The signed Varchenko Determinant for Complexes of Oriented Matroids

Varchenko [17] considered a symmetric matrix which may be viewed as a bilinear form on the vector space of linear forms of the set of regions of a hyperplane arrangement $\mathcal{A}$ over some ordered field $\mathbb{K}$. The value of the product of the characteristic vectors of regions $Q_{i}$ and $Q_{j}$ is given by a product $\prod_{e\in S(Q_{i},Q_{j})}w_{e}$, where the $w_{e}$ are weights on the hyperplanes $H_{e}$ of the arrangements and $S(Q_{i},Q_{j})$ is the set of hyperplanes that have to be crossed on a shortest path from $Q_{i}$ to $Q_{j}$. The corresponding Varchenko Matrix $B_{\mathcal{A}}$ has entries of the form $\prod_{e\in S(Q_{i},Q_{j})}w_{e}$ for any pair of regions $Q_{i}$ and $Q_{j}$, also see Theorem 2.0. In order to determine when the bilinear form is degenerate, Varchenko [17] gave an elegant factorization of the determinant of that matrix, considering the weights as variables.

After the original proof of Varchenko there were several approaches to provide cleaner proofs of this result. Denham and Henlon [6] sketched an elegant alternative way to prove the result. Gente [7] provided some more details for that proof and generalized their approach to cones, which are also called topcones or in our notation supertopes, i.e. convex sets of regions. This method was generalized by Hochstättler and Welker [9] to oriented matroids, which form a combinatorial model for hyperplane arrangements reflecting their local linear structure but allowing for some global non-linearities. Aguiar and Mahajan [1] generalized the original proof of Varchenko to a signed version of the matrix and also derived the result for topcones. In the signed case one considers an oriented hyperplane arrangement and the entries of the signed Varchenko matrix depend on which side of a hyperplane a cell lies, see Definition 2.0. Randriamaro [12] generalized their proof to oriented matroids and proved the factorization formula also for topoplane arrangements [13]. In his habilitation thesis he furthermore showed very recently that topoplane arrangements form a topological representation for complexes of oriented matroids (COMs) [14], leading to our result from a different perspective. Note that Varchenko and Brylawski [5] already generalized another, simpler bilinear form from hyperplane arrangements to matroids.

Bandelt et al. generalized oriented matroids to COMs by relaxing the global symmetry while maintaining convexity and local symmetry. This framework captures a variety of classes beyond oriented matroids, e.g., distributive lattices, CAT(0)-cube complexes, lopsided sets, linear extension graphs, and affine oriented matroids, see [2]. Hochstättler and Welker proved the factorization formula not only for the full oriented matroid but also for supertopes, i.e. topal fibers in oriented matroids. Every supertope is a COM and it has even been conjectured in [2, Conjecture 1] and [10, Conjecture 1] that any COM can be realized as a supertope of an oriented matroid. Also see Problem 6.0. However, that conjecture seems out of reach at the moment. A big part of the motivation for the work in the present paper is to support that conjecture by explicitly proving the factorization formula for general COMs. The presentation as well as the proof follow the lines of Hochstättler and Welker [9]. We furthermore achieve a generalization to the signed version of the Varchenko matrix, thus generalizing Randriamaro [12].

The paper is organized as follows. In Section 2 we introduce the considered structures. In Section 3 we present some tools from algebraic topology that we need for the proof of the main theorem. The latter is presented in Section 4. We give some examples and applications in Section 5 and conclude the paper with some further remarks in Section 6.

Before we introduce the Varchenko Determinant, we need to get familiar with COMs. COMs have been introduced in [2] as a common generalization of oriented matroids, affine oriented matroids, and lopsided sets. We will use the notation from [2] and [4]. Note that the symbols $+,-$ and $0$ act like $1,-1$ and $0$ when it comes to negation and multiplication. We start with the following definitions and axioms.

Let us first present OMs as special COMs.

Let ${\mathcal{M}}=(E,\mathcal{L})$ be a COM. In the following we assume that ${\mathcal{M}}=(E,\mathcal{L})$ is simple, i.e.

In this setting the sign-vectors in $\mathcal{L}$ of full support are called topes and their collection is denoted by ${\mathcal{T}}$.

The restriction of a sign-vector $X\in\{0,+,-\}^{E}$ to $E\backslash A$, $A\subseteq E$, denoted by $X\backslash A\in\{0,+,-\}^{E\backslash A}$, is defined by $(X\backslash A)_{e}=X_{e}$ for all $e\in E\backslash A$. We also write $X|_{E\backslash A}$. The deletion of a COM is defined by $(E\backslash A,\mathcal{L}\backslash A)$, where $\mathcal{L}\backslash A=\{X\backslash A,\;X\in\mathcal{L}\}$, also written as $\mathcal{L}|_{E\backslash A}$.

Let ${\mathcal{M}}=(E,\mathcal{L})$ be a COM and $S^{+},S^{-}\subseteq E$ such that there exists a tope $T\in{\mathcal{T}}$ where $S^{+}$ is a subset of the positive elements of $T$ and $S^{-}$ is part of the negative elements of $T$. The topal fiber $\rho_{(S^{+},S^{-})}({\mathcal{L}})$ is defined by the covectors $\{X\mid X\in\mathcal{L},X_{e}=+\,\forall\,e\in S^{+},X_{e}=-\,\forall\,e\in S^{-}\}$. We denote by ${\mathcal{T}}(S^{+},S^{-})$ the set of topes of $\rho_{(S^{+},S^{-})}({\mathcal{L}})$. Since all covectors of a topal fiber have the same entries on $S^{+}\cup S^{-}$, we usually suppress the redundant coordinates in $S^{+}\cup S^{-}$, to obtain a simple COM on the groundset $E\backslash(S^{+}\cup S^{-})$. We will make use of the fact (shown in [2]) that the class of simple COMs is closed under deletion and under taking topal fibers.

