MAT-freeness in Hyperplane Arrangements

• MAT-freeness is a combinatorial property defined by the existence of a MAT-partition that extends the Multiple Addition Theorem to guarantee the freeness of hyperplane arrangements.
• It connects the algebraic theory of logarithmic derivations with combinatorial structures such as MAT-labeled graphs and vines, offering a unified framework for analyzing graphic arrangements.
• This property highlights key issues like field dependence, explicit enumeration, and open problems in poset characterizations, thereby advancing the structural classification of free arrangements.

MAT-freeness is a combinatorial property of hyperplane arrangements introduced to formalize and generalize powerful sufficient criteria for freeness derived from the Multiple Addition Theorem (MAT) of Abe–Barakat–Cuntz–Hoge–Terao. It plays a central role in the structural classification of free arrangements and connects the algebraic theory of logarithmic derivations with combinatorial objects such as vines and edge-labeled graphs, especially in the context of graphic arrangements.

1. The Module of Logarithmic Derivations and MAT-freeness

Let $V \cong \Bbb K^\ell$ be a finite-dimensional vector space over a field $\Bbb K$ of characteristic zero, with coordinate algebra $S = \Sym(V^*) = \Bbb K[x_1,\dots,x_\ell]$. The $S$-module of $\Bbb K$-derivations is

$\Der(S) = \bigoplus_{i=1}^\ell S\,\partial_{x_i}.$

A central hyperplane arrangement is a finite set $\A = \{H_1,\dots,H_n\}\subset V$, each $H = \ker(\alpha_H)$ for some linear form $\alpha_H\in V^*$. The defining polynomial is $Q(\A) = \prod_{H\in\A} \alpha_H \in S$. The module of logarithmic derivations along $\A$ is

$D(\A) = \left\{\theta \in \Der(S) \mid \theta(Q(\A)) \in Q(\A)\,S \right\}.$

$\A$ is free if $D(\A) \cong \bigoplus_{i=1}^\ell S(-e_i)$, and the multiset $\{e_1,\dots,e_\ell\}$ are its exponents.

The Multiple Addition Theorem (MAT) provides a method to build free arrangements from existing ones by simultaneously adding several hyperplanes under strict numerical and combinatorial conditions. This leads to the notion of an MAT-partition: a sequence $\pi = (\pi_1, \dots, \pi_n)$ partitioning $\A$ such that, for each $k$:

• (MP1) The set $\pi_k$ consists of linearly independent hyperplanes,
• (MP2) No $H\in \pi_k$ contains the intersection of all hyperplanes in $\A_{k-1}$,
• (MP3) For each $H\in \pi_k$,

$|\A_{k-1}| - |(\A_{k-1} \cup \{H\})^H| = k-1,$

where $(\A_{k-1} \cup \{H\})^H$ denotes the restriction onto $H$.

An arrangement is MAT-free if it admits such a partition. Every MAT-free arrangement is free, but the converse fails in general (Tran et al., 2023, Cuntz et al., 2019, Mücksch et al., 2020, Abe et al., 2018).

2. MAT-labeled Graphs and Graphic Arrangements

Given a simple graph $G = (V, E)$ with $|V| = \ell$, the associated graphic arrangement is

$\A_G = \{x_i - x_j = 0 : \{i, j\} \in E\} \subset \Bbb K^\ell.$

The arrangement $\A_G$ is free if and only if $G$ is chordal, and MAT-free if and only if $G$ is strongly chordal (Tran et al., 2023).

A key combinatorial encoding for MAT-freeness in graphic arrangements is the MAT-labeling of edges. An edge-labeled graph $(G, \lambda)$, with $\lambda: E \to \Bbb Z_{>0}$, is MAT-labeled if for all $k$:

• (ML1) No edge in $\pi_{\le k-1}$ forms a cycle together with an edge in $\pi_k = \lambda^{-1}(k)$,
• (ML2) Each $e\in\pi_k$ lies in exactly $k-1$ triangles whose other two edges are from $\pi_{<k}$.

Theorem: $\A_G$ is MAT-free if and only if $(G, \lambda)$ admits an MAT-labeling. This equivalence underpins the structural analysis of MAT-freeness within graphic arrangements (Tran et al., 2023).

3. Categorical Equivalence with (Locally) Regular Vines

Vines, introduced in probabilistic graphical modeling, can be recast in terms compatible with hyperplane arrangement theory. A vine is a sequence of forests $F_1, F_2, \dots$ such that $F_{i+1}$ is defined on the edge set of $F_i$, and each $F_i$ is a forest. Alternatively, a vine corresponds to a graded poset $P = \bigsqcup_{i=1}^n P_i$ (with $|P_1| = \ell$), where:

• Each non-minimal element covers exactly two lower elements,
• Each bipartite ‘level graph’ between $P_i$ and $P_{i+1}$ is a forest.

