This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference: http://www.math.harvard.edu/cdm/.

Here are two examples of the kind of combinatorial abstractions of geometric objects referred to in the title of this question:

  • Coxeter groups. These are abstractions of Weyl groups. Weyl groups have geometry coming from Lie theory: they are finite reflection groups associated to a crystallographic root system. Weyl groups (or perhaps finite reflection groups, or including Weyl groups associated to affine lie algebras, etc.) are then the "realizable" Coxeter groups.
  • Matroids. These are abstractions of collections of vectors in some vector space. The matroids coming from collections of vectors in some vector space (over some field, say) are again the "realizable" matroids.

Here is what I mean by "behave so well":

Often it happens that we can associate some interesting polynomial invariant to the combinatorial object in question. Some examples are:

  • The Kazhdan-Lusztig (KL) polynomial associated to a Coxeter system.
  • The characteristic polynomial associated to a matroid.
  • The recent KL polynomial associated to a matroid (see https://arxiv.org/abs/1412.7408).

And these polynomials have surprising and deep properties (positivity or unimodality/log-concavity of coefficients) that are not at all obvious from their definitions. A recurring theme is that these properties can be established in the "realizable" cases by appealing to algebraic geometry, specifically, to some suitable cohomology theory. However, the properties continue to hold for the general, nonrealizable objects for which there is no underlying geometry. The proofs of the general result are usually more "elementary" in so far as they avoid any algebraic geometry; but chronologically they come after the realizable results.

For instance, the coefficients of KL polynomials associated to a Coxeter system are positive. This was a famous conjecture of Kazhdan-Lusztig, proved a few years ago by Elias and Williamson (https://arxiv.org/abs/1212.0791). However, positivity was known for realizable Coxeter groups much earlier by interpreting the polynomials as Poincaré polynomials for the intersection cohomology of certain Schubert varieties.

Similarly, it is conjectured that the KL polynomial of a matroid has positive coefficients (see https://arxiv.org/abs/1611.07474); and this conjecture is known to be true when the matroid is realizable, again by interpreting the coefficients as dimensions of intersection cohomology spaces on certain varieties.

Or for the characteristic polynomial of a matroid: we know that the coefficients of this polynomial are log-concave, as was recently proved in the remarkable work of Adiprasito-Huh-Katz (https://arxiv.org/abs/1511.02888). Again, this result was preceded by the same result for the realizable case, due to Huh-Katz (https://arxiv.org/abs/1104.2519), interpreting the coefficients as intersection numbers for some toric variety.

So we come to my question:

Why do combinatorial abstractions of geometric objects behave so well, even in the absence of any underlying geometry?

EDIT: At around the 50 minute mark of his plenary talk at ICM 2018 (on Youtube here: https://www.youtube.com/watch?v=-3q6C558yog), Geordie Williamson asks a roughly similar question, and suggests that it may be a "mystery for the 21st century."

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    Remarkable question! My hope is that cohomology theories used for the realizable structures are in fact partial cases of more general theories, which work in the same way for more general combinatorial structures. – Fedor Petrov Nov 27 '16 at 19:23
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    I disagree with the claim that general Coxeter groups are "objects for which there is no underlying geometry". I recommend reading the book "The Geometry and Topology of Coxeter Groups" by Mike Davis. – Uri Bader Nov 27 '16 at 22:05
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    @UriBader: it's a fair response. Perhaps one answer could be exactly that all these objects are "realizable" in some generalized sense (e.g., all matroids are realizable over hyperfields, see: arxiv.org/abs/1601.01204). But still, in the general case, the geometry is far from clearly explaining why these deep properties of the associated polynomials hold. – Sam Hopkins Nov 27 '16 at 22:11
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    Is there anything wrong with the following point of view? What we are learning is that the correct "explanation" of positivity and log-concavity is combinatorial, rather than geometric. We tend to think of geometry as "explaining" these things only because that's the familiar, easier case, but once we get more familiar with combinatorics, we'll come to see the combinatorial proof as the natural one. – Timothy Chow Nov 28 '16 at 16:38
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    I'm not an expert on this by any means, but in the Adiprasito-Huh-Katz paper that proves some log-concavity conjectures for general matroids, what's roughly happening (and this is not really discussed in the paper, the terminology is more combinatorial) is that there is in fact underlying geometry. In this case, the geometry is tropical. Matroids in some sense are exactly the tropical varieties of degree 1. (This can be seen as the generalized sense 'of "realizable" you mentioned). – Rasta Mouse Sep 19 '17 at 17:07

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