# Research Goals

The first two parts of this document, dealing with Scott's more immediate goals, are somewhat technical in nature. The last part, entitled "Future directions and applications," contains expository material and is suitable for both mathematical and non-mathematical readers.

Scott has, often in collaboration, provided an answer or breakthrough for a number of open mathematical questions of some standing, cf. [25], [39], [56], [62], as well as [19], [20], [22], and [33]. In more recent years a main direction has been provided by a central conjecture in group representation theory, due to George Lusztig at MIT.

More than thirty years ago Scott explained [29], using an analogy with continuous group theory, how the main obstacles to understanding finite group permutation actions would eventually reduce to hard problems in the linear case, cf. also [74]. The Lusztig conjecture is such a problem, or at least a serious piece of one, though phrased in terms of "semisimple algebraic groups," rather than finite groups. Nevertheless, together with prior results of Steinberg, the conjecture represents a first step toward broadly understanding how finite linear groups embed one into another. Suppose G(q) is a finite group of Lie type (that is, an analog of a continuous group G, but parameterized by a finite coefficient field with q elements, which must be a power of a prime p). Then the conjecture provides (by way of Steinberg's theory) a capsule description, or character formula, for the most important irreducible representations, provided p satisfies a size requirement. The conjecture is known to be true--through the efforts of Kazhdan-Lusztig, Kashiwara-Tanisaki, and Andersen-Jantzen-Soergel--for very large values of p, depending on G. But it is a pure existence proof, without giving bounds on p; later bounds produced (due to Peter Fiebig) are still extemely large. In a 2009 survey article, Jantzen reports that this known bound is a 40 digit number for the groups of type A8 (corresponding to SL9(q), the group of 9 x 9 matrices of determinant 1, with coefficents in a field of q elements, q a power of p), "whereas one expects that p =9 should be good enough."

The precise size requirement on p of the original conjecture has been put into question by a recent preprint of Geordie Williamson, indicating the original size guess, which may be formulated as p = n (with equality also allowed) for SLn (q), is wrong, and suggesting that a hypothesis p = f(p) with f an exponential function of n might be required. However, it should be pointed out that n! is an exponential function of n, not nearly in the "huge" range of the Fiebig bounds, so it would remain worthwhile to know if the Lusztig conjecture were true for p>n! style='mso-spacerun:yes'> Such a bound could be generally formulated for all finite groups of Lie type, using as a general version of n! the order of the "Weyl" group, which happens to be the symmetric group on n letters for SLn(q). Perhaps more importantly, any proof of the Lusztig conjecture with a bound expressed meaningful in terms of familiar group parameters is likely to help formulate future character formula conjectures for smaller primes.

The conjecture is known in all "rank" cases and also for type A3, with Lusztig's original requirement on p. "Type A3" refers to familiar groups of invertible 4 x 4 matrices of determinant 1. In the late 90s, using computer programs developed with the assistance of Mike Konikoff and Chris McDowell, then both Virginia undergraduates, Scott showed empirically that the conjecture held as stated, and even in a stronger form proposed by Kato, in the first open cases, with G the group of 5 x 5 matrices of determinant 1, and p = 5 and 7. (The p=5 case was independently treated earlier by Buch and Lauritzen.) Scott and his collaborators, especially Ed Cline and Brian Parshall, have been working on the Lusztig conjecture and related questions for some time. (Cline is now deceased.) Each of the papers [40], [41], [44], [45], [47], [48], [49], [55], [57], [58], [59], [60], [63], [66], [67], [68], [77], [81], [87], [95], [97], [100], [102] is relevant, though many develop general theories of broader significance. A main focus of [87], for example, is to provide general homological and structural consequences that hold when the Lusztig conjecture is true. This both increases the impact of the conjecture and strengthens prospects for a future inductive proof. (Inductions often work best when addressing stronger formulations.) Another paper, [102] with Parshall, has a similar flavor, but in new directions often involving "forced gradings," an attractive way of obtaining graded structures, but involving difficult proofs. It was first shown to be viable in [95]. Some applications of the method do not require any Lusztig conjecture hypothesis, but stronger results are often obtained with it. Other work with Parshall [92], [93] gives new applications to bounding cohomology dimensions of the Lusztig conjecture, and also addresses, along with the paper [90] with Xi, general growth rates for dimensions of extension and cohomology groups with respect to the size (rank) of the underlying group. The papers with Parshall grew directly out of the last paper [88] with Cline. The paper with Xi shows dimensions of first extension groups between irreducible modules can be arbitrarily large if the rank of the underlying group is allowed to grow. This is unknown (Guralnick's conjecture) for the corresponding 1-cohomology case, though [80], a 2003 result of Scott's collaborations with undergraduates Konikoff and McDowell, showed for the first time that the dimensions could be as large as 3, and in infinitely many cases. In particular McDowell had calculated by computer some coefficients of Kazhdan-Lusztig polynomails (arising in the statement of the Lusztig conjecture) which were relevant to 1-cohomology dimensions in the algebraic groups case. More recent computer calculations of Frank Luebeck, inspired by those of Tim Sprowl, also a Scott undergraduate student, found much bigger dimensions. Before he graduated, Sprowl found the current biggest known value for these 1-cohomology group dimensions for groups of type A. It is 36672. See the web page detailing Scott's research with undergraduates, which also gives more recent history, including especially the contributions of Frank Luebeck. The understanding of 1-cohomology groups of finite groups of Lie type with coefficients in irreducible modules is another of the "hard problems" mentioned above that are required for a constructive theory of nonlinear (permutation goup) actions, as mentioned in the above paragraph. The relationship between algebraic group cohomology and that of finite groups of Lie type was the subject of a classical paper of Cline-Parshall-Scott-van der Kallen [22]. See also [76] and the recent joint work [98] with Brian Parshall and David Stewart.

