Modular Construction of Free Hyperplane Arrangements
In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the grap...
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| description | In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 080, 19 pages
Modular Construction
of Free Hyperplane Arrangements
Shuhei TSUJIE
Department of Education, Hokkaido University of Education, Hokkaido, Japan
E-mail: tsujie.shuhei@a.hokkyodai.ac.jp
URL: https://sites.google.com/view/tsujieshuheimath/
Received January 29, 2020, in final form August 13, 2020; Published online August 22, 2020
https://doi.org/10.3842/SIGMA.2020.080
Abstract. In this article, we study freeness of hyperplane arrangements. One of the
most investigated arrangement is a graphic arrangement. Stanley proved that a graphic
arrangement is free if and only if the corresponding graph is chordal and Dirac showed that
a graph is chordal if and only if the graph is obtained by “gluing” complete graphs. We
will generalize Dirac’s construction to simple matroids with modular joins introduced by
Ziegler and show that every arrangement whose associated matroid is constructed in the
manner mentioned above is divisionally free. Moreover, we apply the result to arrangements
associated with gain graphs and arrangements over finite fields.
Key words: hyperplane arrangement; free arrangement; matroid; modular join; chordality
2020 Mathematics Subject Classification: 52C35; 05B35; 05C22; 13N15
1 Introduction
A (central) hyperplane arrangement A over a field K is a finite collection of subspaces of codi-
mension 1 in a finite dimensional vector space K`. A standard reference for arrangements is [17].
Let S denote the polynomial algebra K[x1, . . . , x`], where (x1, . . . , x`) is a basis for the dual
space
(
K`
)∗
. Let Der(S) denote the module of derivations of S, that is,
Der(S) := {θ : S → S | θ is S-linear and θ(fg) = fθ(g) + θ(f)g for any f, g ∈ S}.
The module of logarithmic derivations D(A) is defined by
D(A) := {θ ∈ Der(S) | θ(αH) ∈ αHS for all H ∈ A},
where αH is a linear form such that ker(αH) = H.
Definition 1.1. An arrangement A is called free if D(A) is a free S-module.
Although the definition of free arrangements is algebraic, Terao’s celebrated factorization
theorem [26, Main Theorem] shows a solid relation between algebra, combinatorics, and topology
of arrangements. Terao’s conjecture asserts that the freeness of an arrangement is determined
by its combinatorial property and it is still widely open.
One of typical family of arrangements is graphic arrangements. Let Γ = ([n], EΓ) denote
a simple graph, where [n] := {1, . . . , n}. Define a graphic arrangement A(Γ) by
A(Γ) := {{xi − xj = 0} | {i, j} ∈ EΓ}.
This paper is a contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji
Saito for his 77th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Saito.html
mailto:tsujie.shuhei@a.hokkyodai.ac.jp
https://sites.google.com/view/tsujieshuheimath/
https://doi.org/10.3842/SIGMA.2020.080
https://www.emis.de/journals/SIGMA/Saito.html
2 S. Tsujie
A simple graph is chordal if every cycle of length at least 4 has a chord, which is an edge con-
necting nonconsecutive vertices of the cycle. Freeness of graphic arrangements is characterized
in terms of graphs as follows.
Theorem 1.2 (Stanley, see [9, Theorem 3.3] for example). A graphic arrangement A(Γ) is free
if and only if Γ if chordal.
A vertex of a simple graph is called simplicial if its neighborhood form a clique.
Theorem 1.3 (Dirac [7, Theorems 1, 2, and 4]). The class of chordal graphs coincides with the
smallest class C of graphs satisfying the following conditions.
(i) The null graph is belongs to C.
(ii) Suppose that a simple graph Γ has a simplicial vertex v and Γ \ v ∈ C. Then Γ ∈ C.
(iii) Let Γ be a simple graph on V = V1 ∪ V2. Suppose that the induced subgraph Γ[V1 ∩ V2] is
complete (including the null graph) and EΓ = EΓ[V1] ∪ EΓ[V2]. If Γ[V1] ∈ C and Γ[V2] ∈ C,
then Γ ∈ C.
Note that every complete graph Kn belongs to C by the condition (i) and (ii), and the
condition (ii) is unified with the condition (iii) for non-complete graphs. Thus a chordal graph
is constructed by gluing complete graphs.
The purpose of this paper is to generalize the class consisting of chordal graphs in terms of
matroids with conditions described in Theorem 1.3 and associate it with freeness of hyperplane
arrangements. See Oxley [18] for basic terminologies of matroids.
There are some generalizations of chordality for matroids in terms of circuits and chords
(see [20]). These are different from our generalization.
Let L be a geometric lattice. An element X is called modular if
r(X) + r(Y ) = r(X ∧ Y ) + r(X ∨ Y ) for all Y ∈ L,
where r denotes the rank function of L. A flat X of a simple matroid M is called modular if X
is modular in L(M), the lattice of flats of M .
Definition 1.4. A simple matroid M on the ground set E is said to be a modular join if there
exist two proper modular flats E1 and E2 of M such that E = E1 ∪ E2. We also say that M
is the modular join over X, denoted M = PX(E1, E2) = PX(M1,M2), where Mi := M |Ei for
i = 1, 2 and X := E1 ∩ E2.
Remark 1.5. Ziegler [38] introduced a modular join, which is a special case of a generalized
parallel connection or a strong join investigated by Brylawski [6] and Lindström [15]. Our
definition of a modular join is different from Ziegler’s one. However, they are equivalent (see
[38, Propositioin 3.3] and [6, Proposition 5.10]). In addition, note that X is modular in M1, M2,
and M .
Definition 1.6. A matroid M is called round (or nonsplit) if the ground set is not the union
of two proper flats. A subset S of the ground set of M is round if the restriction M |S is round.
It is well known that the graphic matroid of a simple graph without isolated vertices is round
if and only if the graph is complete (see [5, Theorem 4.2] for example) and that any induced
subgraph isomorphic to a complete graph corresponds to a modular flat of a graphic matroid
(see [18, Proposition 6.9.11 and below] for example). Our generalization of chordal graphs is
defined as follows.
Definition 1.7. Let ME be the minimal class of simple matroids which satisfies the following
conditions.
Modular Construction of Free Hyperplane Arrangements 3
(i) The empty matroid is a member of ME .
(ii) If a simple matroid M has a modular coatom X and M |X ∈ME , then M ∈ME .
(iii) Let M be a modular join of M1 and M2 over a round flat. If M1,M2 ∈ME , then M ∈ME .
We say that a simple matroid in ME is modularly extended.
Comparing the conditions in Theorem 1.3 and Definition 1.7, we can prove that a simple
graphic matroid is modularly extended if and only if the associated graph is chordal.
Recent studies [25] and [29] treat a similar class for signed graphs and their associated ar-
rangements. Moreover, we will study modularly extended matroids associated with gain graphs
in Section 4.
The linear dependence of an arrangement A determines a simple matroid M(A) on itself.
Namely, a subset {H1, . . . ,Hn} ⊆ A is defined to be independent if the codimension of the
intersection H1 ∩ · · · ∩ Hn is equal to n. The main theorem of this paper is as follows (see
Definitions 2.12 and 2.13 for the definition of divisional flags and divisional freeness).
Theorem 1.8. Every modularly extended matroid has a divisional flag. In particular, if the
linear dependence matroid M(A) on an arrangement A is modularly extended, then A is divi-
sionally free.
A simple matroid is supersolvable if it has a saturated chain consisting of modular flats,
or equivalently it belongs to the minimal class satisfying (i) and (ii) in Definition 1.7. It is
known that supersolvable arrangements are inductively free by [12, Theorem 4.2], and hence
divisionally free by [1, Theorem 1.6]. Clearly, the classME contains all supersolvable matroids.
Furthermore, we have the following theorem.
Theorem 1.9. The class ME coincides with the minimal class which contains all supersolvable
simple matroids and is closed under taking modular joins over round flats.
A graphic arrangement is supersolvable if and only if the corresponding graph is chordal [24,
Proposition 2.8]. Therefore freeness and supersolvability are equivalent in the class of graphic
arrangements.
As Ziegler [38] mentioned, there exists a modular join of supersolvable matroids which is not
supersolvable. Therefore the classME is strictly larger than the class of supersolvable matroids.
In order to see an example, let M ∈ME be a non-supersolvable matroid which is a modular
join M = PX(M1,M2) of supersolvable matroids M1 and M2. The rank of M is computed by
r(M) = r(M1)+r(M2)−r(X) by Brylawsky [6, Propositioin 5.2]. Since M is not supersolvable,
we have r(Mi) ≤ r(M)− 2 for i ∈ {1, 2} and r(X) ≥ 1. Therefore r(M) ≤ 2r(M)− 5 and hence
r(M) ≥ 5. Actually, there exists a non-supersolvable matroid inME whose rank is exactly 5 as
follows.
