## Hyperplane face semigroups and zonotopes

Hyperplane face semigroups have attracted interest in recent years due to applications to Markov chains, descent algebras and buildings. Here I’ll describe an alternative way to think about hyperplane face semigroups from the zonotope view point.

Let me start with zonotopes. A zonotope is a linear projection of a hypercube. In other words, it is an image of $[-1,1]^n$ under some linear map. Alternatively, one can define a zonotope as a Minkowski sum $Z=[-v_1,v_1]+\cdots+[-v_n,v_n]$ of line segments. In this case $Z$ is the image of $[-1,1]^n$ under $(t_1,\ldots, t_n)\mapsto\displaystyle{\sum_{i=1}^n t_iv_i.}.$ Zonotopes are convex polytopes and each face of a zonotope is a zonotope (it is an image of a face of the hypercube).

Zonotopes are dual to central hyperplane arrangements. Let $\mathcal H=\{H_1,\ldots, H_n\}$ be a central hyperplane arrangement in a finite dimensional real inner product space $V$. We assume that they are give by linear forms $f_i=0$, for $i=1,\ldots, n$, with $f_1,\ldots, f_n\in V^*$. The dual zonotope $Z(\mathcal H)$ is the Minkowski sum $[-f_1,f_1]+\cdots+[-f_n,f_n]$ in $V^*$.

Associated to each point $v\in V$ is a sign vector (also called a covector in oriented matroid theory). If $x\in\mathbb R$, then $\mathrm{sgn}(x)$ is $+,-,0$ according to whether $x$ is positive, negative or zero. The sign vector of $v\in V$ is then given by $\sigma(v)=(\mathrm{sgn}(f_1(v)),\ldots, \mathrm{sgn}(f_n(v)))$. For example, if $\mathcal H$ is the coordinate hyperplane arrangement $x_1=0, x_2=0,\ldots, x_n=0$ in $\mathbb R^n$, then the sign vector of $v$ just records the sign of each of its coordinates. The dual zonotope in this case is just the $n$-cube $[-1,1]^n$.

Another important hyperplane arrangement is the braid arrangement in $\mathbb R^n$. It consists of the hyperplanes $x_i-x_j=0$, with $1\leq i. Sign vectors are in bijection with ordered set partitions of $[n]$. For instance, the ordered set partition $(\{1,2\},3,\{4,5\})$ of $[5]$ corresponds to all sign vectors of points $(x_1,\ldots, x_5)\in \mathbb R^5$ with $x_1=x_2. The dual zonotope to the braid arrangement is the permutahedron. It is the convex hull of all vectors in $\mathbb R^n$ of the form $(\sigma(1),\sigma(2),\ldots, \sigma(n))$ with $\sigma\in S_n$.

The faces of $Z(\mathcal H)$ are in bijection with sign vectors of the arrangement $\mathcal H$. Using $V^{**}=V$ we can think of vectors in $V$ as functionals on $V^*$. Then one can check that the functionals associated to two vectors are maximized on the same face of $Z(\mathcal H)$ if and only of they have the same sign vector. If $v=(\sigma_1,\ldots,\sigma_n)$ is a sign vector of the arrangement, the corresponding face of $Z(\mathcal H)$ is the Minkowski sum $\sum_{\sigma_i=0} [-f_i,f_i]+\sum_{\sigma_i=+} \{f_i\}+\sum_{\sigma_i=-}\{-f_i\}$, which is again a zonotope!

In oriented matroid theory there is standard way to multiply sign vectors. First we define a monoid structure on $L=\{+,-,0\}$ as follows. $0$ is the identity element. The elements $+,-$ are left zeroes, that is $+\cdot x=+$ and $-\cdot x=-$ for all $x\in \{+,-,0\}$. Now we can view the set of $n$-dimensional sign vectors $L^n$ as a monoid. A beautiful observation of Tits is that the set $F(\mathcal H)$ of sign vectors associated to
$F(\mathcal H)$ is a submonoid of $L^n$. To see this let $x,y\in V$. Then $\sigma(x)\sigma(y)=\sigma(z)$ where $z$ is the result of making a small movement on the line segment from $x$ to $y$.

I want to give an action of the monoid $F(\mathcal H)$ on the zonotope by cellular maps.

Let’s consider first the coordinate hyperplane arrangement $\mathcal H_n$ in $\mathbb R^n$. This case is very easy and explicit. Recall that $F(\mathcal H_n)=L^n$ and $Z(\mathcal H_n)=[-1,1]^n$. Consider the following three functions $\mathbb R\to \mathbb R$: $r_0$ is the identity: $r_+$ is the constant map to $1$; and $r_-$ is the constant map to $-1$. Then the assignment $\sigma=(\sigma_1,\ldots,\sigma_n)\to r_{\sigma_1}\times\cdots\times r_{\sigma_n}=r_\sigma$ gives an isomorphism of $F(\mathcal H_n)$ with a monoid of cellular mappings on $Z(\mathcal H_n)$.

I’ve not been able to generalize this explicit construction to arbitrary arrangements. Instead, I use the fact that a zonotope $Z$, like any polytope or regular CW complex, is homeomorphic to the geometric realization $\|\Delta(F(Z))\|$ of the order complex $\Delta(F(Z))$ of its face poset $F(Z)$.
Moreover, if we give $\|\Delta(F(Z))\|$ the CW structure whose closed balls are the geometric realizations of principal order ideals, then the above homeomorphism is a combinatorial isomorphism of CW complexes.

The beautiful fact is that if $\mathcal H$ is a hyperplane arrangement, the face poset of the zonotope $Z(\mathcal H)$ is Green’s $\mathcal R$-order on $F(\mathcal H)$! This is a standard fact in oriented matroid theory and can be easily checked from the above correspondence between sign vectors and faces. The reader should check that the $\mathcal R$-preorder on $F(\mathcal H)$ is an order and boils down to $F\leq G$ if and only if $GF=F$.

Now the $\mathcal R$-order is stable for left multiplication for any semigroup and so $F(\mathcal H)$ acts on the face poset of $Z(\mathcal H)$ by order-preserving maps. Thus it acts on $\|\Delta(F(Z(\mathcal H)))\|$ by cellular maps with respect to the CW structure defined above. As this is a combinatorially isomorphic CW complex with $Z(\mathcal H)$ we have found the desired action.

I would very much like a direct description of this action which avoids going through face posets and order complexes. There should be something as simple as in the case of coordinate hyperplane arrangements!

One application of this action is that the augmented cellular chain complex for $Z(\mathcal H)$ is a minimal length projective resolution of the trivial $\mathbb ZF(\mathcal H)$-module and so the cohomological dimension of $F(\mathcal H)$ is $\dim Z(\mathcal H)$.

All this can be generalized to affine hyperplane arrangements, oriented matroids and complex hyperplane arrangements. This is part of forthcoming work with Margolis and Saliola.