# Is there a characterization of hyperplane arrangement intersection posets?

For a hyperplane arrangement $\mathcal{A}$ over a vector space $V$, we define its intersection poset, $L(\mathcal{A})$, as the set of all nonempty intersections of hyperplanes in $\mathcal{A}$ ordered by reverse inclusion. The empty intersection, $V$ itself, is the unique minimal element of $L(\mathcal{A})$.

It is known that $L(\mathcal{A})$ is a ranked meet-semilattice, and moreover, any interval $[x,y]$ in $L(\mathcal{A})$ is a geometric lattice. But these properties alone are not sufficient for some poset $P$ to be the intersection poset of a hyperplane arrangement. Consider the following poset:

If this were the intersection poset of some arrangement, then $a$ would be parallel to $d$ and to $c$, $b$ would be parallel to $d$, and thus $b$ and $c$ would be parallel. But $b$ and $c$ have nonempty intersection, so this is nonsense.

Is there a known characterization of hyperplane arrangement intersection posets?

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Chapters 4 and 8 of Oriented Matroids By Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, Gunter M. Ziegler reviews the big face lattice of oriented matroids, and when that is realizable as a hyperplane arrangement. Chapter 4 is self contained and I think you can skip to chapter 8 from there (unfortunately no e-book available)

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Thank you for the reference, I will get a copy of this book. I assume that the answer is in general rather complicated? – Sam Hopkins Jan 24 '13 at 4:31
Yes indeed; it's complicated to the point where I have started to think that arrangements of pseudo-spheres is the "right" class to be talking about rather than the less general hyperplane arrangements. That's because the face lattice of pseudo-sphere arrangements has a nice characterisation in terms of shelability discovered by Björner. – Rabee Tourky Jan 24 '13 at 7:32
Also if you do find something interesting resolving this please do write it up here. – Rabee Tourky Jan 24 '13 at 7:34
Just the question of when a geometric lattice is the intersection lattice of a linear hyperplane arrangement is quite complicated. It is equivalent to asking when a matroid can be represented over a field. – Richard Stanley Jan 25 '13 at 1:03
@Richard Stanley yes, however, if it is was easy to know when matroid lattice can be realized as a linear hyperplane arrangement, then it would also be easy to know when an oriented matroid face lattice can. But not the converse. – Rabee Tourky Jan 25 '13 at 7:33

The intersection poset of a (not necessarily central) hyperplane arrangement is a geometric semi-lattice, as defined by Bjorner and Wachs, who show that every such poset is isomorphic to the subposet of $x \not \geq a$ of a geometric lattice and an atom a. This corresponds geometrically to putting the arrangement into projective space, and adding the hyperplane at infinity, which corresponds to $a$. I haven't looked at the definition, but I suspect your example above is not a geometric semi-lattice; if it were it should be "realizable" by an arrangement of pseudo-hyperplanes, but your argument shows it cannot.

Given a geometric semi-lattice, it is realizable by an arrangement of affine hyperplanes if and only if the associated geometric lattice is realizable by linear hyperplanes, which is equivalent to the matroid-realizability question as Richard points out above. For this you need to specify a field - some lattices are realizable over $\mathbb C$ or over a finite field, but not over $\mathbb R$. Oriented matroids and arrangements of pseudo-spheres are relevant only to realizability over $\mathbb R$.

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