For a covector $X\in\mathcal{L}$, the set $F(X)=\{X\circ Y\mid Y\in\mathcal{L}\}$ is usually called the face of $X$. We define ${\mathrm{star}}(X)=\{T\in{\mathcal{T}}\,|\,X\leq T\}$, where the componentwise ordering with respect to $0<+,-$ is used. Note that ${\mathrm{star}}(X)\backslash\underline{X}$ is the set of topes of $(E\backslash\underline{X},F(X)\backslash\underline{X})$, which is well-known and easily seen to be an oriented matroid.

Let us look at a special OM which we will need in the next chapter.

It can easily be checked that $\mathcal{C}_{n}$ is the set of covectors of an OM. We use $\mathcal{C}_{3}$ as an example:

We define the signed Varchenko matrix for COMs analogously to this matrix for hyperplane arrangements in [1]. For this purpose we introduce two variables $x_{e}^{+}$, $x_{e}^{-}$ for each element $e\in E$. Let $\mathbb{K}$ be a field and let $\mathbb{K}[x_{e}^{*}\,|\,*\in\{+,-\},e\in E]$ the polynomial ring in the set of variables $x_{e}^{*}$, $*\in\{+,-\}$, $e\in E$.

Note that the diagonal entries ${\mathfrak{V}}_{P,P}$ of the matrix are equal to 1. Let us illustrate this definition with the graphic OM of a directed triangle.

In this work we will prove the following theorem.

A corollary of this result, namely the case where $x_{e}^{-}=x_{e}^{+}$, which is the original version of the Varchenko matrix, has been already proven for OMs in [9]. We formulate it for COMs.

We start with some basics about partially ordered sets $\mathcal{P}$ (posets). For an introduction we recommend [18]. One can associate an abstract simplicial complex $\Delta(\mathcal{P})$, called order complex, to every poset. The elements of $\mathcal{P}$ are the vertices of this complex and the chains (i.e. totally ordered subsets) the faces. Two posets are homotopy equivalent if their order complexes are homotopy equivalent. A poset is called contractible if its order complex is homotopy equivalent to a point. Clearly a poset is contractible if it has a unique minimal or a unique maximal element, since this element is contained in every maximal chain and consequently in every maximal face of the order complex. For details see [18]. We introduce now the Möbius function $\mu$ of a poset:

The bounded extension $\hat{\mathcal{P}}$ of a poset is the poset together with a new maximal element $\hat{1}$ and a new minimal element $\hat{0}$. The Möbius number of $\mathcal{P}$ is defined by

where the right-hand-side is evaluated in $\hat{\mathcal{P}}$.

It follows from the following fact that the Möbius number is a topological invariant with respect to homotopic equivalence.

In particular we get the following corollary, whose second part follows from the definition of contractability and Example 3.0.

For a poset $\mathcal{P}$ and $p\in\mathcal{P}$ we denote by $\mathcal{P}_{\leq p}$ the subposet $\{q\in\mathcal{P}~|~q\leq p\}$.

We will now associate posets with COMs, so let ${\mathcal{M}}=(E,\mathcal{L})$ be a COM and let $R\in\{+,-\}^{E}$ be a fixed sign vector. We consider ${\mathcal{T}}$ as a poset with order relation

We write ${\mathcal{T}}_{R}$ if we consider ${\mathcal{T}}$ with this partial order and we call $R$ the base pattern of the poset.

Now we will introduce a theorem which will help us with our crucial Lemma 3.3.

Note that the restriction $R$ in the statement of Theorem 3.0 is not necessarily a tope of ${\mathcal{M}}$. In order to apply the Quillen Fiber Lemma in the proof of Theorem 3.0 we need the following lemma.

We need two preparatory results for the proof of Theorem 3.0. For the first one also see [8, Lemma 10]. We reprove that result here, since in the presentation in [8] the signs are chosen the opposite way. Recall from Example 2.0 that $\mathcal{C}_{n}$ is the set of covectors of the OM of the directed cycle on $n$ vertices.

Now we are in position to prove Theorem 3.0.

We now introduce ${\mathcal{T}}_{R,e}$, which is a truncated version of ${\mathcal{T}}_{R}$. For $e\in E$ and $R\in{\mathcal{T}}$ we define ${\mathcal{T}}_{R,e}$ as the poset $\{T\in{\mathcal{T}}~|~T_{e}=-R_{e}\}\cup\{\hat{0}\}$ with $\hat{0}$ as its artificial least element and the remaining poset structure induced from ${\mathcal{T}}_{R}$. For $P\in{\mathcal{T}}_{R,e}$ we write $(\hat{0},P)_{R,e}$ for the interval from $\hat{0}$ to $P$ in ${\mathcal{T}}_{R,e}$. The following result will help us later in the main proof to obtain a factorization of the Varchenko matrix.

We conclude this section with another result on contractability needed in the proof of our main theorem. We start with a lemma:

For $e\in E$ and $P\in{\mathcal{T}}$ we say that $e$ defines a proper face of $P$ if there is a covector $X\in\mathcal{L}$ with $X\leq P$ and $X_{e}=0$ with $X\neq\mathbf{0}$. Note that in this case there is a unique maximal such covector, namely the composition of all of them. Otherwise, we say that $e$ does not define a proper face of $P$.