Example 1.10. Let A1 be an arrangement over R consisting of the following 7 hyperplanes
{z = 0}, {x1 = 0}, {x2 = 0}, {x1 − z = 0}, {x2 − z = 0}, {x1 − x2 = 0}, {x1 + x2 = 0}.
Then A1 is supersolvable with modular coatom {x1 = x2 = 0}. Let A2 be an isomorphic copy
of A1 and take a modular join of A1 and A2 over {z = 0}, that is, an arrangement A consisting
of the following 13 hyperplanes
{z = 0}, {x1 = 0}, {x2 = 0}, {x1 − z = 0}, {x2 − z = 0}, {x1 − x2 = 0}, {x1 + x2 = 0},
{y1 = 0}, {y2 = 0}, {y1 − z = 0}, {y2 − z = 0}, {y1 − y2 = 0}, {y1 + y2 = 0}.
Then r(A) = 5, M(A) ∈ ME , and by Theorem 1.8, A is divisionally free. Furthermore, one
can deduce that A is not supersolvable by showing that it has no modular flats of rank 4 (or see
Example 4.12).
4 S. Tsujie
The organization of this paper is as follows. In Section 2, we recall basic properties about
simple matroids and geometric lattices, including modularity. In addition, we introduce divi-
sional atoms and study them for modular joins. In Section 3, we give a proof of Theorem 1.8.
Finally, in Section 4, we give applications to arrangements corresponding to gain graphs and
arrangements over finite fields.
2 Preliminaries
2.1 Simple matroids and geometric lattices
In order to avoid confusion, we do not use geometrical terminology such as “point” and “hy-
perplane” for matroids and we call an element of a matroid an atom and a flat of corank 1
of a matroid a coatom. This is lattice theoretic terminology due to the following well-known
theorem.
Theorem 2.1 (see [30, p. 54, Theorem 2] for example). The correspondence between a simple
matroid and its lattice of flats is a bijection between simple matroids and geometric lattices.
Thus any properties about simple matroids are translated into properties of geometric lattices,
and vice versa. For example, the contraction and the restriction of matroids are just intervals
in the lattice of flats as follows.
Proposition 2.2 ([18, Proposition 3.3.8]). Let X be a flat of a matroid M . Then
(1) L(M/X) '
[
X, 1̂
]
= {F ∈ L(M) |X ≤ F}.
(2) L(M |X) '
[
0̂, X
]
= {F ∈ L(M) |F ≤ X}.
Note that the contraction of a simple matroid is not simple in general. However, we can
associate the simple matroid si(M) with a matroid M and this operation does not affect the
lattice of flats, that is, L(M) ' L(si(M)).
If A be an arrangement and M(A) the simple matroid on A, then for any hyperplane H ∈ A,
we have M
(
AH
)
' si(M(A)/H), where AH denotes the restriction defined by
AH := {K ∩H |K ∈ A \ {H}}.
Note that the restriction of an arrangement does not correspond to the restriction of a ma-
troid but the simplification of the contraction. The restriction of a matroid corresponds to the
localization of an arrangement.
2.2 Characteristic polynomials
Let L be a geometric lattice. The characteristic polynomial χ(L, t) ∈ Z[t] is defined by
χ(L, t) :=
∑
X∈L
µ(X)tr(1̂)−r(X),
where r denotes the rank function of L and µ : L→ Z denotes the one-variable Möbius function
of L defined recursively by
µ(X) :=
1 if X = 0̂,
−
∑
Y <X
µ(Y ) otherwise.
The characteristic polynomial of M is defined by χ(M, t) := χ(L(M), t).
Modular Construction of Free Hyperplane Arrangements 5
The intersection lattice L(A) of a central arrangement A is defined by
L(A) :=
{ ⋂
H∈B
H
∣∣∣∣B ⊆ A}
with a partial order by reverse inclusion. Note that L(A) is a geometric lattice and naturally
isomorphic to L(M(A)). The characteristic polynomial χ(A, t) ∈ Z[t] is defined by
χ(A, t) :=
∑
X∈L(A)
µ(X)tdimX .
Since the rank function of L(A) is given by the codimension, χ(A, t) = t`−rχ(M(A), t), where `
is the dimension of the ambient space and r is the rank of L(A). When ` = r we say that A is
essential. It is well known that there exists an essential arrangement A0 for every arrangement A
such that L(A) = L(A0) and A is free if and only if A0 is free (see [17] for details).
2.3 Modularity
We excerpt some conditions equivalent to modularity from Brylawski [6].
Proposition 2.3 (Brylawski [6, Theorem 3.3]). Let X be an element of a geometric lattice L.
Then the following conditions are equivalent.
(1) X is modular.
(2) For Y ≤ Z in L, Y ∨ (X ∧ Z) = (Y ∨X) ∧ Z.
(3) For all Y ∈ L, [X ∧ Y,X] ' [Y,X ∨ Y ].
(4) For any atom e 6≤ X, [0̂, X] ' [e,X ∨ e] and X ∨ e is modular in
[
e, 1̂
]
.
Theorem 2.4 (Brylawski [6, Theorem 3.11] (the modular short-circuit axiom)). Let M be
a simple matroid on the ground set E and X ⊆ E a nonempty subset. Then X is a modular flat
of M if and only if for every circuit C of M and an atom e ∈ C \X there exist an atom x ∈ X
and a circuit C ′ such that e ∈ C ′ ⊆ {x} ∪ (C \X).
Theorem 2.5 (Brylawski [6, Corollary 3.4]). A coatom X of a simple matroid on E is modular
if and only if for any distinct two atoms e, e′ ∈ E \X there exists e′′ ∈ X such that {e, e′, e′′}
forms a circuit.
We give some properties of modularity required in this article.
Proposition 2.6 (Jambu–Papadima [11, Lemmas 1.3 and 1.9]). Let X be a modular coatom
of a simple matroid M on E. Then for any two distinct atoms a, b ∈ E \ X, there exists
unique f(a, b) ∈ X such that a, b, f(a, b) form a circuit. Moreover, for any three distinct atoms
a, b, c ∈ E \X, the atoms f(a, b), f(a, c), f(b, c) form a circuit.
Proposition 2.7 (Brylawski [6, Proposition 3.5]). Let X be a modular flat of a simple ma-
troid M and Y a modular flat of the restriction M |X, then Y is a modular flat of M .
Proposition 2.8 (Brylawski [6, Proposition 3.6]). Let X and Y be modular flats of a simple
matroid. Then X ∩ Y is a modular flat.
Proposition 2.9 (Probert [20, Corollary 4.2.8]). Every modular flat of a round matroid is
round.
Theorem 2.10 (Stanley [23, Theorem 2]). If X is a modular element of a geometric lattice L,
then χ
([
0̂, X
]
, t
)
divides χ(L, t).
6 S. Tsujie
2.4 Divisionality
The following theorem plays an important role in this article.
Theorem 2.11 (Abe [1, Theorem 1.1 (division theorem)]). An arrangement A is free if there
exists H ∈ A such that AH is free and χ
(
AH , t
)
divides χ(A, t).
Theorem 2.11 leads to the concepts of a divisional flag, which is originally defined for ar-
rangements. However, we define it for simple matroids as follows.
Definition 2.12. A divisional flag of a simple matroid M of rank n is a sequence of flats
∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn = E
such that r(Xi) = i for i ∈ {0, . . . , n} and χ(M/Xi+1, t) |χ(M/Xi, t) for i ∈ {0, . . . , n− 1}.
Note that since χ(M/Xn, t) = 1 and χ(M/Xn−1, t) = t − 1 in the definition, the conditions
χ(M/Xn, t) |χ(M/Xn−1, t) and χ(M/Xn−1, t) |χ(M/Xn−2, t) are always satisfied.
Definition 2.13. An arrangement called divisionally free if the matroid M(A) has a divisional
flag.
Note that, by Theorem 2.11, every divisionally free arrangement is free. In order to find
a divisional flag, define a divisional atom as follows.
Definition 2.14. An atom e of a simple matroid M is called divisional if χ(M/e, t) divides
χ(M, t).
Proposition 2.15. A nonempty simple matroid M has a divisional flag if and only if there
exists a divisional atom e such that si(M/e) has a divisional flag.
Proof. Suppose that M has a divisional flag ∅ = X0 ⊆ X1 ⊆ · · · ⊆ Xn = E. Then e := X1
is a divisional atom and the images of X1, . . . , Xn under the isomorphism [e, 1̂] ' L(si(M/e))
given by Proposition 2.2(1) form a divisional flag of si(M/e). The converse holds by a similar
argument. �
Abe [2, Proposition 5] proved that every supersolvable arrangement is divisionally free by
constructing a divisional flag from a saturated chain of modular flats. The following lemma is
a generalization for simple matroids.