It was a wonderful and somewhat surprising consequence of the Lusztig conjecture theory that it could also provide, through the theory of [22] mentioned at the end of the above paragraph, estimates for the sizes of some 1-cohomology group dimensions for finite groups of Lie type. In this way the theory has led to counterexamples to a 50+ year old conjecture in permutation groups, (More precisely, the conjecture, due to G. E. Wall in 1961, involves maximal subgroups of finite groups; these lead to "primitive" permutation groups by the associated group action on cosets.) All these counterexamples presently involve groups of "huge" size in the sense of Fiebig's bounds for the Lusztig conjecture, though they fit into a very natural theoretical framework. See the the website www.aimmath.org of the American Institute of Mathematics, where the counterexamples are announced. Whatever further progress is or is not made on the Lusztig conjecture, it has already helped to strikingly demonstrate how "Lie theory" (more precisely, the theory of semismple algebraic groups), especially as supported by computer calculations of the combinatorial Kazhdan-Lusztig polynomials used in the statement of the conjecture, can penetrate into problems in finite group theory not previously accessible.

Beyond the Lusztig conjecture

Most immediate prospects for a proof involve a reduction to previous known geometric calculations (on so-called perverse sheaves). However, it remains a possibility that there is a purely algebraic proof. The same should be true of related, already proved conjectures due to Kazhdan and Lusztig in the continuous case. Such a proof would have a considerable impact on all of Lie theory, where currently geometric methods are perhaps over-emphasized. Though Scott has an appreciation of geometry, and some expertize with it, he believes that, instead, finite-dimensional algebras could be the principal organizational theme, with geometry appearing only in the inspiration of some of the algebra.

In addition, a purely algebraic proof of any form of the characteristic p Lusztig conjecture itself, might make it easier to extend the theory to the nondefining characteristic (or cross-characteristics also called nondescribing characteristic) representation theory of finite groups of Lie type. Scott has already begun, with Jie Du as well as and Parshall, the study of algebras appropriate to these nondefining characteristic representations [68], [69], [70], [71], [72], [73], [75], [76]. Ed Cline was originally involved in this work. For a history of nondefining characteristic representation theory, see [74]. Here one is studying representations of these groups over finite fields of coefficients, but the coefficient systems are quite different from those used to define the group. Still, these representations must be understood if one is to understand all the linear representations. As a bonus, one would get information, likely decisive, for nonlinear representations (permutation group actions). The relationship of defining, nondefining, and nonlinear representations was the subject of Scott's article [74], the lead survey article for the Proceedings of the 1997 Newton Institute program on representations of algebraic groups and related finite groups.

It remains problematic what to do with the smaller coefficient fields, both in defining and nondefining characteristics, those not large enough even for the original formulation of the Lusztig conjecture to apply. If the Lusztig conjecture and similar results in nondefining characteristic could be established, it would at least be possible to handle some of the smaller characteristics fairly immediately with computer calculations, and have complete tables of character formulas for modest size ranks.

