Technical Details

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 E₈ are known. How many are there? Before the software was written, we didn't know. We expected about 696,729,600 the order of the Weyl group. (This is what small examples suggest.) In fact the number is 453,060. In retrospect we should have known this: this number is closer to the number of involutions in W. Writing the software, and doing this calculation, made us understand the mathematics better.

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 E₈ will provide a precise list of Arthur's unipotent representations: perhaps 200 of the 453,060. This is an example of the mathematical consequences of this work.

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 E₈. Why? First of all, because it was there. Secondly, because what we really want to do is compute the unitary dual of real Lie groups. The computations required for this are orders of magnitude harder than the calculation of E₈ polynomials. If we want to have any chance of computing the unitary dual of F₄, we better be able to compute KLV polynomials for E₈. We needed to push the technology to the limit. It would not have been enough to simply find a big enough computer.

What is the importance of this result? First of all, computing KLV polynomials for any group, including E₈, is a step on the way to computing the unitary dual. Secondly, the atlas software has changed the way we think about mathematics. Thirdly, the atlas software is a wonderful tool for studying representation theory, from graduate students, to researchers in related fields, to expert in real groups.

Unitary representations play an important role in many branches of mathematics, including number theory and quantum mechanics. The split real form of E₈ in particular plays a role in M-theory in physics. We are not saying that knowing the unitary dual of E₈ will have applications to M-theory. But it might; this is basic research, and it is difficult to know where advances will lead.

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 E₈. See below.

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 E₈.

The Atlas software

Fokko 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_xm(x,y)π(x) (the sum runs over x in P) in the Grothendieck group. This equality can be inverted: π(y)=Σ_xM(x,y)I(x).

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. E₆ takes less than one second and E₇ about 3 minutes. Computing KLV polynomials for the split real form of E₈ naively requires a computer with about 256 gigabytes of memory (all accessible from a single processor). The number of representation (the size of the parameter set P) is 453,060, so we're computing a matrix of size 453,060x453,060 (this matrix has ones on the diagonal and is upper triangular).

The E₈ calculation

Fokko du Cloux, David Vogan and Marc van Leeuwen began trying to compute KLV polynomials for the split real form of E₈. Eventually they started using sage, a computer at the University of Washington, run by William Stein. Ultimately it was necessary to do the calculation mod 251, 253, 255, and 256, and combine the answers using the Chinese Remainder Theorem. For some statistics, more information on the computation and some history see the atlas web site.

Kazhdan-Lusztig polynomials

The Kazhdan-Lusztig-Vogan Polynomials, which give the multiplicities discussed above, are defined in terms of some geometry which we have yet to introduce. We start by considering Category O, which is the setting of the original Kazhdan-Lusztig polynomials.

We change notation and G be a connected, complex reductive group, and B a Borel subgroup. Then B has a finite number of orbits on G/B, parametrized by the Weyl group W. Fix a regular integral infinitesimal character. For any element w in W there is a Verma module L(w) (this is like the standard module above), with the given infinitesimal character, containing a unique irreducible submodule π(w). There is a decomposition L(w)=Σ_y m(y,w)π(y). Again this can be inverted, to give π(w)=Σ_yM(y,w)I(y).

The integers m(y,w) are given by Kazhdan-Lusztig polynomials. These are defined in terms of the flag variety G/B, and are related to singularities of, and closure relations between, the orbits of B on G/B. If w,y are elements of W, then the Kazhdan-Lusztig polynomial P_x,y is a polynomial in q, defined in terms of the orbits corresponding to x and y. Then M(x,y)=P_x,y(1) up to (an explicitly computed) sign.

Kazhdan-Lusztig-Vogan polynomials

In place of B orbits on G/B we now consider K orbits on G/B. This is also a finite set, but not parametrized by W. In the category O case, a Verma module is attached to an orbit of B on G/B. Note that the stabilizer of a point in G/B is connected. In the G(R) case, a representation is associated to a pair (O,S) where O is an orbit of K on G/B, and S is a K-invariant local system on O. Specifying the local system amounts to specifying a character of the component group of the stabilizer of a point of O. This is a finite 2-group, but not necessarily trivial, so there can be more representations than orbits. The parameter space P referred to above is the set of pairs (O,S). Associated to each such parameter are standard and irreducible modules. The multiplicity formulas are defined in terms of perverse sheaves on these orbits. These are the Kazhdan-Lusztig-Vogan polynomials.

For example suppose G(R) is the split real form of E₈. The number of orbits of K on G/B is 320,206, and the number of representations (with given infinitesimal character) is 453,060.

Acknowledgements

Jeffrey Adams, 3/22/07, jda@math.umd.edu

Addendum

The goal of the Atlas of Lie Groups and Representations is to classify the unitary dual of a real Lie group G by computer. A step in this direction is to compute the admissible representations of G, including their Kazhdan-Lusztig-Vogan polynomials. The computation for E₈ was an important test of the technology. While an impressive achievement, it is but a small step on the way towards the unitary dual, and not remotely as important as the original work of Kazhdan, Lusztig, Vogan, Beilinson, Bernstein et. al.

Nevertheless, because of the nature of the result, the Atlas team and the American Institute of Mathematics decided this would be an excellent opportunity to educate the public about research in pure mathematics. The intended audience of it was the general public, and this was undertaken for the benefit of mathematics awareness as a whole, and not for the Atlas project itself. We are happy to have been successful in raising awareness of mathematics research worldwide.

3/25/07

Atlas Web Site

E₈ page at AIM