Lemma 2.16. Let M be a simple matroid on the ground set E and X a modular coatom of M .
Then every atom e ∈ E \X is divisional and si(M/e) 'M |X. In particular, every supersolvable
matroid has a divisional flag.
Proof. Take an atom e ∈ E\X, and e is a complement of X, that is, X∧e = 0̂ and X∨e = 1̂. By
Proposition 2.3(3), we have
[
0̂, X
]
'
[
e, 1̂
]
, which implies M |X ' si(M/e). Moreover, by Theo-
rem 2.10, the characteristic polynomial χ(M/e, t) = χ(si(M/e), t) = χ(M |X, t) = χ
([
0̂, X
]
, t
)
divides χ(M, t) and hence e is divisional.
When M is supersolvable, si(M/e) is supersolvable. Therefore M has a divisional flag by
induction and Proposition 2.15. �
Modular Construction of Free Hyperplane Arrangements 7
2.5 Modular joins
We review some properties of modular joins and will show a relation between modular joins and
divisional atoms.
Proposition 2.17 (see also Ziegler [38, Lemma 3.10]). Let X be a minimal flat of a simple
matroid M such that M is a modular join M = PX(M1,M2) over X. If M1 has a modular
coatom, then M1 has a divisional atom not belonging to X.
Proof. Let Z ⊆ E1 be a modular coatom of M1, where E1 denotes the ground set of M1. By
Lemma 2.16, every element in E1 \Z is a divisional atom of M1. Assume that E1 \Z ⊆ X. Then
Z ∩X ( X and M is a modular join M = PZ∩X(M |Z,M2) over Z ∩X, which is a contradiction
to the minimality of X. Hence E1 \Z 6⊆ X and every element E1 \(Z∪X) is a desired atom. �
Theorem 2.18 (Brylawski [6, Theorem 7.8]). Let M = PX(M1,M2) be a modular join. Then
χ(M, t) =
χ(M1, t)χ(M2, t)
χ(M |X, t)
.
The following proposition is essentially due to Brylawski for generalized parallel connections.
However, since we treat the special case of modular joins, we give a proof of the proposition
below.
Proposition 2.19 (Brylawski [6, Theorem 5.11.4]). Let M be a modular join M = PX(M1,M2)
and e an atom of M1 not belonging to X. Then si(M/e) is isomorphic to a modular join of
si(M1/e) and M2, and
χ(si(M/e), t) =
χ(si(M1/e), t)χ(M2, t)
χ(M |X, t)
.
Proof. Let E1 and E2 be the ground sets of M1 and M2, which are modular flats of M .
Take an atom e ∈ E1 \ X. The matroid si(M/e) corresponds the interval
[
e, 1̂
]
of L(M)
under the correspondence mentioned in Proposition 2.1. Note that E1 is modular in
[
e, 1̂
]
by
Proposition 2.3(2) and E2 ∨ e is modular in
[
e, 1̂
]
by Proposition 2.3(4).
The atoms of si(M/e) are identified with the atoms of the interval
[
e, 1̂
]
. These atoms
coincide with {e∨ e′ | e′ ∈ E \ {e}}, where E = E1 ∪E2 denotes the ground set of M . If e′ ∈ E1,
then e ∨ e′ ≤ E1. Suppose that e′ ∈ E2. Then e ∨ e′ ≤ e ∨ E2. Thus si(M/e) is a modular
join of matroids corresponding to [e, E1] and [e, E2 ∨ e]. The matroid corresponding to [e, E1]
is isomorphic to si(M1/e). By Proposition 2.3(3), [e, E2 ∨ e] '
[
0̂, E2
]
. Hence the matroid
corresponding to [e, E2 ∨ e] is isomorphic to M2. Thus si(M/e) is isomorphic to a modular join
of si(M1/e) and M2.
By Proposition 2.3(2), E1 ∧ (E2 ∨ e) = (E1 ∧ E2) ∨ e = X ∨ e. Using Theorem 2.18 and
Proposition 2.3(3), we have
χ(si(M/e), t) =
χ([e, E1], t)χ([e, E2 ∨ e], t)
χ([e,X ∨ e]), t
=
χ([e, E1], t)χ
([
0̂, E2
]
, t
)
χ
([
0̂, X
])
, t
=
χ(si(M1/e), t)χ(M2, t)
χ(M |X, t)
. �
Lemma 2.20. Let M be a modular join M = PX(M1,M2). Every divisional atom of M1 not
belonging to X is a divisional atom of M .
Proof. Let e be a divisional atom of M1 such that e 6∈ X. Then there exists an integer a such
that χ(M1, t) = (t− a)χ(si(M1/e), t). Using Proposition 2.19, we have
χ(M, t) =
χ(M1, t)χ(M2, t)
χ(M |X, t)
=
(t− a)χ(si(M1/e), t)χ(M2, t)
χ(M |X, t)
= (t− a)χ(si(M/e), t).
Thus e is a divisional atom of M . �
8 S. Tsujie
3 Proof of main theorems
3.1 Proof of Theorem 1.8
Lemma 3.1. The class ME is closed under taking restrictions to modular flats.
Proof. Let M ∈ME and X a modular flat of M . We proceed by induction on the rank of M .
The case r(M) = 0 is trivial. Hence we suppose that r(M) ≥ 1.
First assume that M has a modular coatom Z such that M |Z ∈ ME . If X ⊆ Z, then X is
a modular flat of M |Z. By the induction hypothesis, M |X = (M |Z)|X ∈ME . Assume X 6⊆ Z.
Then X ∨ Z = E, the ground set of M . By Proposition 2.8, X ∩ Z is a modular flat of M and
hence M |Z. By the induction hypothesis, M |(X ∩ Z) ∈ME . Moreover, by the modularity,
r(X)− r(X ∩ Z) = r(X ∨ Z)− r(Z) = r(E)− r(Z) = 1.
Therefore X ∩ Z is a modular coatom of M |X and hence M |X ∈ME .
Next we suppose that M is a modular join M = PY (E1, E2) over a round flat Y with
M |Ei ∈ME for i = 1, 2. If X ⊆ Ei for some i, then M |X = (M |Ei)|X ∈ME by the induction
hypothesis. Otherwise, Xi := X∩Ei 6= ∅ is a proper subset of X and M |Xi = (M |Ei)|Xi ∈ME
by the induction hypothesis for i = 1, 2. Since both X1 and X2 are modular by Proposition 2.8
and X = X1 ∪ X2, it follows that M |X is a modular join of M |X1 and M |X2. Moreover
X1 ∩X2 = X ∩ Y is round by Proposition 2.9. Therefore M |X ∈ME . �
Theorem 3.2. Every nonempty simple matroid M ∈ ME has a divisional atom e such that
si(M/e) ∈ME.
Proof. We will proof the following claims by induction on the rank of M .
(i) If M has a modular coatom, then there exists a divisional atom e such that si(M/e) ∈ME .
(ii) If X is a minimal round flat of M such that M is a modular join M = PX(E1, E2). Then,
for each i = 1, 2, there exists a divisional atom ei ∈ Ei \X such that si(M/ei) ∈ME .
First suppose that r(M) = 1, that is, the ground set of M is a singleton. Then only the
case (i) occurs and the atom of M satisfies the assertion.
Now suppose that r(M) ≥ 2. If M has a modular coatom X, then every atom e ∈ E \X is
divisional and si(M/e) 'M |X ∈ME by Lemmas 2.16 and 3.1. Thus the assertion holds.
Next we suppose that M is a modular join. We assume that X is a minimal round flat of M
such that M = PX(M1,M2). Since every modular flat in X is also round by Proposition 2.9,
X is a minimal flat such that M is a modular join over X.
We will show that M1 has a divisional atom e1 not belonging to X such that si(M1/e1) ∈ME .
Note that M1 is a member ofME by Lemma 3.1. Assume that M1 has a modular coatom. Then,
by Proposition 2.17, M1 has a divisional atom not belonging to X such that si(M1/e1) ∈ ME .
Hence we may assume that M1 has a minimal round flat Y such that M1 is a modular join
M1 = PY (F, F ′). Since X is round, we have F ⊇ X or F ′ ⊇ X. Without loss of generality, we
may assume that F ′ ⊇ X. By the induction hypothesis, M1 has a divisional atom e1 ∈ F \ Y
such that si(M1/e1) ∈ ME . Assume that e1 ∈ X. Then e1 ∈ F ′ and hence e1 ∈ F ∩ F ′ = Y ,
which contradicts e1 6∈ Y . Thus M1 has a divisional atom e1 not belonging to X such that
si(M1/e1) ∈ME .
