Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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: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?

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

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)

share|improve this answer
    
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
1  
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
add comment

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 $\mathbb R$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.