Opinion:Alperin on group theory
This opinion page is based on an interview/discussion with Professor Jonathan Lazare Alperin, on topics in group theory.
On Alperin's fusion theorem
Q: What led you to the result that's now called Alperin's fusion theorem?
A: I was fiddling around with the alternating group of degree eight, and observed that the fusion in the whole group was determined by what's happening with the local subgroups.
I had started with the following question: take an involution (element of order two) in the center of the 2-Sylow subgroup of a simple group. Now consider its centralizer in the whole group. Question: If this centralizer is isomorphic to the centralizer of some involution in the center of the 2-Sylow subgroup of the alternating group, is that simple group isomorphic to the alternating group of degree eight? While solving this, I noticed that fusion (whether or not two elements in the Sylow subgroup are conjugate in the whole group) is determined completely by the local subgroups. So I thought, perhaps this result holds for all finite groups.
When Gorenstein heard of this, he said that it was highly unlikely that such a result could be proved for any finite group. At the time, the dominant thinking was that we need to use the classification of finite simple groups to prove any result about an arbitrary finite group. I felt that the result should be true, because if it were true, then a lot of stuff done earlier, dating back to work by Burnside, would fit in very well.
I persevered at the problem for about a year. One day, while sitting down for dinner at my parents' house, I realized what I needed to do. That night, I sat down and wrote it out.
Q: What were the repercussions of the fusion theorem?
A: Even before the fusion theorem was proved, the importance of fusion was generally recognized. People just weren't expecting a result like this to be true in generality. After the publication of the fusion theorem, more progress was made in the area of fusion.
In the same journal issue where I published the fusion theorem, Gorenstein discussed some interesting consequences of this result. The idea of fusion being determined locally was used in many other results leading to the classification of finite simple groups, including work of George Glauberman. More recently, the notion of fusion system has been developed, to describe things that behave like finite groups, even though they are not groups. Fusion systems are of interest to algebraic topologists. Unfortunately, I don't get any royalties for it.
Q: What led you to generalize the McKay conjecture?
A: When I saw the McKay conjecture, I thought it had a flavor of number theory to it, so blocks came to my mind. So I considered some natural generalizations to the block case, and did numerical computations to verify the conjectures. The McKay conjecture basically said that you do two apparently completely unrelated computations and they yield the same result. What my block version did was to break up each number as a sum of pieces, and say that each of the pieces on both sides is the same.
I felt that if my generalization is false, a counterexample should be easy to find. So I checked it on certain groups, and didn't find counterexamples. After I published my version of the conjecture, people checked it against many other groups, determined to find a counterexample, but every example they checked satisfied the conjecture.
Today, block theory is a commonplace tool, so nobody would have their name attached to generalizing a result to block theory. At the time, however, it wasn't natural to generalize any conjecture to the block context, so I got my name attached to it.
Q: Do you think the McKay conjecture is true?
A: There is an overwhelming body of evidence in support of the conjecture. To hang a person for a crime, the jury needs to be convinced beyond a reasonable doubt that the person has committed the crime. The evidence in favor of the McKay conjecture (and many related conjectures) goes beyond reasonable doubt.
The conjecture has been checked for symmetric groups, a number of the linear groups, as well as solvable groups. There have been many generalizations of the McKay conjecture, all of which make the equality of the numbers all too likely to be a coincidence. There are a number of approaches to proving the McKay conjecture and other related conjectures like Brauer's height zero conjecture and Broue's Abelian defect group conjecture. Some recent work has shown that if a strong collection of facts can be proved about all simple groups, then the McKay conjecture holds in general.
Q: What would be the repercussions of proving the McKay conjecture or related conjectures?
A: There may not be many direct applications of the conjecture. More important is the hope that in developing the tools to prove these conjectures, we'll get a better understanding of groups and representations. Having specific problems to work on, specially problems like there, can be more fruitful than having a general goal of collecting information.
Fermat's last theorem is a result that doesn't have many applications. But the tools and machinery developed to solve the theorem are extremely important.
On representation theory
Q: How and why did you move from group theory to representation theory?
A: Around the 1970s, I did some long and painful work in group theory. My last work on group theory, written in collaboration with Gorenstein and others, was 261 pages. Around that time, some of the questions I had in group theory led me to look at representations. This was the beginning of my shift from group theory. You could also say that I was exhausted with groups.
On making conjectures and proving them
Q: What are the factors that lead you to a conjecture or theorem? Is it based on what's important? Or what you think can be fiddled around with?
A: Both. To some extent, it's important to know what is important. But it is also important to be able to fiddle around and try to make bold conjectures.
When a conjecture is of the sort that two numbers coming from two apparently unrelated contexts are always equal, then one know that if the conjecture is false, a counterexample shouldn't be too hard to find. A good collection of initial test cases for conjectures about finite groups is the symmetric groups, finite groups of Lie type, and solvable groups. That's a reasonably representative and diverse collection of finite groups. If two numbers come out to be equal for all of these cases, then there had better be something going on.
Q: How has the use of computers helped in verifying or disproving conjectures?
A: At the time the McKay conjecture was made, we didn't have computers to assist us, so we did all the computations by hand. A computer is magical -- you just feed in the inputs, and it gives you the answers. So having computers can help with some computations (for instance, computing numerical invariants for the sporadic groups).
However, proving conjectures for infinite families of groups cannot be done solely using a computer. It isn't possible to check the McKay conjecture for all symmetric groups by computer. So the group theorists are not out of business.
On group theory and other branches of mathematics
Q: How does one use the classification of finite simple groups in the rest of mathematics? Is it important to know the details of the classification or can one just treat it as a black box?
A: As a representation theorist, I use the classification of finite simple groups both as a black box and with various of its details. When proving that some result holds for all finite simple groups, it isn't just enough to know the list of the finite simple groups; I also need to know other facts that have been proved about the structure of these groups.
People in geometric group theory and topology use the classification.
Also, some of the techniques we used for the classification are being used in algebraic topology; for instance, the notion of fusion system as I mentioned earlier.