By Lemma 2.20, e1 is a divisional atom of M . Moreover, by Proposition 2.19, si(M/e1) is
isomorphic to a modular join of si(M1/e1) and M2, and hence si(M/e1) ∈ME . �
Proof of Theorem 1.8. Use Theorem 3.2 and Proposition 2.15. �
Modular Construction of Free Hyperplane Arrangements 9
3.2 Proof of Theorem 1.9
Lemma 3.3. Let X be a modular coatom of a simple matroid M on E such that M |X is
a modular join M |X = PY (F1, F2). Then M is a modular join PY (F ′1, F2) or PY (F1, F
′
2),
where F ′i is some flat of M such that Fi is a modular coatom of M |F ′i for each i.
Proof. Recall that, for e, e′ ∈ E \ X, f(e, e′) denotes a unique element in X such that
e, e′, f(e, e′) form a circuit (see Proposition 2.6). Assume that there exist three distinct atoms
a, b, c ∈ E \ X such that f(a, b) ∈ F1 \ F2 and f(a, c) ∈ F2 \ F1. By Proposition 2.6, the
atoms f(a, b), f(a, c), f(b, c) form a circuit. Therefore if f(b, c) ∈ F1, then f(a, c) ∈ F1, and if
f(b, c) ∈ F2, then f(a, b) ∈ F2. The both cases contradict the assumption. Hence without loss
of generality, we may assume that f(a, b) ∈ F1 for any two distinct two atoms a, b ∈ E \X. Let
F ′1 := (E \X) ∪ F1.
Now we will show that M = PY (F ′1, F2). Clearly, F ′1∩F2 = F1∩F2 = Y . Hence it is satisfied
to show that the subset F ′1 is a modular flat of M . We will prove it by using Theorem 2.4. Let C
be a circuit of M and take an atom e ∈ C \F ′1. We will construct a desired circuit by induction
on m := |C ∩ (E \X)|.
First, consider the case m = 0. By modularity of F1, we have an atom x ∈ F1 ⊆ F ′1 and
a circuit C ′ such that e ∈ C ′ ⊆ {x} ∪ (C \ F1) = {x} ∪ (C \ F ′1), which is a desired circuit.
Second, suppose that m = 1. Let C ∩ (E \X) = {a}. Since C \ {a} ⊆ X and C is a circuit,
we have a ∈ clM (X) = X, which is a contradiction. Hence the case m = 1 does not occur.
Finally, assume that m ≥ 2. Let a, b ∈ C∩(E\X) be distinct atoms. By Proposition 2.6, T :=
{a, b, f(a, b)} is a circuit. Using the strong circuit elimination axiom (see [18, Proposition 1.4.12]
for example), we obtain a circuit C1 such that e ∈ C1 ⊆ (C∪T )\{a} = (C\{a})∪{f(a, b)}. Note
that C1\F ′1 ⊆ C\F ′1 since f(a, b) ∈ F1. Furthermore, the circuit C1 satisfies |C1∩(E\X)| ≤ m−1.
Therefore, by the induction hypothesis, we have an atom x ∈ F ′1 and a circuit C ′ such that
e ∈ C ′ ⊆ {x} ∪ (C1 \ F ′1) ⊆ {x} ∪ (C \ F ′1), which is a desired circuit. Thus F ′1 is a modular flat
and hence M = PY (F ′1, F2). Moreover F1 is a modular flat of M |F ′1 by Theorem 2.5. �
Proof of Theorem 1.9. It suffices to show that every non-supersolvable matroid M ∈ME is
a modular join M = PY (M ′1,M
′
2) over a round flat Y such that M ′1,M
′
2 ∈ ME . We proceed
by induction on the rank r(M). If r(M) ≤ 2, then M is supersolvable and we have nothing to
prove. Assume that r(M) ≥ 3.
We may assume that M has a modular coatom X such that M |X ∈ ME . If M |X is
supersolvable, then so is M , which is a contradiction. Therefore M |X is not supersolvable
and, by induction, there are simple matroids M1,M2 ∈ ME and a round flat Y such that
M |X = PY (M1,M2). By Lemma 3.3, M is also a modular join M = PY (M ′1,M
′
2) with
M ′1,M
′
2 ∈ME . �
4 Applications
4.1 Arrangements associated with gain graphs
Gain graphs yield two important classes of arrangements. One includes the Weyl arrangements
of type A, B, and D and the other includes the Catalan, Shi, and Linial arrangements. In this
subsection, we study modularly extended matroids associated with gain graphs.
4.1.1 Basic notions
A gain graph is a tuple Γ = (VΓ, EΓ, LΓ, GΓ), where
� VΓ is a finite set,
10 S. Tsujie
� LΓ is a subset of VΓ,
� GΓ is a group,
� EΓ is a finite subset of {(u, v, g) ∈ VΓ × VΓ × GΓ |u 6= v} divided by the equivalence
relation ∼ generated by (u, v, g) ∼
(
v, u, g−1
)
.
Let {u, v}g denote the equivalence class containing (u, v, g) and hence {u, v}g = {v, u}g−1 . Ele-
ments of VΓ, EΓ, and LΓ are called vertices, edges, and loops of the gain graph Γ respectively
and GΓ is called the gain group of Γ. We quite simplify the notion of gain graphs. See Za-
slavsky [33] for a general treatment. Note that every simple graph can be regarded as a loopless
gain graph over the trivial group.
A cycle of a gain graph Γ a loop or a subset of EΓ consisting of edges
{v1, v2}g1 , {v2, v3}g2 , . . . , {vm−1, vm}gm−1 , {vm, v1}gm
with distinct vertices v1, . . . , vm (m ≥ 2), where {v1, v2}g1 6= {v2, v1}g2 if m = 2. The cycle above
is said to be balanced if g1g2 · · · gm = 1. Note that whether or not the value equals the identity
is independent of indexing the vertices of the cycle and hence being balanced is well-defined.
Every loop is defined to be unbalanced.
A subset S ⊆ EΓ t LΓ is called balanced if every cycle in S is balanced (and hence S has no
loops). The set S is said to be contrabalanced if S has no balanced cycles. Moreover, S is called
balance-closed if
{e ∈ EΓ \ S | there exists a balanced cycle C such that e ∈ C ⊆ S ∪ {e}} = ∅.
A path on distinct vertices v1, . . . , vm (m ≥ 1) is a subset of EΓ consisting of edges
{v1, v2}g1 , {v2, v3}g2 , . . . , {vm−1, vm}gm−1 .
A tight handcuff is the union of two cycles C1 and C2 such that C1 and C2 have just one common
vertex. A loose handcuff is the union of two cycles C1 and C2, and a path P from v1 to v2 of
positive length such that P and Ci meet only at vi and the cycles C1 and C2 does not share
vertices. A handcuff is a tight or loose handcuff. A theta is the union of three paths meeting
only at their endvertices.
Suppose that G is a finite group. Let KG
n denote the loopless gain graph on the vertex set
[n] = {1, . . . , n} with gain group G and edges
{{i, j}g | i, j ∈ [n] with i 6= j and g ∈ G}
and let K̊G
n denote the gain graph KG
n together with all possible loops. Note that both of KG
0
and K̊G
0 mean the null graph.
A gain graph is connected if there exists a path between every pair of vertices of the graph.
If a gain graph is disconnected, then it is decomposed into the connected components in a usual
manner. A connected component of a subset S ⊆ EΓ tLΓ is a connected component of the gain
graph (VΓ, S ∩ EΓ, S ∩ LΓ, GΓ).
Let W ⊆ VΓ. A subgraph induced by W is a gain graph Γ[W ] = (W,EΓ[W ],W ∩ LΓ, GΓ),
where EΓ[W ] := {{u, v}g ∈ EΓ |u, v ∈ W}. An induced subgraph of Γ is a subgraph induced by
some subset of VΓ. Moreover, Γ \ v := Γ[VΓ \ {v}].
4.1.2 Frame matroids and the associated arrangements
Theorem 4.1 (Zaslavsky [34, Theorem 2.1]). Let Γ be a gain graph. Then the following condi-
tions define the same matroid on EΓ t LΓ.
Modular Construction of Free Hyperplane Arrangements 11
(a) A subset of EΓ t LΓ is independent if and only if every connected component of it has no
balanced cycles and at most one unbalanced cycle.
(b) A subset of EΓtLΓ is a circuit if and only if it is a balanced cycle, a contrabalanced handcuff,
or a contrabalanced theta.
We call the matroid the frame matroid of Γ, denoted by M×(Γ).
The frame matroid M×
(
K̊G
n
)
is known as the Dowling geometry Qn(G) introduced by Dow-
ling [8]. When n ≥ 3, the Dowling geometry Qn(G) is representable over a field K if and only
if G is isomorphic to a subgroup of K× [8, Theorem 9].