One interest very new direction in the research of Scott and Parshall, first briefly suggested by the papers [101] and [102], is the the theory of Q-Koszul algebras. These algebras resemble Koszul algebras, the latter a highly specialized but powerful kind of graded algebra arising in characteristic 0 Kazhdan-Lusztig theory. A main distinguishing feature of Q-Koszul algebras is that they need only be "quasi-hereditary" and not "semisimple" in grade 0. This allows for the possibility that many finite dimensional algebras associated with semisimple algebraic groups in characterstic p = 0 might be Q-Koszul, at least after applying appropriate "forced grading" constructions. This could give a structure theory for representations of these groups quite independent of the validity of any character formulas such as the Lusztig conjecture. And indeed, it is easy to give examples of such algebras in algebraic group contexts where the Lusztig conjecture is known not be true.

Future directions and applications

Though Scott is a pure mathematician, at least in his published work, he has also participated in many interdisciplinary seminars and believes that mathematics ultimately should be applied. Scott is coinventor on four patents, US 7236953 and three follow-ups, all dealing with probability distributions in finance, and hopes to publish academic articles in finance and possibly areas of computer science. These are both subjects in which he has often taught courses of an applied mathematical nature. Scott intends to be active in pure and applied research for many years to come. Here is a view of how the theoretical world of group representations relates to more applied areas, and a glimpse of what might become future research directions after today's current problems are solved. For a related general discussion of group representation theory and its applications, see the web page describing Scott's research with undergraduates (accessible from the home page).

Finite permutation group actions are the fundamental building blocks of all finite symmetries and dynamical systems: Abstract finite groups, when equipped with a selected set of generators and a selected action of these generators on a set, can produce all possible patterns of symmetry, even in spaces of almost unimaginably high dimensions. This large number of dimensions is critical for applications, if one is thinking of more than just producing pretty pictures: Each dimension might be viewed as describing a numerical parameter of a complicated physical system, so that the number of dimensions would correspond to the number of parameters required to describe the system. One can then think of the system as a point or particle in a very large dimensional space. Such a particle, constrained to move by only following the movements of a sequence of generators for a group, may follow a path that may be remarkably confined and understandable--that is, the production of symmetry by the group translates into constraints in the behavior of the system. There are other ways group actions can impose constraints, like the spinning of a top imposes an axis of rotation--any directional motion which is compatible with the spinning (commutes with it) must proceed along the axis. Many past applications have been in the domain of physics, e.g., the constraints of the Lorentz group on relativistic space-time, or of the Gell-Mann eight-fold way on fundamental particle interactions. But, in the information age, group actions have been used in communication codes and disk drive error correction. Here all the symmetry is in bit patterns of 0's and 1's. Theoretical results in automata theory, such as the Krohn-Rhodes theorem, suggest applications to computing will continue.

Roughly, the Krohn-Rhodes theorem says that every finite-state dynamical system may be understood in terms of a combinatorial arrangement of finite groups and their actions, the latter intimately related to the reversible movements of particles or sets of particles allowed by the system. To repeat, the particle terminology is suggested by physics, but it is meant as a purely abstract term. Indeed, it might well be more appropriate to think of the rapidly changing bit pattern in a computing or communication device, rather than a particle moving in a traditional physical space.

Scott regularly lectured on the Krohn-Rhodes theorem in courses, and it is part of his future research plans (as well as one of the long-term justifications of his current work on group actions, since the latter play the role of fundamental building blocks in Krohn-Rhodes theory). It seems likely the theory could benefit from more canonical decompositions. Such a program was begun in a grand two-volume treatise Automata, Languages and machines, by the deceased topologist Samuel Eilenberg. Rhodes saw potential applications of his work with Krohn to biological processes at the molecular level, possibly as an insightful classifying mechanism. However, the computational issues are quite serious for even very standard processes such as the Krebs cycle, and it is not clear how best to get decompositions in the sense of Krohn-Rhodes that might be called canonical. There are opportunities here for theoretical advances (e.g., more canonical decompositions) to have practical consequences. Scott expects some discrete version of Klein's Erlangen program to provide a unifying theme here, in which the traditional roles of continuous geometry is re-invented in terms of symmetry and groups, and transformed into considerations of finite groups and their permutation actions.