Locally regular vines (LR-vines) and regular vines (R-vines) add proximity conditions: every principal ideal is itself an R-vine, and “proximity” holds (two elements sharing a parent at a higher level also share a child at a lower level).

An explicit equivalence of categories is established:

$$\{\text{MAT-labeled graphs}\} \simeq \{\text{locally regular vines}\}$$

and, restricting to complete graphs,

$$\{\text{MAT-labeled complete graphs}\} \simeq \{\text{R-vines}\}$$

Functorial correspondences $\Psi$ and $\Omega$ between the two categories are inverses up to natural isomorphism (Tran et al., 2023).

4. Root Poset Characterization and Ideals in Vines

A significant result is a root-poset-style characterization of MAT-freeness for graphic arrangements. For a simple graph $G$,

$$G \text{ is strongly chordal} \iff \exists \ \ P: \text{LR-vine, } \min(P) = V(G), \text{ and } v\mapsto C_v = \{i, j\} \text{ matches an MAT-labeling of } G$$

That is, every MAT-free graphic arrangement arises as an ideal in the poset of some vine—a direct generalization of the classical root poset structure in Coxeter theory (Tran et al., 2023).

For the D-vine (type $A$), this recovers the classical root poset of $A_{\ell-1}$, and the unique MAT-labeled complete graph encodes this explicitly.

5. Classification, Field Dependence, and Categorical Properties

For complex reflection arrangements, MAT-freeness and MAT2-freeness coincide: the only irreducible complex reflection arrangements which are not MAT-free are those associated with $G(e,e,\ell)$ ($e > 2, \ell > 2$) and the seven exceptional non-real reflection groups ($$G_{24}, G_{27}, G_{29}, G_{31}, G_{32}, G_{33}, G_{34}$$). All irreducible real reflection arrangements are MAT-free except possibly in these exceptional cases (Cuntz et al., 2019).

A crucial recent development demonstrates MAT-freeness is not, in general, a combinatorial (i.e., intersection lattice-determined) property. Explicit examples show two arrangements over distinct fields with the same intersection lattice can differ in MAT-freeness status; in particular, MAT-freeness depends on the ground field except when restricted to arrangements over infinite fields, where it is combinatorial (Hoge et al., 4 Dec 2025).

MAT-freeness is strictly weaker than inductive or additive freeness and does not share their field independence or robustness under lattice isomorphism (Hoge et al., 4 Dec 2025, Cuntz et al., 2019).

6. Counting and Categorical Enumeration

The enumeration of non-isomorphic MAT-labelings of the complete graph $K_\ell$ is governed by counts of R-vines:

$$E_1 = E_2 = E_3 = 1,\qquad E_\ell = \frac{A_\ell + B_\ell}{2} \quad (\ell \ge 4),$$

where

$$A_\ell = 2^{\frac{(\ell-2)(\ell-3)}{2}}, \quad B_\ell = \sum_{k=1}^{\lfloor \ell/2\rfloor - 1} A_\ell c_k 2^{-k+\sum_{i=0}^{k-1}(\ell-4-2i)},$$

with $c_k = 1$ except $c_{\lfloor \ell/2\rfloor-1}=2$ (Tran et al., 2023).

7. Applications, Accuracy, and Open Problems

MAT-freeness underlies various advances, notably providing a uniform and case-free proof of the Orlik–Solomon–Terao conjecture for ideal subarrangements of Weyl arrangements and for extended Shi and Catalan arrangements. Every MAT-free arrangement is “accurate” in the sense that its exponent sequence can be realized by free restrictions corresponding to suitable intersections, extending structural results for Coxeter and Weyl arrangements (Mücksch et al., 2020).

A sequence of open questions remains, including: the full relationship between MAT-freeness and inductive/ divisional freeness, the possibility for an intrinsic poset characterization of MAT-freeness beyond graphic cases, and the detailed nature of field dependence—such as characterization of field extensions preserving MAT-freeness and the behavior of MAT-like freeness properties under localization and deletion (Cuntz et al., 2019, Hoge et al., 4 Dec 2025).

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References: For foundational definitions and categorical equivalences, see (Tran et al., 2023). For the combinatorial and field-dependence properties, see (Hoge et al., 4 Dec 2025, Mücksch et al., 2020, Cuntz et al., 2019, Abe et al., 2018). For applications to accuracy and Coxeter/Weyl arrangements, see (Mücksch et al., 2020).