When Γ is a gain graph on [n] whose gain group G is a subgroup of the multiplicative
group K× of a field K, we may associate Γ with an arrangement A×(Γ) in Kn defined by the
following
A×(Γ) := {{xi − gxj = 0} | {i, j}g ∈ EΓ} ∪ {{xi = 0} | i ∈ LΓ}.
Example 4.2. When G = {1} ⊆ K×, the arrangement A×(Γ) is the graphic arrangement. Es-
pecially, A×
(
K
{1}
n
)
is the braid arrangement, also known as the Weyl arrangement of type An−1.
If G = {±1} ⊆ R×, then the arrangements A×
(
K̊
{±1}
n
)
and A×
(
K
{±1}
n
)
are known as the Weyl
arrangements of type Bn and Dn. More specifically,
A×
(
K{1}n
)
= {{xi − xj = 0} | 1 ≤ i < j ≤ n},
A×
(
K̊{±1}
n
)
= {{xi ± xj = 0} | 1 ≤ i < j ≤ n} ∪ {{xi = 0} | 1 ≤ i ≤ n},
A×
(
K{±1}
n
)
= {{xi ± xj = 0} | 1 ≤ i < j ≤ n}.
Theorem 4.3 (Zaslavsky [36, Theorem 2.1(a)]). The linear dependence matroid on A×(Γ) is
isomorphic to the frame matroid M×(Γ).
Note that if Γ is a simple graph, then M×(Γ) and A×(Γ) coincide with the graphic matroid
and arrangement. Moreover, recall that the graphic matroids associated with complete graphs
are round. Here we have a generalization for frame matroids.
Proposition 4.4. Suppose that G is a finite group. Then the frame matroids M×
(
KG
n
)
and
M×
(
K̊G
n
)
are round except for M×
(
K
{±1}
2
)
.
Proof. First we prove that M×
(
KG
n
)
is round except for M×
(
K
{±1}
2
)
. Let E be the ground set
of M×
(
KG
n
)
and suppose that E = F1 ∪ F2.
Now consider the case n = 2. If |G| = 1, then M×
(
KG
2
)
is trivially round. Assume that
|G| ≥ 3. Then we may suppose that |F1| ≥ 2. Since every contrabalanced handcuff is a circuit
by Theorem 4.1(b), we have F1 = E and hence M×
(
KG
2
)
is round.
Suppose that n ≥ 3. Define a relation ∼ on the vertex set KG
n by u ∼ v if u = v or
there exists an edge in F1 connecting u and v. Since every balanced triangle is a circuit by
Theorem 4.1(b), the relation ∼ is an equivalence relation. Assume that there exist two or more
distinct equivalence classes. Consider the equivalence relation ≈ among vertices defined by u ≈ v
if u = v or there exists an edge in F2 between u and v. If u 6∼ v, then u ≈ v since E = F1 ∪ F2.
When u ∼ v, take a vertex w such that u 6∼ w and v 6∼ w. Then u ≈ w and v ≈ w. Therefore
u ≈ v. Namely, all vertices are equivalent under the relation ≈. Hence without loss of generality
we may assume that every two distinct vertices are connected by an edge in F1.
If |G| = 1, then F1 = E and hence M×
(
KG
2
)
is round. Therefore we assume |G| ≥ 2. Suppose
there exists a pair of vertices such that there exist two different edges in F1 connecting them.
Since a contrabalanced theta and a contrabalanced tight handcuff are circuits by Theorem 4.1(b),
12 S. Tsujie
we have F1 = E and M×
(
KG
2
)
is round. Hence we may assume that there exist no such pairs,
that is, there exists exactly one edge in F1 between each pair of vertices.
If |G| ≥ 3, then F2 has at least two edges between every pair of vertices, which implies
F2 = E. Therefore M×
(
KG
2
)
is round. Hence we may assume that |G| = 2. Focus on a triangle
in F1. If the triangle is unbalanced, then there exists another edge in F1 such that it forms
a balanced triangle with two edges in the triangle since every balanced triangle is a circuit.
Therefore F1 has a pair of vertices such that there exist at least two edges between them, which
contradicts to the assumption of F1. Therefore the triangle is balanced and hence F2 has an
unbalanced triangle. By the same argument, F2 has a pair of vertices such that there exist at
least two edges between them. This implies that F2 = E. Thus M×
(
KG
2
)
is round.
Next, we prove that M×
(
K̊G
n
)
is round. Let S be the subset of the ground set E of M :=
M×
(
K̊G
n
)
corresponding to the subgraph K̊
{1}
n . Then M |S = M×
(
K̊
{1}
n
)
' M(Kn+1), which
is round. Every edge {i, j}g of K̊G
n forms a contrabalanced handcuff with the loops attached
to the endvertices i and j. Since a contrabalanced handcuff is a circuit by Theorem 4.1(b), we
have clM (S) = E. The assertion holds by the following proposition. �
Proposition 4.5 (Kung [14, Lemma 4.1], Probert [20, Lemma 4.2.7]). Let S be a subset of the
ground set of a matroid M . If S is round, then clM (S) is round.
Example 4.6. The matroids on Weyl arrangements A×
(
K
{1}
n
)
of type An−1 and A×
(
K̊
{±1}
n
)
of type Bn are round. The matroid on Weyl arrangement A×
(
K
{±1}
n
)
of type Dn (n ≥ 3) is also
round.
In the case of simple graphs, recall that a subgraph isomorphic to a complete graph corre-
sponds to a modular flat. Here is a generalization for frame matroids.
Proposition 4.7. Let Γ be a gain graph with a finite gain group G. Suppose that Γ has an
induced subgraph isomorphic to K̊G
n . Then the corresponding flat of M×(Γ) is modular.
Proof. Let X denote the corresponding flat. We will prove modularity of X by using Theo-
rem 2.4. Let C be a circuit and take an atom e ∈ C \ X. Suppose that S is the connected
component of C \X containing e. We may assume that C ∩X 6= ∅. Then S is an independent
set. By Theorem 4.1(a), S has no balanced cycles and at most one unbalanced cycle. Moreover S
has at least one vertex of the subgraph K̊G
n since C ∩X 6= ∅.
First, assume that S has an unbalanced cycle containing e (including the case e itself is
a loop). Then the unbalanced cycle and the loop x ∈ X of a vertex belonging to both S
and K̊G
n with the path connecting them form a handcuff C ′, which is a desired circuit since
e ∈ C ′ ⊆ {x} ∪ S ⊆ {x} ∪ (C \X).
Second, suppose that S has an unbalanced cycle not containing e. If we delete the unbalanced
cycle, then the remaining graph is a forest. Hence we can obtain a path which contains e and
connects the unbalanced cycle and a leaf. The leaf is a vertex of the subgraph K̊G
n since every
circuit has no leaves by Theorem 4.1(b). Then the loop x ∈ X of the leaf and the unbalanced
cycle with the path form a handcuff, which is a desired circuit.
Finally, consider the case S has no unbalanced cycle. In this case, S is a tree. Therefore we
can obtain a path which contains e and connecting leaves of S. This path contains at least 3
vertices since e 6= X. The endvertices of the path belong to the subgraph K̊G
n . We can choose
x ∈ X between the endvertices such that x and the path form a balanced cycle, which is a desired
circuit and hence we can conclude X is modular. �
Remark 4.8. Contrary to Proposition 4.7, KG
n (|G| ≥ 2) may yield a non-modular flat. For ex-
ample, see Fig. 1. The edges between the middle and right vertices denote the flat corresponding
to KG
2 . Choose two edges in it and consider the contrabalanced loose handcuff formed with the
Modular Construction of Free Hyperplane Arrangements 13
Figure 1. The flat corresponding to KG
2 is not modular.
loop and the edge between the left and middle vertices, which is a circuit by Theorem 4.1(b).
Using Theorem 2.4, we can conclude that the flat corresponding to KG
2 is not modular. In
a similar way, we can construct a frame matroid in which the flat corresponding to KG
n (n ≥ 3)
is not modular.
Next, we introduce bias-simplicial vertices, which is a generalization of simplicial vertices of
simple graphs.
Definition 4.9. A vertex v in a gain graph Γ is called bias simplicial if the following conditions
hold.
(i) If {u, v}g, {v, w}h ∈ EΓ, then {u,w}gh ∈ EΓ.
(ii) If {u, v}g, {u, v}h ∈ EΓ, and g 6= h, then u ∈ LΓ.
(iii) If {u, v}g ∈ EΓ and v ∈ LΓ, then u ∈ LΓ.
Zaslavsky [35, Theorem 2.1] characterized modular coatoms of frame matroids. The following
theorem is an excerpt. (Note that one type of modular coatom is missing in the classification.
See Koban [13, Theorem 2.1′] for the complete classification.)
