The project continues to be supported by NSF and, originally, was funded by their REU program (Research Experience for Undergraduates). It is also funded by our UVa Mathematics Department's REU program. The first undergraduate involved was Mike Konikoff, now graduated, and his place was taken by Chris McDowell, now also graduated. A search for yet another undergraduate is currently underway (10/26/99).
The first goals were to check the conjecture for the finite groups and Lie algebras of 5 x 5 matrices over the finite fields of 5 and 7 elements, respectively. To do this, one had to understand further huge matrix representations of these smaller systems, already as large as 10,000,000 x 10,000,000 in the case of the field of 5 elements, and 300,000,000 x 300,000,000 in the p=7 case. Brute force computing was out of the question, and the problems were barely accessible by using all available theory. But, In the past six years we developed computer programs that have solved these problems. For p=5 we share credit with a Danish team, Lauritzen and Buch, but the p=7 case was inaccessible to them, and is ours alone.
The debugging process has been incredibly formidable. What do you do with code that runs successfully on matrices that are smaller than 100 by 100, but then begins to go wrong? Check it by hand in the 100 by 100 case? Hardly. The approach we have used effectively is to write secondary programs independently performing some of the tasks of the first (with different algorithms), and compare results. This is time consuming, but has worked for us. Using two and sometimes three indedpendent programs, we were both able to debug our programs and be confident in the answers. Many of the calculations were done on the University's 24-node IBM SP platform, though PC's and Silicon Graphics workstations were used as well.
To summarize results to date: The Lusztig conjecture and an extension of the conjecture proposed by Kato are true both for p=5 and p=7 in the 5 x 5 matrix case. In addition, data incidental to the computations suggest new properties of the so-called Kazhdan-Lusztig polynomials, that certain of their coefficients that compute extensions between irreducible modules are extremely small. They are never bigger than 2 for the 5 x 5 case, in keeping with an analogous conjecture of R. Guralnick, USC, who proposed that all 1-cohomology groups of finite groups with coefficients in faithful irreducible modules have dimesnion at most 2. Unfortunately, this doesn't hold up for the 6 x 6 case, as we discovered in early 1999 with some Kazhdan-Lusztig polynomial calculations. The numbers only grow to 4, though, and only to 3 in the 1-cohomology case relevant to Guralnick's conjecture. So there still appears to be something quite interesting going on, and that the spirit of the conjecture of Guralnick (that the dimensions are really small) is still quite viable.
Next, we hope and expect that the programs developed will provide further empirical insight into the area, and suggest avenues for an eventual mathematical proof of the Lusztig conjecture. In that direction, which is perhaps the most important of all, much remains to be done.
Data links:
