A few thoughts on the Donovan conjecture
A few thoughts on the Donovan conjecture (by Peter Donovan, August 2020)
Richard Brauer was primarily interested in developing block theory as a means of better understanding group theory using the complex character theory. He had the concepts of a defect group, a modular irreducible, the decomposition matrix and the Cartan matrix but not of Morita equivalence. He had a bound on the number of isotypes of irreducibles in a block with defect group isomorphic to a fixed defect group and a possible upper bound on the decomposition numbers that could occur, again with prescribed isotype of defect group. This latter was in fact incorrect.
I had worked earlier with Freislich on extending Gelfand's work on the four subspace problem to the other extended Dynkin diagrams. The text was somewhat long and of necessity repetitive and emerged only as semi-published lecture notes. So I was well aware of the finite-tame-wild trichotomy and so ultimately got the idea of reversing the natural order of things and classifying tame symmetric algebras with at most three isotypes of irreducibles. This led to my Dihedral Defect Groups paper, which (sort of) showed that the bounding of the Cartan matrix (and so the decomposition matrix) was possible in this case. Erdmann has since completed this analysis.
With the cyclic defect group case very well understood and now (for the prime 2) a few other cases turning up, the Donovan conjecture on finiteness of the number of possibilities for the structure of blocks with given isotype of defect group seemed reasonable, at least at the oral level. However, for odd characteristic and with the defect group being the product of two cyclic groups, what seemed (and still seems) necessary is to define a Brauer whatsit that describes at least the modular data. After this, research perhaps could be extended to the general complexity two case. I agonised over what the product of two Brauer trees must be. My proposed method of proof was to use the evident (after Quillen's work, summarised by Benson in his admirable pair of books) extra structure on block algebras. My Spectral Duality paper has enough hints on what I had in mind.
This is not the place to comment on more recent work.
Time has rolled on with my watching developments and writing some reviews for MR/mathscinet. I have maintained contact with two old friends working in the subject.
Old men forget (Shakespeare, Henry V) but here are some recollections of my mathematical development. My 1968 PhD thesis, supervised by Atiyah at Oxford, covered a [math]p[/math]-adic Brauer character formula for finite groups acting on algebraic varieties in characteristic [math]p\gt 0[/math]. It at least introduced the canonical [math]p[/math]-adic trace for endomorphisms of vector spaces. I was influenced by Mackey's lectures at Oxford. My work with Karoubi was one of the earliest papers on non-commutative geometry. Atiyah's last paper (joint with Segal) augmented and redid this material many years later. More recently I rediscovered the cryptology involved in two of the major communication intelligence coups of the Pacific part of World War II. See the Springer Donovan-Mack book.
Now, as an aging grandfather sheltering with the grandmother at home from the current pandemic, I am re-examining Grothendieck's SB149 work to find a duality Riemann-Roch formula, re-examining the Donovan conjecture material (and Linckelmann's two-volume work helps here) and (very difficult indeed) have an eye on some work of Connes. One out of these three would be wonderful. Meanwhile a data base on Pacific cryptology is being augmented as opportunity turns up.