Theorem 4.10 (Zaslavsky [35, Theorem 2.1(1)]). Let Γ be a gain graph and v a bias-simplicial
vertex. Then the flat of M×(Γ) corresponding to the induced subgraph Γ \ v is modular.
One can show that the frame matroid M×
(
K̊G
n
)
is supersolvable for any finite group G by
Theorem 4.10.
Zaslavsky [35, Theorem 2.2] characterized supersolvability of frame matroids as the minimal
class of gain graphs which satisfies the conditions (i)–(v) in the following theorem and is closed
under taking disjoint unions. We can replace disjoint unions with modular joins for modular
extendedness (condition (vi)).
Theorem 4.11. Let G be a finite subgroup of the multiplicative group of a field K and CG× the
minimal class of gain graphs with gain group G which satisfies the following conditions.
(i) The null graph is a member of CG× .
(ii) KG
2 ∈ CG× .
(iii) If {±1} ⊆ G, then K
{±1}
3 ∈ CG× .
(iv) If {±1} ⊆ G, then every connected loopless gain graph Γ over {±1} such that the positive
edges form a chordal graph, the negative edges form a star {u, v1}−1, . . . , {u, vr}−1, and
v1, . . . , vr form a clique consisting of positive edges is a member of CG× .
(v) If Γ has a bias-simplicial vertex v and Γ \ v ∈ CG× .
(vi) If there exists a decomposition VΓ = V1 ∪ V2 such that Γ[V1],Γ[V2] ∈ CG× , EΓ = EΓ[V1] ∪
EΓ[V2], and Γ[V1 ∩ V2] ' K̊G
n for some n ≥ 0, then Γ ∈ CG× .
14 S. Tsujie
Figure 2. The signed graph ./ (Dashed line segments denote negative edges).
Then for every Γ ∈ CG× the corresponding arrangement A×(Γ) is divisionally free.
Proof. By Propositions 4.4, 4.7, and Theorem 4.10, the frame matroid M×(Γ) is modularly
extended. By Theorems 1.8 and 4.3 we can conclude that A×(Γ) is divisionally free. �
Example 4.12. Let ./ be a signed graph described in Fig. 2, where a signed graph is a gain
graph with gain group {±1}. Let .̊/ denote the signed graph ./ with the loops attached to every
vertex. Then .̊/ ∈ C{±1}
× and the arrangement A×(.̊/) is the arrangement A in Example 1.10,
that is, an arrangement consisting of the following hyperplanes
{z = 0}, {x1 = 0}, {x2 = 0}, {x1 − z = 0}, {x2 − z = 0}, {x1 − x2 = 0}, {x1 + x2 = 0},
{y1 = 0}, {y2 = 0}, {y1 − z = 0}, {y2 − z = 0}, {y1 − y2 = 0}, {y1 + y2 = 0}.
The signed graph .̊/ is not of type (i)–(iv) in Theorem 4.11. Moreover, .̊/ has no bias-simplicial
vertex. Therefore A×(.̊/) is not supersolvable but divisionally free.
Remark 4.13. Since K
{1}
n , K̊
{±1}
n ∈ C{±1}
× , the Weyl arrangements of type An−1 and Bn are
divisionally free (actually these are supersolvable).
If n ≥ 4, then K
{±1}
n 6∈ C{±1}
× . Actually the frame matroid M×
(
K
{±1}
n
)
is not modularly
extended since it is round by Proposition 4.4 and has no modular coatoms by [35, Theorem 2.1]
and [13, Theorem 2.1′].
However, it is well known that every Weyl arrangement is free by Saito [21, 22], including the
Weyl arrangement A×
(
K
{±1}
n
)
of type Dn, which is also inductively free (see [12, Example 2.6]).
Thus the Weyl arrangement of type Dn (n ≥ 4) is inductively free (and hence divisionally free)
but not modularly extended.
Question 4.14. Does there exist a modularly extended arrangement which is not inductively
free?
4.1.3 Extended lift matroids and associated arrangements
Theorem 4.15 (Zaslavsky [34, Theorem 3.1]). Let Γ be a loopless gain graph. Then the following
conditions define the same matroid on EΓ t {∞}.
(a) A subset of EΓ t {∞} is independent if and only if has no balanced cycle and contains at
most either ∞ or one unbalanced cycle.
(b) A subset of EΓ t {∞} is a circuit if and only if it is a balanced cycle, a contrabalanced tight
handcuff, a contrabalanced theta, the union of two vertex-disjoint unbalanced cycles, or the
union of {∞} and an unbalanced cycle.
(c) A subset X ∈ EΓt{∞} is a flat if and only if X satisfies the one of the following conditions.
(i) X 63 ∞ and X is balanced and balance-closed.
(ii) X 3 ∞ and X \ {∞} is the union of the edge sets of the induced subgraphs Γ[W1], . . . ,
Γ[Wr], where W1, . . . ,Wr are mutually disjoint subsets of VΓ.
We call the matroid the extended lift matroid of Γ, denoted M+(Γ).
Modular Construction of Free Hyperplane Arrangements 15
When Γ is a loopless gain graph on [n] whose gain group is a subgroup of the additive
group K+ of a field K, we may associate Γ with an arrangement A+(Γ) in Kn+1 defined by the
following
A+(Γ) := {{z = 0}} ∪ {{xi − xj = gz} | {i, j}g ∈ EΓ},
where z, x1, . . . , xn denote the coordinate of Kn+1. This arrangement is the cone over the
affine arrangement consisting of hyperplanes corresponding to edges of Γ and the element ∞
corresponds to the hyperplane at infinity {z = 0}.
Remark 4.16. Consider gain graphs with gain group Z. For a positive integer a, the arrange-
ments A+(Γ) with edge sets
{{i, j}g | 1 ≤ i < j ≤ n, g ∈ {−a,−a+ 1 . . . , a}},
{{i, j}g | 1 ≤ i < j ≤ n, g ∈ {−a+ 1,−a+ 2, . . . , a}},
{{i, j}g | 1 ≤ i < j ≤ n, g ∈ {−a+ 2,−a+ 3, . . . , a}},
are the cones of the extended Catalan, Shi, and Linial arrangements. The cones of the extended
Catalan and Shi arrangements are known to be free [3, 10, 31].
Postnikov and Stanley [19] computed the characteristic polynomials of the deformations of
Weyl arrangement of type A, including these arrangements. Moreover, all roots of the character-
istic polynomial of the extended Linial arrangement have real part (2a−1)n/2 [19, Theorem 9.12].
Combining Terao’s factorization theorem [26], we have that if n = 3, then the cone over the ex-
tended Linial arrangement is not free and thus every cone over the extended Linial arrangement
is not free since it contains the extended Linial arrangement of dimension 3 as a localization
(see, for example [17, Theorem 4.37]).
Recently, Nakashima and the author [16] give explicit formulas for the number of flats of
extended Catalan and Shi arrangements with theory of gain graphs and combinatorial species.
Theorem 4.17 (Zaslavsky [36, Theorem 3.1(a)]). The linear dependence matroid on A+(Γ) is
isomorphic to the extended lift matroid M+(Γ).
Note that if Γ is a simple graph, then M+(Γ) and A+(Γ) are the graphic matroid and
arrangement with an extra element independent from the other elements. Recall again that
a subgraph isomorphic to a complete graph yields a round and modular flat. The following
propositions are generalizations for extended lift matroids.
Proposition 4.18. Suppose that G is a non-trivial finite group. Then the extended lift matroid
M+
(
KG
n
)
is round.
Proof. When n = 1, the assertion is trivial since M+
(
KG
1
)
= {∞}. Hence we suppose that
n ≥ 2 and assume that there exist two proper flats X and Y such that X ∪ Y = EΓ t {∞}.
We may assume that X 3 ∞. Then there exist mutually disjoint subset W1, . . . ,Wr of VΓ
such that X \ {∞} is the union of the edge sets of the induced subgraphs Γ[W1], . . . ,Γ[Wr] by
Theorem 4.15(c). Note that r ≥ 2 since X is a proper flat.
If Y 63 ∞, then Y is balanced by Theorem 4.15(c). However, Y must contain all edges
between a vertex in W1 and a vertex in W2, and hence Y is unbalanced since the gain group G
is non-trivial, which is a contradiction.
Now suppose that Y 3 ∞. Let Z be the flat consisting of the edges of the subgraph K
{1}
n .
Then we have Z = (X ∩ Z) ∪ (Y ∩ Z). This contradicts the roundness of the graphic matroid
of the complete graph. Hence we can conclude that the assertion holds. �
Proposition 4.19. Let Γ be a loopless gain graph with a finite gain group G. Suppose that G
has an induced subgraph isomorphic to KG
n . Then the corresponding flat of M+(Γ) is modular.
