## Technical DetailsThis page is intended for mathematicians who want a more detailed explanation of the Atlas project and the E_{8} calculation.
## What is the big deal?The most important thing that we've done is written an algorithm which converts some very difficult abstract mathematics, the representation theory of real groups, into combinatorics which can be computed. This is a substantial accomplishment. The starting point is some very deep mathematics, due to Harish-Chandra, Langlands, Kazhdan, Lusztig, Vogan and others. Making this into explicit algorithms was a major achievement of Fokko du Cloux. In the process of taking these known mathematical results, and converting them into a computer program, we have deepened our understanding of the mathematics. Secondly, we have implemented this algorithm on a computer. The program allows you to input the data for any connected complex (equivalently, algebraic) group, and any real form G of that group. You can then list the Cartan subgroups of the group, compute Weyl groups, and other structure theory. You can also compute the representations of G with any fixed integral infinitesimal character.
This information was already in some sense "known". The significance
of the software is that it has changed our attitude about what the
word "known" means. In principle, the set of irreducible
representations of the split real form of E Here is an anecdote to illustrate how valuable this has been. When Fokko du Cloux was writing the software, he would stop by David Vogan's office periodically and show him an unusual example. David would say no, that is impossible, there must be a bug. In virtuallly every case Fokko was right, and we learned something about the mathematics we hadn't known before.
Jim Arthur has made conjectures describing the possible residues of
Eisenstein series in the theory of automorphic forms. His conjectures
imply the existence of some extraordinarily interesting unitary
representations called "unipotent" representations.
One difficulty in working with his conjectures is that the unipotent
representations are characterized by some abstract properties that are
difficult to verify for a particular representation. The calculation
that we've made for E The software also computes Kazhdan-Lusztig-Vogan (KLV) Polynomials. Kazhdan, Lusztig and Vogan gave an algorithm for computing these in the 1980's. This was very deep mathematics, much more significant than anything we have done. Again, having an algorithm to compute something, and actually computing it, are two different things.
We computed the Kazhdan-Lusztig-Vogan polynomials for
E
What is the importance of this result?
First of all, computing KLV polynomials for any group, including
E
Unitary representations play an important role in many branches of
mathematics, including number theory and quantum mechanics. The split
real form of E
This leaves the question of why this story took off in the press. For
us, that is harder to understand than the Kazhdan-Lusztig-Vogan Polynomials
for E
## The Atlas of Lie Groups and Representations
The Atlas of Lie Groups and
Representations is a project to compute the unitary dual of any
real reductive Lie group. This is a major unsolved problem in
mathematics. A second major goal is to make software available for
computing structure and representation theory of real groups. This is
intended both for educational and research use. This is the software
we used to compute Kazhdan-Lusztig-Vogan polynomials for
E
## The Atlas softwareFokko du Cloux wrote the atlas software, which computes the irreducible admissible representations of G. More precisely, fix a regular integral infinitesimal character for G. The software computes the irreducible representations of G with this infinitesimal character, which is a finite set. (Later versions of the software will remove the "regular" and "integral" restrictions, neither of which are serious.) The software is available at the atlas software page. It is still a very early version (0.2.6.2). Here is a little more information about what this software does. First of all it allows you to enter the data to describe an arbitrary, connected, reducted complex group G(C). Then you can define a particular real form G. The software computes structural information about G, in particular Cartan subgroups and their ("real") Weyl groups. It then computes the irreducible representations of G with a given regular integral infinitesimal character. Now suppose we have fixed G(C), G, and a regular integral infinitesimal character for G. There is a parameter set P, which is a finite set. For each element x of P there are two natural admissible representations of G, with the given infinitesimal character. First of all there is a "standard" module I(x). I(x) is the full induced representation of a discrete series representation on a parabolic subgroup. Secondly there is an irreducible representation π(x), which is a distinguished subrepresentation of I(x). The representation π(x) is typically smaller than I(x), interesting and difficult to understand. The unitary dual problem is to determine which irreducible representations π(x) are unitary.
Every standard representation I(x) can be decomposed into a "sum" of
irreducible representations. It is not the case that I(x) is
completely reducible; it has a Jordan-Holder series in which each
irreducible subquotient is irreducible. In other words there is an
equality I(y)=Σ It is important to compute the non-negative integers m(x,y) and the integers M(x,y). The latter is what the Kazhdan-Lusztig-Vogan polynomials do. (See below for more on the Kazhdan-Lusztig-Vogan polynomials, and their relation to Kazhdan-Lusztig polynomials.)
The atlas software computes the KLV polynomials for any real
group. For any group of rank 7 or less the computation is very
fast. E ## The E |