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Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$. It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no pentagon sublattice and pentagons are non-quadratic). Note that being Koszul implies that an algebra is quadratic.

Question 1: Which modular lattice $L$ have the property that $KL$ is a Koszul algebra? (Does it depend on the field?)

All distributive lattices should be Koszul.

Question 2: Is there a nice ring-theoretic/combinatorial interpretation of the Koszul dual algebra of the incidence algebra of a modular lattice that is Koszul (other than using the quadratic perp)?

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    $\begingroup$ D. Woodcock. Cohen-Macaulay complexes and Koszul rings. J. London Math. Soc. (2), 57(2):398–410, 1998 and P. Polo. On Cohen-Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties. Bull. London Math. Soc., 27(5):425–434, 1995. $\endgroup$
    – Benjamin Steinberg
    Commented May 30, 2022 at 13:35
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    $\begingroup$ @BenjaminSteinberg Thanks, I wonder whether this leads to an elementary order theoretic characterisation when modular lattices are Koszul. $\endgroup$
    – Mare
    Commented May 30, 2022 at 13:40
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    $\begingroup$ Every interval of a modular (even semimodular) lattice is Cohen-Macaulay, so by the discussion above all modular lattices are Koszul. $\endgroup$
    – Richard Stanley
    Commented May 30, 2022 at 14:53
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    $\begingroup$ @Mare, being modular is not necessary. Semimodular is enough. And I’m not sure that is necessary $\endgroup$
    – Benjamin Steinberg
    Commented May 30, 2022 at 19:38
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    $\begingroup$ Look at the semimodular but nonmodular in en.wikipedia.org/wiki/Semimodular_lattice. The Pentagon is embedded as a sublattice but the Pentagon does not embed in the hasse diagram, which is the relevant issue for a Quiver presentation. There are some elements in the lattice between various elements of the Pentagon $\endgroup$
    – Benjamin Steinberg
    Commented May 30, 2022 at 19:56

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Here is a summary of what I know, with @RichardStanley's remark taken into account. It was shown independently by Polo and by Woodcock that the incidence algebra of a graded poset is Koszul iff each open interval of the poset is Cohen-Macaulay (over the field in question). See P. Polo. On Cohen-Macaulay posets, Koszul algebras and certain modules associated to Schubert varieties. Bull. London Math. Soc., 27(5):425–434, 1995 and D. Woodcock. Cohen-Macaulay complexes and Koszul rings. J. London Math. Soc. (2), 57(2):398–410, 1998.

A semimodular lattice has Cohen-Macaulay open intervals (as pointed out by @RichardStanley) and so has a Koszul incidence algebra. This includes modular lattices.

Being modular is not necessary for being quadratic. The issue is the Hasse diagram need not contain a pentagon for a non-modular lattice. The pentagon is just a sublattice but it is not "convexly" embedded.

A fairly general class of graded posets with quadratic incidence algebras (and maybe they are exactly the class) are those which are what is sometimes called strongly connected, meaning that the order complex of each closed interval is a chamber complex (meaning any two facets are connected by a gallery as people study them in the theory of buildings). This means that any two maximal chains with the same top and bottom can be connected by a sequence of moves where you change only one element of the chain at each step. Semimodular lattices are strongly conencted, for example. We prove in Lemma 7.4 of https://www.ams.org/books/memo/1345/memo1345.pdf (you can also find it on ArXiv) that the incidence algebra of a strongly connected poset is quadratic. We also observe the quadratic dual is defined simply by reversing the Hasse diagram and taking the relation which is the sum of all paths of length 2. This is the quadratic perp but it is a nicer description in this case.

We show that for geometric lattices associated to oriented matroids (including central hyperplane arrangements), the Koszul dual of the incidence algebra is the monoid algebra of the monoid of covectors of the oriented matroid. But the proof involves a nontrivial topological argument to show that this monoid has a quiver presentation matching the quadratic perp. We prove a similar thing for intersection semilattices of affine hyperplane arrangements and affine oriented matroids, as well as some other CW complexes that have a left regular band multiplication on its face poset including finite CAT(0) cube complexes. Again the incidence algebra of a certain strongly connected meet semilattice is Koszul dual to the semigroup algebra of the left regular band of the CW complex.

I don't know of a general combinatorial description of the Koszul dual.

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  • $\begingroup$ Thank you very much. Is the arxiv version identitcal in terms of content to the book version? $\endgroup$
    – Mare
    Commented May 30, 2022 at 20:30
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    $\begingroup$ I think so. Pretty close if not exact $\endgroup$
    – Benjamin Steinberg
    Commented May 30, 2022 at 22:59
  • $\begingroup$ I'm fairly confident for graded posets quadratic incidence algebra is equivalent to strongly connected. I think if it is quadratic the admissible ideal must actually be generated by all differences of paths of length 2. Then I think writing the difference of paths of length n as belonging to this ideal should produce a gallery. $\endgroup$
    – Benjamin Steinberg
    Commented May 30, 2022 at 23:54

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