This is an extraordinary story which I learned about only last week.

In the 1960s, while Donald Higman in the USA was developing the theory of coherent configurations, Boris Weisfeiler was introducing the equivalent notion of cellular algebras in the USSR. The motivation was a bit different: Higman used his constuction to find restrictions on numerical parameters for finite permutation groups, but Weisfeiler was considering the graph isomorphism problem. Cellular algebras arise in the “partition refinement” approach to graph isomorphism, which is the engine behind such packages as Brendan McKay’s `nauty`, and was also used by Laszlo Babai along with probabilistic methods for his elementary bound for the orders of uniprimitive permutation groups.

The first paper of which I am aware is entitled “Reduction of a graph to a canonical form and an algebra which appears in this process, by Weisfeiler and A. A. Leman in 1968. The subject was taken up by other Soviet mathematicians, notably Misha Klin. Since then however, the term “cellular algebras” has been used with an entirely different meaning in the works of Graham and Lehrer and others.

This is a topic which I will certainly have to discuss here in the future!

Anyway, Weisfeiler left the USSR for the USA in 1975. While there, he worked on algebraic groups and linear groups. In two papers in the 1980s, he used the recently announced Classification of Finite Simple Groups (CFSG) in two remarkable pieces of work: a strong approximation theorem for general linear groups, and a dramatic improvement to Jordan’s Theorem on linear groups.

This theorem says that a finite subgroup of GL_{n}(*K*) contains an abelian normal subgroup with index bounded by a function of *n* alone. For a simple example, suppose that the field *K* is the complex numbers, and *G* is the group generated by the diagonal matrices whose diagonal entries are *k*th roots of unity and the permutation matrices. Then the order of *G* is *k ^{n}*·

*n*!. The diagonal matrices form a normal subgroup whose order is

*k*, which is arbitrarily large since

^{n}*k*is unrestricted; but its index is

*n*!, a function of

*n*.

Mathematicians such as Brauer and Feit worked on improving the bound, but in the words of Alex Lubotzky, “Weisfeiler’s result is still, twenty years later, the best known result. Unfortunately, a detailed proof has never appeared.”

The reason for the non-appearance is bizarre and tragic. In January 1985, Boris Weisfeiler went hiking in Chile and disappeared, leaving the manuscript incomplete. The reason for his disappearance has not been established; theories range from a hiking accident, through an accidental shooting by the Chilean military, to capture and execution by a Nazi group based in Colonia Dignidad.

The story has been told in many places, and there is even a film, but I was unaware of this.

Last week, I had an email from Boris’ sister Olga. She, with Tamara Kurdyaeva, Roman Bezrukavnikov, and Michael Finkelberg, have converted the incomplete manuscript into LaTeX and posted it on the arXiv. Olga sent me the link since there is a reference to me in the bibliography, and it was not clear which paper it is. She also asked me to let her know of any misprints I find.

I am sure that contributions to this project from anyone would be welcome. So take a look, read the paper, and send Olga your comments!

Further reading about Boris Weisfeiler is here or here.

I should point out that the absolute upper bound (normally S_{n+1} acting on n-dimensional space, but sometimes a product of small linear groups when n is small, was found by Michael Collins a few years ago. So Alex’s quotation is a little out of date!

It was in 2005. Michael Collins’ work is cited in the preprint that has just appeared on the arXiv.

Minor correction: “Since then however, the term “coherent algebras” has been used with an entirely different meaning in the works of Graham and Lehrer and others.” should be “… the term “cellular algebras” has been used …”. (Indeed I’m more familiar with the Graham-Lehrer setting so this post was a little odd to read at the start.

oops… – thanks, now fixed.

The story is back in the news. A Chilean judge has ruled that Boris Weisfeiler’s disappearance was not a human rights violation, but a common crime (to which a statute of limitations applies) and the case should now be closed …