16 S. Tsujie
Proof. Let X denote the corresponding flat and we show X is modular by using Theorem 2.4.
Let C be a circuit and e ∈ C \ X. We may assume that C ∩ X 6= ∅. Then C \ X 63 ∞
and it is independent has no balanced cycle and contains at most one unbalanced cycle by
Theorem 4.15(a).
Assume that C \X has an unbalanced cycle containing e. Then the unbalanced cycle and ∞
form a circuit by Theorem 4.15(b), which is a desired circuit.
Now suppose that there exists no unbalanced cycle containing e. By Theorem 4.15(b), there
exists a cycle in C containing e. Therefore we can find a path in C \ X containing e whose
endvertices belong to the subgraph KG
n Choose a suitable edge between the endvertices, and
we obtain a balanced cycle, which is a desired circuit. Thus we can conclude that the assertion
holds true. �
We introduce link-simplicial vertices which are another generalization of simplicial vertices
of simple graphs and are fit to loopless gain graphs and extended lift matroids.
Definition 4.20. A vertex v in a loopless gain graph Γ is called link simplicial if the following
condition holds:
� If {u, v}g, {v, w}h ∈ EΓ, then {u,w}gh ∈ EΓ.
Zaslavsky characterized modular coatoms of extended lift matroids. We excerpt from the
theorem.
Theorem 4.21 (Zaslavsky [35, Theorem 3.1]). Let Γ be a loopless gain graph and v a link-
simplicial vertex. Then the flat of M+(Γ) corresponding to the induced subgraph Γ \ {v} is
modular.
From this theorem, one can show that the extended lift matroid M+
(
KG
n
)
is supersolvable
for any finite group G.
Zaslavsky [35, Theorem 3.2] also characterized supersolvability of extended lift matroids as
the minimal class satisfying the conditions (i) and (ii) in the following theorem. We can consider
the additional condition (iii) for modular extendedness.
Theorem 4.22. Let G be a finite subgroup of the additive group of a field K and CG+ the minimal
class of loopless gain graphs with gain group G which satisfies the following conditions.
(i) The null graph is a member of CG+ .
(ii) If Γ has a link-simplicial vertex v and Γ \ v ∈ CG+ , then Γ ∈ CG+ .
(iii) If there exists a decomposition VΓ = V1 ∪ V2 such that Γ[V1],Γ[V2] ∈ CG+ , EΓ = EΓ[V1] ∪
EΓ[V2], and Γ[V1 ∩ V2] ' KG
n for some n, then Γ ∈ CG+ .
Then for every Γ ∈ CG+ the corresponding arrangement A+(Γ) is divisionally free.
Proof. The extended lift matroid M+(Γ) is modularly extended by Propositions 4.18, 4.19,
and Theorem 4.21. Using Theorems 1.8 and 4.17, we can conclude that A+(Γ) is divisionally
free. �
Example 4.23. Let ./ be a signed graph described in Fig. 2. Here we regard the gain group {±1}
the additive group of F2. Then ./ ∈ CF2
+ and the arrangement A+(./) consists of the following
9 hyperplanes
{z = 0}, {x1 + z = 0}, {x2 + z = 0}, {x1 + x2 = 0}, {x1 + x2 = z},
{y1 + z = 0}, {y2 + z = 0}, {y1 + y2 = 0}, {y1 + y2 = z}.
Since ./ has no link-simplicial vertices, A+(./) is not supersolvable but divisionally free.
Modular Construction of Free Hyperplane Arrangements 17
Remark 4.24. Theorem 4.22 requires that the gain group G is finite. Hence we cannot say
anything about the extended Catalan and Shi arrangements, where the gain group is Z, although
they are free.
4.2 Arrangements over finite fields
Free arrangements over finite fields are investigated by Ziegler [37] and Yoshinaga [32]. Recently,
Palezzato and Torielli [28] show relations between freeness of arrangements over Q and freeness
of arrangements over finite fields.
Let A(n, q) denote the hyperplane arrangement consisting of all hyperplanes in n-dimensional
vector space over the finite field Fq. The linear dependence matroid of A(n, q) is known as the
projective geometry PG(n− 1, q). Since the lattice L(PG(n− 1, q)) is the lattice of subspaces of
the vector space Fn
q , every flat of PG(n− 1, q) is modular. Therefore PG(n− 1, q) and A(n, q)
are supersolvable.
Proposition 4.25. The linear dependence matroid on A(n, q), that is, the projective geometry
PG(n− 1, q) is round.
Proof. Assume that the ground set of PG(n− 1, q) is written as the union of two flats F1, F2.
Let V1 and V2 be the subspaces of Fn
q corresponding to the flats F1 and F2. Then Fn
q = V1 ∪ V2.
This yields V1 ⊆ V2 or V2 ⊆ V1. Thus PG(n− 1, q) is round. �
Proposition 4.26 (Oxley [18, Corollary 6.9.6]). Let M be a simple matroid representable
over Fq. Suppose that X is a flat of M such that M |X is isomorphic to the projective geometry
PG(n, q). Then X is modular.
Theorem 4.27. Let Fq be the minimal class consisting of simple matroids representable over Fq
satisfying the following conditions.
(i) Every supersolvable matroids over Fq belongs to Fq.
(ii) Fq is closed under taking modular joins over the projective geometry PG(n, q).
If the linear dependence matroid of an arrangement A over Fq belongs to Fq, then A is division-
ally free.
Proof. It follows that Fq is a subclass of ME by Propositions 4.25, 4.26, and Theorem 1.9.
Therefore, by Theorem 1.8, every arrangement in Fq is divisionally free. �
Example 4.28. Ziegler [38, Example 4.3] constructed a binary matroid which is not super-
solvable but obtained by taking a modular join of supersolvable matroids. We show that the
binary arrangement corresponding to the matroid is divisionally free. The arrangement A is
constructed as follows. Let A1 be an arrangement of rank 4 in F4
2 consisting of the following 11
hyperplanes
{z1 = 0}, {z2 = 0}, {z1 + z2 = 0},
{x1 = 0}, {x2 = 0}, {x1 + x2 = 0}, {x1 + z1 = 0}, {x2 + z1 = 0},
{x1 + x2 + z1 = 0}, {x1 + z1 + z2 = 0}, {x2 + z1 + z2 = 0}.
Then we can prove that
{x1 = 0} ⊆ {x1 = x2 = 0} ⊆ {x1 = x2 = z1 = 0} ⊆ {x1 = x2 = z1 = z2 = 0}
is a saturated chain consisting of modular flats in L(A1) by Theorem 2.5 and Proposition 2.7.
Therefore A1 is supersolvable (see also [27, Corollary 2.17] and [4, Theorem 4.3]). Let A2 be an
18 S. Tsujie
isomorphic copy of A1 and A the modular join of A1 and A2 over {{z1 = 0}, {z2 = 0}, {z1 +z2 =
0}} ' PG(1, 2), that is, A consists of the following 19 hyperplanes
{z1 = 0}, {z2 = 0}, {z1 + z2 = 0},
{x1 = 0}, {x2 = 0}, {x1 + x2 = 0}, {x1 + z1 = 0}, {x2 + z1 = 0},
{x1 + x2 + z1 = 0}, {x1 + z1 + z2 = 0}, {x2 + z1 + z2 = 0},
{y1 = 0}, {y2 = 0}, {y1 + y2 = 0}, {y1 + z1 = 0}, {y2 + z1 = 0},
{y1 + y2 + z1 = 0}, {y1 + z1 + z2 = 0}, {y2 + z1 + z2 = 0}.
By Theorem 4.27, A is divisionally free. According to Ziegler [38, Example 4.3], A is not
supersolvable.
Acknowledgments
I greatly appreciate N. Nakashima, D. Suyama, and M. Torielli for valuable discussions, which
provide a basis of Section 4. I also owe my deepest gratitude to the anonymous referees whose
comments are very helpful to polish the paper.
References
[1] Abe T., Divisionally free arrangements of hyperplanes, Invent. Math. 204 (2016), 317–346, arXiv:1502.07520.
[2] Abe T., Restrictions of free arrangements and the division theorem, in Perspectives in Lie Theory, Springer
INdAM Ser., Vol. 19, Editors F. Callegaro, G. Carnovale, F. Caselli, C. De Concini, A. De Sole, Springer,
Cham, 2017, 389–401, arXiv:1603.03863.
[3] Athanasiadis C.A., On free deformations of the braid arrangement, European J. Combin. 19 (1998), 7–18.
[4] Björner A., Edelman P.H., Ziegler G.M., Hyperplane arrangements with a lattice of regions, Discrete Com-
put. Geom. 5 (1990), 263–288.
[5] Borissova S., Regular round matroids, Master Thesis, California State University, 2016, available at https:
//scholarworks.lib.csusb.edu/etd/423.
