Hilbert schemes and tableaux

I’m flying to Japan next week to spend the summer there and so this will likely be my final post on American soil for a while. My friend Jeremy mentioned to me a while ago that the Bridgeland-King-Reid (BKR) theorem – which is the derived version of the McKay correspondence – is used in Haiman’s proof of the Macdonald positivity conjecture. While I found this intriguing, most of the literature regarding the proof is rather lengthy and so I didn’t initially follow up on his comment. There were celebrations for the graduating PhD students in the department last week that involved short talks describing the theses of the graduates, one of which expounded on Macdonald polynomials and the geometry of Hilbert schemes, which are currently inseparable in terms of the proof methods in the field. During the talk, I suddenly realised how one can associate a tableau to the toric zero-strata (or, alternatively, the affine pieces) of \text{Hilb}^n\mathbb{A}^2, and hence found a much needed access-point into the literature.

Anecdotes aside, Haiman’s proof of the Macdonald positivity conjecture is long and complicated, but seems to have a nice narrative structure that his survey articles emphasise. I was reminded of comment of Ian Stewart that I read last week about the seven (or so) storylines that populate all of Hellenistic drama, and how this resembles the relatively few plots of math proofs, however long they are…

My notes above try to sketch the route the proof follows dwelling on the details that I found most interesting and that were reasonably quick to digest. The result of BKR – which deals with equivariant Hilbert schemes – is applicable because of the fact that the standard Hilbert scheme \text{Hilb}^n\mathbb{A}^2 is isomorphic to the equivariant Hilbert scheme S_n\text{-Hilb}\,\mathbb{A}^{2n} where S_n acts by permuting pairs of coordinates; in this sense, really \mathbb{A}^{2n}=(\mathbb{A}^2)^n. It’s not so hard to see this isomorphism: tuples in \mathbb{A}^{2n} considered up to the action of S_n are just sets of n points in \mathbb{A}^2 with some care taken at tuples with multiple copies of the same point.

The Macdonald positivity conjecture states that the coefficients (which are only a priori rational functions with rational coefficients) in expressing certain combinatorially/algebraically interesting polynomials indexed by partitions as a linear combination of the Schur basis for the algebra of symmetric functions are actually polynomials with positive integral coefficients. A standard means of accomplishing such a feat is to realise the original polynomial as some sort of dimension count; in this case, as the Frobenius series of a bigraded S_n-module. This module turns out to be the fibre of the tautological bundle over S_n\text{-Hilb}\,\mathbb{A}^{2n} at the zero-stratum corresponding to the relevant partition.

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