Whitney Encounters of the Second Kind

I’m speaking tomorrow in the AMS Current Events Bulletin about the work of Adiprasito, Huh, and Katz on the Rota-Welsh conjecture and Hodge theory for matroids.   See this previous post for an introduction to their work.  My write-up for the Current Events Bulletin is included in this booklet.

Here’s an excerpt from the last section of my slides which I may or may not have time to discuss in tomorrow’s talk.  It concerns this recent paper of June Huh and Botong Wang.  (Note added: As anticipated I did not have time to cover this material!  Here are the slides themselves: ceb_talk)

Let M be a simple matroid of rank r with lattice of flats L (see this post for the relevant definitions).  Let W_k(M) be the number of flats in M of rank k; these numbers are called the Whitney numbers of the second kind for M.  (The Whitney numbers of the first kind are the coefficients of the characteristic polynomial of M, which Adiprasito, Huh, and Katz prove form a log-concave sequence.)  The reason for the terminology is that if M is the matroid associated to the complete graph K_n on n vertices, the Whitney numbers of the first (second) kind coincide with the Stirling numbers of the first (second) kind.

The following two conjectures remain open:

Conjecture (Welsh): The Whitney numbers of the second kind for M form a log-concave, and hence unimodal, sequence.

Conjecture (Dowling and Wilson’s “Top-Heavy” conjecture): For all k < r/2 we have W_k(M) \leq W_{r-k}(M).

The top-heavy conjecture can be viewed as a vast generalization of the de Bruijn-Erdos theorem that every non-collinear set of points E in the plane determines at least |E| lines.

There has been significant recent progress by Huh and Wang on the representable case of the above conjectures.  They prove:

Theorem (Huh–Wang, 2016): For all simple matroids M representable over some field:

  1. The first half of the sequence of Whitney numbers of the second kind is unimodal, i.e., W_1(M) \leq W_2(M) \leq \cdots \leq W_{\lfloor r/2 \rfloor}(M).
  2. The top-heavy conjecture is true.

Unlike the situation with Whitney numbers of the first kind (e.g. in the work of Huh-Katz), the projective algebraic variety Y_M which one associates to M in this case is highly singular; thus instead of invoking the Kahler package for smooth projective varieties, Huh and Wang have to use analogous but much harder results about intersection cohomology which are formulated and proved using perverse sheaves. (Specifically, they require the Bernstein–Beilinson–Deligne–Gabber decomposition theorem for intersection complexes and the Hard Lefschetz theorem for l-adic intersection cohomology of projective varieties.)

To extend the arguments of Huh and Wang to the non-representable case, the first of several serious challenges would be to construct a combinatorial model for the intersection cohomology of the variety Y_M.

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