[6] Brylawski T., Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975),
1–44.
[7] Dirac G.A., On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76.
[8] Dowling T.A., A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973),
61–86.
[9] Edelman P.H., Reiner V., Free hyperplane arrangements between An−1 and Bn, Math. Z. 215 (1994),
347–365.
[10] Edelman P.H., Reiner V., Free arrangements and rhombic tilings, Discrete Comput. Geom. 15 (1996), 307–
340.
[11] Jambu M., Papadima S., A generalization of fiber-type arrangements and a new deformation method,
Topology 37 (1998), 1135–1164.
[12] Jambu M., Terao H., Free arrangements of hyperplanes and supersolvable lattices, Adv. Math. 52 (1984),
248–258.
[13] Koban L., Comments on: “Supersolvable frame-matroid and graphic-lift lattices” [European J. Combin. 22
(2001), 119–133] by T. Zaslavsky, European J. Combin. 25 (2004), 141–144.
[14] Kung J.P.S., Numerically regular hereditary classes of combinatorial geometries, Geom. Dedicata 21 (1986),
85–105.
[15] Lindström B., On strong joins and pushout of combinatorial geometries, J. Combin. Theory Ser. A 25
(1978), 77–79.
[16] Nakashima N., Tsujie S., Enumeration of flats of the extended Catalan and Shi arrangements with species,
arXiv:1904.09748.
https://doi.org/10.1007/s00222-015-0615-7
https://arxiv.org/abs/1502.07520
https://doi.org/10.1007/978-3-319-58971-8_14
https://arxiv.org/abs/1603.03863
https://doi.org/10.1006/eujc.1997.0149
https://doi.org/10.1007/BF02187790
https://doi.org/10.1007/BF02187790
https://scholarworks.lib.csusb.edu/etd/423
https://scholarworks.lib.csusb.edu/etd/423
https://doi.org/10.2307/1997065
https://doi.org/10.1007/BF02992776
https://doi.org/10.1016/s0095-8956(73)80007-3
https://doi.org/10.1007/BF02571719
https://doi.org/10.1007/BF02711498
https://doi.org/10.1016/S0040-9383(97)00092-X
https://doi.org/10.1016/0001-8708(84)90024-0
https://doi.org/10.1016/S0195-6698(03)00122-7
https://doi.org/10.1007/BF00147534
https://doi.org/10.1016/0097-3165(78)90035-3
https://arxiv.org/abs/1904.09748
Modular Construction of Free Hyperplane Arrangements 19
[17] Orlik P., Terao H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften,
Vol. 300, Springer-Verlag, Berlin, 1992.
[18] Oxley J., Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, Vol. 21, Oxford University Press,
Oxford, 2011.
[19] Postnikov A., Stanley R.P., Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A
91 (2000), 544–597, arXiv:math.CO/9712213.
[20] Probert A.M., Chordality in matroids: in search of the converse to Hliněný’s theorem, Ph.D. Thesis, Victoria
University of Wellington, 2018.
[21] Saito K., On the uniformization of complements of discriminant loci, in Hyperfunction and Linear Differential
Equations, RIMS Kôkyûroku, Vol. 287, Res. Inst. Math. Sci. (RIMS), Kyoto, 1977, 117–137.
[22] Saito K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo
Sect. IA Math. 27 (1980), 265–291.
[23] Stanley R.P., Modular elements of geometric lattices, Algebra Universalis 1 (1971), 214–217.
[24] Stanley R.P., Supersolvable lattices, Algebra Universalis 2 (1972), 197–217.
[25] Suyama D., Torielli M., Tsujie S., Signed graphs and the freeness of the Weyl subarrangements of type B`,
Discrete Math. 342 (2019), 233–249, arXiv:1707.01967.
[26] Terao H., Generalized exponents of a free arrangement of hyperplanes and Shepherd–Todd–Brieskorn for-
mula, Invent. Math. 63 (1981), 159–179.
[27] Terao H., Modular elements of lattices and topological fibration, Adv. Math. 62 (1986), 135–154.
[28] Torielli M., Palezzato E., Free hyperplane arrangements over arbitrary fields, J. Algebraic Combin. 52
(2020), 237–249, arXiv:1803.09908.
[29] Torielli M., Tsujie S., Freeness of Hyperplane arrangements between Boolean arrangements and Weyl ar-
rangements of type B`, Electron. J. Combin. 27 (2020), P3.10, 15 pages, arXiv:1807.02432.
[30] Welsh D.J.A., Matroid theory, Dover Books on Mathematics, Dover Publications, Mineola, N.Y., 2010.
[31] Yoshinaga M., Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math.
157 (2004), 449–454.
[32] Yoshinaga M., Free arrangements over finite field, Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 179–182,
arXiv:math.CO/0606005.
[33] Zaslavsky T., Biased graphs. I. Bias, balance, and gains, J. Combin. Theory Ser. B 47 (1989), 32–52.
[34] Zaslavsky T., Biased graphs. II. The three matroids, J. Combin. Theory Ser. B 51 (1991), 46–72.
[35] Zaslavsky T., Supersolvable frame-matroid and graphic-lift lattices, European J. Combin. 22 (2001), 119–
133.
[36] Zaslavsky T., Biased graphs. IV. Geometrical realizations, J. Combin. Theory Ser. B 89 (2003), 231–297.
[37] Ziegler G.M., Matroid representations and free arrangements, Trans. Amer. Math. Soc. 320 (1990), 525–541.
[38] Ziegler G.M., Binary supersolvable matroids and modular constructions, Proc. Amer. Math. Soc. 113 (1991),
817–829.
https://doi.org/10.1007/978-3-662-02772-1
https://doi.org/10.1093/acprof:oso/9780198566946.001.0001
https://doi.org/10.1006/jcta.2000.3106
https://arxiv.org/abs/math.CO/9712213
https://doi.org/10.1007/BF02944981
https://doi.org/10.1007/BF02945028
https://doi.org/10.1016/j.disc.2018.09.029
https://arxiv.org/abs/1707.01967
https://doi.org/10.1007/BF01389197
https://doi.org/10.1016/0001-8708(86)90097-6
https://doi.org/10.1007/s10801-019-00901-x
https://arxiv.org/abs/1803.09908
https://doi.org/10.37236/9341
https://arxiv.org/abs/1807.02432
https://doi.org/10.1007/s00222-004-0359-2
https://doi.org/10.3792/pjaa.82.179
https://arxiv.org/abs/math.CO/0606005
https://doi.org/10.1016/0095-8956(89)90063-4
https://doi.org/10.1016/0095-8956(91)90005-5
https://doi.org/10.1006/eujc.2000.0418
https://doi.org/10.1016/S0095-8956(03)00035-2
https://doi.org/10.2307/2001687
https://doi.org/10.2307/2048620
1 Introduction
2 Preliminaries
2.1 Simple matroids and geometric lattices
2.2 Characteristic polynomials
2.3 Modularity
2.4 Divisionality
2.5 Modular joins
3 Proof of main theorems
3.1 Proof of Theorem 1.8
3.2 Proof of Theorem 1.9
4 Applications
4.1 Arrangements associated with gain graphs
4.1.1 Basic notions
4.1.2 Frame matroids and the associated arrangements
4.1.3 Extended lift matroids and associated arrangements
4.2 Arrangements over finite fields
References
|
| id | nasplib_isofts_kiev_ua-123456789-210768 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T05:19:27Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tsujie, Shuhei 2025-12-17T14:31:31Z 2020 Modular Construction of Free Hyperplane Arrangements. Shuhei Tsujie. SIGMA 16 (2020), 080, 19 pages 1815-0659 2020 Mathematics Subject Classification: 52C35; 05B35; 05C22; 13N15 arXiv:1908.01535 https://nasplib.isofts.kiev.ua/handle/123456789/210768 https://doi.org/10.3842/SIGMA.2020.080 In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields. I greatly appreciate N. Nakashima, D. Suyama, and M. Torielli for valuable discussions, which provide the basis of Section 4. I also owe my deepest gratitude to the anonymous referees whose comments are very helpful in polishing the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Construction of Free Hyperplane Arrangements Article published earlier |
| spellingShingle | Modular Construction of Free Hyperplane Arrangements Tsujie, Shuhei |
| title | Modular Construction of Free Hyperplane Arrangements |
| title_full | Modular Construction of Free Hyperplane Arrangements |
| title_fullStr | Modular Construction of Free Hyperplane Arrangements |
| title_full_unstemmed | Modular Construction of Free Hyperplane Arrangements |
| title_short | Modular Construction of Free Hyperplane Arrangements |
| title_sort | modular construction of free hyperplane arrangements |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210768 |
| work_keys_str_mv | AT tsujieshuhei modularconstructionoffreehyperplanearrangements |