Let $\mathcal A$ be a central hyperplane arrangement in $\mathbb R^d$ and let's assume that it is essential, meaning the hyperplanes in $\mathcal A$ intersect in the origin. The intersection lattice $L(\mathcal A)$ consists of those subspaces of $\mathbb R^d$ that are intersections of hyperplanes from $\mathcal A$ (including the empty intersection which is interpreted as $\mathbb R^d$) ordered by reverse inclusion. So notice that $\mathbb R^d$ is the bottom of $L(\mathcal A)$ and $\{0\}$ is the top. Let $\mu$ be the Mobius function of $L(\mathcal A)$.

Consider the quantity $$Q_{\mathcal A}=\sum_{W\in L(\mathcal A)}|\mu (W,\{0\})|.$$

Does $Q_{\mathcal A}$ have a natural interpretation in terms of the combinatorics of the arrangement? Note that Zaslavsky's theorem gives $$\sum_{W\in L(\mathcal A)}|\mu(\mathbb R^d,W)|$$ is the number of chambers of the arrangement, but this is, in some sense that is imprecise, a dual quantity.

I know that when $\mathcal A$ is the braid arrangement in $\mathbb R^n$, $Q_{\mathcal A}$ counts the number of necklaces of set partitions of $n$ (i.e., you look at ordered set partitions of $\{1,\ldots,n\}$ up to cyclic permutations).

  • $\begingroup$ You mean "central," not "essential." $\endgroup$ May 14 '21 at 20:57
  • $\begingroup$ @RichardStanley I meant both but only so I could call the intersection of the hyperplanes 0. Maybe my terminology is slightly off $\endgroup$ May 14 '21 at 21:16

Let $\Delta$ be a matroid complex, i.e., an abstract simplicial complex whose faces are the independent sets of a matroid $M$. Let $K[\Delta]$ denote the face ring (aka "Stanley-Reisner ring") of $\Delta$ over a field $K$. Let $\beta_i(K[\Delta])$ denote the Betti numbers of a minimal free resolution of $K[\Delta]$, regarded as a module (in fact, quotient ring) over $K[x_1,\dots,x_n]$, where $x_1,\dots,x_n$ are the vertices of $\Delta$. Then $$ \beta_i(K[\Delta]) = \sum |\mu(W,\{0\})|, $$ where $W$ ranges over all flats of $M$ of corank $i$. In particular, the sum of all the Betti numbers is the sum requested. See Theorem 9 of http://math.mit.edu/~rstan/pubs/pubfiles/34.pdf.

  • $\begingroup$ Thanks. Let me assimilate this. $\endgroup$ May 14 '21 at 21:23
  • $\begingroup$ Can these betti numbers be computed from the simplicial complex? $\endgroup$ May 14 '21 at 21:48
  • $\begingroup$ See the above reference. For general simplicial complexes, the Betti numbers are sums of dimensions of homology groups of restrictions $\Delta_W$ of $\Delta$ to some subset $W$ of the vertices (Theorem 4). For matroid complexes, the homology of each $\Delta_W$ vanishes except in the top dimension (this property is in fact equivalent to being a matroid complex), so the top dimensional homology is just the reduced Euler characteristic of $\Delta_W$, up to sign. $\endgroup$ May 14 '21 at 23:27
  • $\begingroup$ OK. Thanks. I remember reading about this once long ago in connection with Leray numbers. Ok. I'll try to read the paper. Thanks for your answer. $\endgroup$ May 15 '21 at 0:04

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