manifolds.*Roggenkamp-Scott [31], [32], [37], [39], [42], [50], [53], [61]*

in a ten-year collaboration produced unprecedented progress in understanding the famous finite group ring isomorphism problem (Graham Higman, Oxford thesis, 1940). Their main positive result [39] was the validity of the problem for finite nilpotent groups, and in an especially strong form for finite *p*-groups, which Higman and other experts had considered the *worst case*. However, in subsequent work, cf. [50], [56], Roggenkamp and Scott found a counterexample to a conjecture posed by Hans Zassenhaus, a stronger version of the problem, considered by Zassenhaus to be the key issue. Indeed, recently, M. Hertweck, partly building on the techniques of that counterexample, has now set forth details to a counterexample to the original problem of Higman. Hertweck is a Ph.D. student at U. Stuttgart of a former University of Virginia postdoctoral visitor, Wolfgang Kimmerle.

*Scott-Xi [91]*

starts from Xi’s double-coset approach to the parabolic Kazhdan-Lusztig polynomials for an affine Weyl grup important for algebraic groups. A main result is that the “leading” coefficient can get arbitrarily large, increasing linearly with the rank in affine type A. This difficult result is confirmed in an independent proof using representation theory and a formula of Andersen It has as consequence, using the validity of the Lusztig conjecture for large primes, that first tension groups between irreducible modules for algebraic groups can have dimension growing to infinity with the rank of the underlying group. A similar fact holds for finite groups of Lie type. A corresponding question for 1-cohomology (Guralnick’s conjecture) remains open, though the present paper shows the conjecture cannot be extended to first extension groups. See also the discussion in the "Research with undergraduates" section of this web site.