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@johncarlosbaez hmm, in the abstract, this sounds like the halting problem but for maths. Perhaps property testing frameworks can help here? because technically speaking, modeling this is the hardest thing, afterwards the computer can try bigger and bigger examples of such groups until one doesn't match, and then reduce from there, but that would require a substantial amount of compute and power, if it can be done at all that is

@esoteric_programmer - as a mathematician my natural reaction is that people have found a brute-force proof of McKay's conjecture, but a really good proof still awaits us. A good proof would give a better explanation of why this is true.

There's no guarantee that such a really good proof exists, but in this situation mathematicians keep searching... and often they find one.

@johncarlosbaez O yeah, this is definitely brute force territory, although a slightly more clever one. Hmm, I have a feeling that a real proof in this domain may lead to some interesting cryptography discoveries

@highergeometer - Thanks! It seems this proof relies on the classification of finite simple groups, which makes it pretty scary.

@johncarlosbaez ok, I wondered if it did (I didn't get a chance to look at the paper) because that's the big result in finite groups that gives you an exhaustive case breakdown.

@highergeometer - it's nice to see that this big result is serving to fuel further progress. I still can't tell if they use the whole classification or just a bunch of parts of the classification. For some reason finite simple groups of Lie type were the holdouts.

edit: There's definitely some folks who won't accept a proof unless it's classification-free. It'd certainly help me sleep better at night too...

@soaproot @johncarlosbaez @highergeometer These "counting conjectures" involving characters are usually reduced to the f.s.g. (or quasi-simples) case by finding the right "inductive" conditions - conditions that allow you to reduce the question for the group G to the subquotients appearing in a composition series. The inductive conditions are generally *much* more involved than the original statement. Once the right inductive conditions are found, it's a matter of showing the inductive conditions hold for all the finite simple groups!

For instance, for the blockwise variant, the Alperin-McKay conjecture, Späth determined inductive conditions that only involve f.s.g.s, rather than quasi-simples. This variant is closed for p=2 by Ruhstorfer (building on the work of many others of course) but open for all others, see https://arxiv.org/abs/2204.06373.

Another conjecture in the field was also recently closed, Brauer's Height Zero conjecture! This was closed by Malle, Navarro, Schaeffer-Fry, and Tiep. This one is implied by Alperin-McKay, but the proof used for odd primes was novel, instead relying on a minimal counterexample argument (which is inductive in a way...). See https://arxiv.org/abs/2209.04736 - it was recently accepted into the Annals! I can't say too much more on this - character theory is not my specialty.

@redrot -Thanks! That's fascinating but too technical for me to fully understand. Here is the question I'm wondering about: how much does the proof of McKay's theorem (or the other results you mentioned) require a case-by-case study of the sporadic finite simple groups? E.g. do we need to know there are 26 and handle each one separately? Or can we lump them together?

@soaproot @highergeometer

@johncarlosbaez @soaproot @highergeometer

edit: to be brief - it's a case-by-case analysis but it's not too bad.

The original McKay conjecture was proven for all sporadic simple groups and all primes essentially one by one, by Wilson: https://core.ac.uk/download/pdf/81156842.pdf

Then the (additional) inductive conditions were shown for all simple groups not of Lie type (this includes all sporadics) by Malle in 2008: https://www.tandfonline.com/doi/full/10.1080/00927870701716090#d1e1616

I wasn't actually too familiar with the history of the proof or the reduction techniques used in the final proof - one thing that surprises me is that results and reductions by other researchers from as recent as 2023 were necessary (the proof was announced in 2023, for reference). It really was a massive team effort!

@johncarlosbaez @soaproot @highergeometer

It may also be interesting that in some cases, the "blockwise" variant, the *Alperin*-McKay conjecture, is implied by a categorical conjecture referred to as "Broué's abelian defect group conjecture." This is likely what Radha Kessar was referring to in the Quanta article when she made her comment about structural reasons. Here's a quick rundown:

If $k$ is a field of prime characteristic 𝑝, then the summands of the group algebra $kG$ are called "blocks" and each has its own "defect group," a $p$-subgroup analogous to a Sylow $p$-subgroup. If a block of $kG$ has defect group $D$, then there is a corresponding block of $k{N}_G(D)$ with the same defect group (this is a fundamental result of Brauer). The weaker form of the conjecture (which is enough to imply Alperin-McKay) predicts that if $D$ is abelian, then the two blocks have equivalent derived module categories. The stronger version, which is my interest, also requires permutation modules, for rather surprising reasons!

The abelian defect group conjecture is very, very far from being closed I believe - there is no "inductive" reduction to f.s.g.s, and very well may never be. However, it is proven in a number of cases - a notable one is the proof for all blocks of symmetric groups by Chuang-Rouquier, who used a categorification of ${\mathfrak{s}\mathfrak{l}}_2$, one of the landmark results in categorification techniques (I think). It's also fairly open at the moment what the non-abelian version of this conjecture should be.

@redrot - thanks, this stuff is incredibly cool! I'll probably never get to it, as I'm still trying to learn about cuspidal representations of GL(𝑛,𝔽ₚ), but it's nice to know this stuff is out there.

@redrot - it's a great article. I got informed of it when someone sent me an email claiming that S₄ is a counterexample. He had been informed by Copilot (Microsoft's AI system) that S₄ has a Sylow p-subgroup that is normal. I said "oh?"

@adityakhanna - cool! I'd never heard of the McKay conjecture until yesterday, even though I was friends with McKay.