Sarah Peluse has just proved a result with a very appealing statement. It concerns the character tables of the symmetric groups S_n. It can be shown that the entries of these tables are integers. (To get a feel for why, see e.g.
www-users.math.umn.edu/~tlawson/old/1….) 1/
www-users.math.umn.edu/~tlawson/old/1….) 1/
Experimental evidence appeared to suggest that as n gets large, almost all the entries of the character table are not just integers, but even integers. Peluse shows that this is indeed the case: the proportion of even entries tends to 1. 2/
arxiv.org/abs/2007.06652
arxiv.org/abs/2007.06652
At least from a quick glance, the proof (actually, she gives two proofs) looks very clever, but I don't know enough about the area, so some of the techniques that look surprising to me may be more standard to experts. But given that Peluse has surprised me in areas ... 3/
that I *do* know about, my money is on its being pretty much as cool an argument as it looks.
The story doesn't end here. The argument generalizes to prove that almost all entries are multiples of 3, 5, 7, 11, and 13, but it's conjectured that ... 4/
The story doesn't end here. The argument generalizes to prove that almost all entries are multiples of 3, 5, 7, 11, and 13, but it's conjectured that ... 4/
the same should be true for all primes. You might think that once you'd cracked it up to 13 you'd see the pattern and would be able to do the whole problem. But it turns out that for her argument to work, she needs a certain quantity to be positive, ... 5/
which is possible for p up to 13 but not possible for p greater than or equal to 17. I can't off the top of my head think of any other good examples of arguments with this property. 6/
The property I mean is that they prove in a natural and uniform way that a conjecture that's believed to be true for all n is true up to some quite large n, but they stop working beyond that n. 7/7
