The formal concept of a group was first conceived by Évariste Galois (1811-1832) and Augustin-Louis Cauchy (1789-1857) in the early nineteenth century. Abstractly defined, a group is a set equipped with a binary operation (often called "multiplication") that is associative, has an identity element, and gives each element a (multiplicative) inverse.

Mathematicians study difficult problems by breaking them down into components. A primary tool for performing such an analysis is symmetry. And the language for describing symmetry is the group. Today groups pervade all aspects of mathematics and analytical science, from geometry, number theory, coding theory, cryptography, and differential equations to computing, financial mathematics, and the theory of automata. Understanding groups is fundamental to understanding the world around us.

Of fundamental interest are finite groups--that is to say, groups with finitely many elements. As an example, the group of symmetries of a square in the plane forms a finite group (having 8 elements) consisting of four rotations plus reflections. The building blocks for the finite groups are the so-called simple finite groups which, apart from trivial examples, fall into three types:

• the infinite family A₅, A₆, … of alternating groups--these are the groups of even permutations of a finite set (necessarily having > 4 members);
• the infinite families of finite groups of Lie type--many of these groups have elementary descriptions and, further, continuous analogues; and
• the 26 sporadic groups--these groups essentially defy analytical description.

A finite simple group often arises naturally as the group of symmetries of some, usually important, geometric object--much as in the example of the square described above.

A particularly rich structure arises when the group under study also happens to have some geometric structure of its own. For example, the group of rotations in the plane can be identified naturally with the unit circle, and the unit circle is itself a geometric object. Many other important groups have such a geometry, arising from their realizations as solution sets to systems of polynomial equations, just as a unit circle is the solution set to the equation x² + y² = 1. The algebraic groups are groups of this type. Although algebraic groups generally have infinitely many elements, they may in turn be used to study and understand finite groups of Lie type. Another interesting aspect involves certain objects called quantum groups--which are not really groups at all but are rather a type of enveloping algebra. While "living in" characteristic 0, quantum groups behave like objects in characteristic p, so they can be used to mediate between the two situations. [It should be noted here that a field has characteristic p if the sum of p copies of any element in the field is 0. Characteristic 0 means that there is no such p.]

EXAMPLE: Let k be a field of characteristic p. A good example of such a field is a finite field with p elements, such as is given by the integers modulo p with addition and multiplication modulo p. [In general, a field is a "number system" satisfying certain natural and intuitively appealing axioms; the real numbers R and the complex numbers C, in addition to the integers modulo p, are well-known examples. The complex numbers are an algebraically closed field, in that any polynomial with coefficients in C has roots in C, but the same is not true for the integers modulo p or R.]

Let SL_n(k) be the special linear group of order n on k; this is the group of n × n matrices with entries in k having determinant 1, under matrix multiplication. When, furthermore, k is algebraically closed (e.g., an "algebraic closure" of the integers modulo p), then SL_n(k) provides a standard example of the kind of algebraic group of importance to questions here. We may define the Frobenius homomorphism on SL_n(k) by

X=(x_ij) → X^[p] ≡ ( x_ij^p ) .

Thus, we are taking each entry of the matrix X to its p^th power. Those matrices X for which X^[p] = X are called the fixed points of the homomorphism. The set of such fixed points forms a group which is just the finite group SL_n(p) consisting of all n × n matrices of determinant 1 in the finite field with p elements--an important finite group of Lie type. All finite groups of Lie type arise in this fashion, either from Frobenius homomorphisms or from generalizations thereof.

Representation theory provides a basic tool for studying finite groups. A (linear) representation is a homomorphism from the given group G into the group GL_n(K) of n × n invertible matrices in a field K. Certain fundamental representations, called irreducible representations, provide the building blocks from which we might hope all other representations can be created. In the case that K has characteristic 0, it is quite clear how these building blocks fit together. In the case that K has characteristic p, the structure is much more difficult and less understood, since representations need not decompose into a direct sum of irreducible ones. This difficulty in characteristic p is closely related to the cohomology of finite groups, a topic of the workshop.

Given a finite group G of Lie type, its representation theory naturally breaks into three (related) cases, depending on the characteristic of the field K. The best understood (but still very difficult) case occurs when K has characteristic zero (e.g., K is the complex numbers). The case when k (the defining field for the ambient algebraic group) and K have the same characteristic is called the defining characteristic case. The cross-characteristic case occurs when K has positive characteristic different from that of k. A famous conjecture (as yet unproved, except for very large characteristics) due to George Lusztig gives strong information on the irreducible representations in the defining characteristic case. This conjecture is closely related to the combinatorially and geometrically defined Kazhdan-Lusztig polynomials which then compute cohomology spaces. The validity of the Lusztig conjecture in general, as well as a formulation of it for small primes, i.e., small characteristics, has great interest among group theorists. In addition, the Kazhdan-Lusztig polynomials seem very hard to compute. The workshop has explored the current state of computational methods for them; it has also considered how to apply them in both the describing and cross-characteristic cases.

Thus, enter the computer. A recent triumph of AIM workshops is that the Atlas of Lie Groups workshop (which has met at AIM for the past four years) has calculated all the Kazhdan-Lusztig polynomials for the exceptional Lie group of type E₈. There is hope that the methods, and the computing techniques, developed to analyze E₈ can also be used in the study of finite groups of Lie type that have been discussed above. But the situation for the finite groups of Lie type is considerably more complicated, since the underlying (affine) Coxeter group upon which the Kazhdan-Lusztig polynomials are based is infinite, whereas the Coxeter group required for the calculations made for the $E_8$ case for real Lie groups was finite. There are also computational problems involving Schur algebras, Hecke algebras, support varieties, and other questions on the cohomology of finite groups that we hope will yield to computer assault.

Useful computational packages for performing involved and sometimes subtle algebraic calculations include Magma, GAP, CHEVIE, and LIE; these overlapping packages have various internal strengths and weaknesses for computations in group theory and related cohomology and representation theories. Furthermore, some are freeware (such as GAP, CHEVIE, and LIE, whereas some are not (such as Magma). Considerable discussion at this workshop concerned the relative merits of these packages and other programs, and how they can be applied to the problems at hand.

One of the fascinating developments of modern mathematics is that the computer is playing a more prominent role in theoretical research. No longer is the computer simply a device for crunching numbers. Now the computer can manipulate concepts and explore ideas. The finite groups of Lie type workshop is at the forefront of this new interface between theory and computation.