A group is a set equipped with a binary operation satisfying certain natural properties (associativity, identity element, multiplicative inverse). A Lie group is a group that also has a geometric structure. As an example, the collection of rotations in the plane is clearly a group--the composition of two rotations is another rotation, and the other properties are clear. But the set of rotations can be identified in a natural way with the unit circle in the plane, and the unit circle has geometric structure. See Figure 1.

Figure 1

Lie groups were first identified and studied by Sophus Lie (1842-1899) in the context of differential equations. The subject is now an extensively developed and widely used branch of mathematics that has applications in areas ranging from differential geometry to mathematical physics.

The Atlas of Lie Groups workshop met this week for the fifth time at AIM. Organized by Jeffrey Adams, one of the main focuses of this group is to develop a sophisticated piece of software that will describe all Lie groups and their irreducible unitary representations. Some explanation is in order.

The unitary group U(n) on n letters is the collection of n × n complex matrices with the rows forming an orthonormal set. The group U(n) is well understood, and is considered to be a "model Lie group." We understand another Lie group G by considering mappings--group homomorphisms--from G to U(n) for some n. These are the unitary representations, although we need to allow n = ∞. The irreducible unitary representations are those that cannot be broken down into more elementary pieces. Thus the Atlas of Lie Groups workshop wants to create a piece of software that will calculate the the collection of irreducible unitary representations (we sometimes call this the unitary dual) of any given Lie group.

It is a result from the early 1980s, due in large part to David Vogan (one of the partipants in this workshop), that the unitary dual for any particular Lie group G may be calculated in finitely many steps. This fact makes it in principle possible to have a computer perform the task. It should be stressed, however, that Vogan's result is highly theoretical. There was some doubt as to whether the calculations were actually feasible, or whether a computer with sufficient speed and capacity could be found for the job.

Every big task requires a test case, and for the Atlas of Lie Groups workshop that test case was the group known as E₈. This is the largest "exceptional" Lie group (exceptional according to a well-known classification of Lie groups). In Spring of 2007, this workshop succeeded in calculating the so-called Kazhdan-Lusztig-Vogan polynomials for E₈. These important polynomials are considered to be a key stepping stone to getting to the unitary dual. The calculation was immense--much larger than the mapping of the human genome. The printout would cover the entire island of Manhattan. The polynomials themselves are typically of degree 12, but the character table in which they are logged is of size 453,060 × 453,060. Thus there are 1.2 trillion pieces of data that need to be cataloged and manipulated. Fokko du Cloux, new deceased, was the "hero programmer" of the project. His herculean efforts brought the calculation of the Kazhdan-Lusztig-Vogan polynomials to fruition, and prepares the workshop for its next step.

The next step is to calculate the unitary dual of E₈. David Vogan has said that if the calculation of the Kazhdan-Lusztig-Vogan polynomials is of the order of magnitude of Manhattan, then the calculation of the unitary dual will be like a trip to Mars.

The solution of any good mathematics problem opens new doors and suggests new problems. Thus the calculation of the Kazhdan-Lusztig-Vogan polynomials for E₈ has posed new theoretical difficulties for the Atlas of Lie Groups project. The focus during the current fifth meeting was on these theoretical questions. There are also serious computer-science-theoretic problems. The problem is very difficult to fit on any computer--even a supercomputer. If the calculations are performed naively--not using various tricks to reduce the order of magnitude of the algorithms--then the job will take millions of years and nobody will be here to see the result. Thus the algorithm needs to be made as efficient as possible.

Representations of Lie groups can be grouped into units called cells, and one of the focuses of the present workshop is a consideration of the theory behind cells. Obviously there are advantages to focusing attention on cells, as these are smaller units of study. Computing cells is very difficult; workshop participant Birne Binegar has discovered experimentally an algorithm which appears to make this much easier. It remains to be proved that this algorithm is correct.

The efforts of the Atlas of Lie Groups workshop represent an important new marriage between theoretical mathematics and theoretical computer science. The scientific breakthroughs that are being achieved are important for both disciplines, and have far-reaching implications for the way that mathematics will be practiced in the future.