arXiv:1705.02409v1 [] 5 May 2017

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS
MICHAEL DIPASQUALE
Abstract. We prove that, on a large cone containing the constant multiplicities, the only free multiplicities on the braid arrangement are those identified
in work of Abe, Nuida, and Numata (2009). We also give a conjecture on the
structure of all free multiplicities on braid arrangements.

1. Introduction
Let V ∼
= Kℓ+1 be a vector space over a field K of characteristic zero, V ∗ its
dual space and S = Sym(V ∗ ) ∼
= K[x0 , . . . , xℓ ]. Given a polynomial f ∈ S denote
by V (f ) the zero-locus of f in V . The braid arrangement of type Aℓ ⊂ V is defined as Aℓ = ∪0≤i<j≤ℓ Hij , where Hij = V (xi − xj ). A multiplicity on Aℓ is a
map m : {Hij } → Z>0 ; we will set mij = m(Hij ). The pair (Aℓ , m) is called
a multi-arrangement. The multi-arrangement (Aℓ , m) is free if the corresponding
module D(Aℓ , m) of multi-derivations (i.e., vector fields tangent to Aℓ with multiplicities prescribed by m) is a free module over the polynomial ring K[x0 , . . . , xℓ ].
(See Section 2 for more details.) If (Aℓ , m) is free, we say m is a free multiplicity.
Free multiplicities on braid arrangements have been studied since the introduction of the module of logarithmic differentials by Saito [12], largely due to their
importance in the theory of Coxeter arrangements and later in connection with a
conjecture of Athanasiadis [6]. Terao made a major breakthrough in [14], showing
that the constant multiplicity on any Coxeter arrangement is free and determining the corresponding exponents. Subsequently, many authors studied freeness of
‘almost-constant’ multiplicities on Coxeter and braid arrangements [13, 14, 16, 5].
In the setting of the braid arrangement, this line of inquiry resulted in a paper of
Abe-Nuida-Numata [2], where the authors classify what we shall call ANN multiplicities. Given non-negative integers n0 , . . . , nℓ and integers ǫij ∈ {−1, 0, 1} for all
0 ≤ i < j ≤ ℓ, an ANN multiplicity is a multiplicity satisfying
(1) mij = ni + nj + ǫij and
(2) mij ≤ mik + mjk + 1 for every triple i, j, k.
In [2] ANN multiplicities are classified as free if and only if a corresponding signed
graph is signed-eliminable; we will describe this precisely in § 5. We shall refer
to the set of multiplicities satisfying the inequalities in (2) as the balanced cone of
multiplicities. The reason for this name will be explained in § 3.
In this note we prove that a multiplicity in the balanced cone is free if and only
if it is a free ANN multiplicity. This partially generalizes the recent classification
of all free multiplicities on the A3 braid arrangement [8], which is joint work of the
author with Francisco, Mermin, and Schweig. To state our result more concretely
we shall associate to the multi-braid arrangement (Aℓ , m) an edge-labeled complete
graph (Kℓ+1 , m). The vertices of Kℓ+1 are labeled in bijection with the variables
x0 , . . . , xℓ ∈ S. An edge {vi , vj } corresponds to Hij = V (xi −xj ) and is furthermore
1

2

M. DIPASQUALE

labeled by m(Hij ) = mij . Now suppose C is a four-cycle in Kℓ+1 which traverses
the vertices vi , vj , vs , vt in order. Define m(C) = |mij − mjs + mst − mit |; since
we take absolute value, m(C) is independent of orientation, depending only on the
four cycle and the multiplicity. Let C4 (Kℓ+1 ) be the set of all four cycles of Kℓ+1 .
Given a subset U ⊂ {v0 , . . . , vℓ } of size at least four, the deviation of m over U is
X
DV(mU ) =
m(C)2 .
C∈C4 (Kℓ+1 )
C⊂U

Our main result is the following.
Theorem 1.1. Suppose (Aℓ , m) is a multi-braid arrangement with m in the balanced cone of multiplicities. For a subset U ⊂ {v0 , . . . , vℓ } let mU = {mij |
{vi , vj } ⊂ U } and denote by qU the number of integers {mij + mik + mjk |
{vi , vj , vk } ⊂ U } that are odd. Then the following are equivalent.
(1) (Aℓ , m) is free
(2) DV(mU ) ≤ qU (|U | − 1) for every subset U ⊂ {v0 , . . . , vℓ } where |U | ≥ 4.
(3) m is a free ANN multiplicity. In other words, there exist non-negative
integers n0 , . . . , nℓ and ǫij ∈ {−1, 0, 1} (for 0 ≤ i < j ≤ ℓ) so that
(a) mij = ni + nj + ǫij
−
+
(b) the signed graph G on {v0 , . . . , vℓ } with EG
= {{vi , vj } : ǫij < 0}, EG
=
{{vi , vj } : ǫij > 0} is signed-eliminable in the sense of [2].
Remark 1.2. Notice that DV(mU ) = 0 if and only if mij − mjs + mst − mit = 0 for
every four-tuple (vi , vj , vs , vt ) of distinct vertices in U . These equations cut out the
linear space LU parametrized by mij = ni + nj for {vi , vj } ⊂ U . Thus DV(mU ) can
be viewed as a measure of how far mU is from the linear space LU ; which in turn
measures how far mU ‘deviates’ from being an ANN multiplicity on the sub-braid
arrangement corresponding to U . This is why we call it the deviation of m over U .
The implication (3) =⇒ (1) in Theorem 1.1 is the result of Abe-Nuida-Numata [2].
The quantity DV(mU ) in Theorem 1.1.(2) arises from studying the local and global
mixed products (introduced in [3]) of (Aℓ , m). The main point of our note is to
show that these ‘deviations’ not only detect freeness in the balanced cone but also
interact well with the notion of signed-eliminable graphs.
The structure of the paper is as follows. In Section 2 we give some background
on arrangements. The proof of Theorem 1.1 is split across sections 3 and 5. The
implication (1) =⇒ (2) is proved in Theorem 3.4. We split the proof of (2) =⇒
(3) into two parts. The first part, establishing that m is an ANN multiplicity if
the inequalities in (2) are satisfied, is Proposition 4.2. The second part, showing
that the inequalities in (2) detect when the associated signed graph is not signedeliminable, is Proposition 5.4. The final implication (3) =⇒ (1) is proved in [2].
We finish in Section 6 by introducing the notion of a free vertex and presenting a
conjecture about the structure of all free multiplicities on braid arrangements.
1.1. Examples. We provide some computations using Theorem 1.1. For the braid
arrangement Aℓ , corresponding to the complete graph Kℓ+1 , we label the vertices
of Kℓ+1 by v0 , . . . , vℓ and, given a multiplicity m, we denote by mij the value of
m on the hyperplane Hij = V (xi − xj ). If U ⊂ {v0 , . . . , vℓ }, we denote by AU the
corresponding sub-braid arrangement of Aℓ .

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

3

Example 1.3. First we consider a family of multiplicities on the A3 arrangement.
Given positive integers s, t, define the multiplicity m3s,t by m01 = m12 = m23 = s
and m02 = m03 = m13 = t (this assigns different multiplicities along two edgedisjoint paths of length three).
The multiplicity m3s,t is in the balanced cone of multiplicities if and only if
s ≤ 2t + 1 and t ≤ 2s + 1. Assuming m3s,t is in the balanced cone of multiplicities,
we now compute the deviation DV(m3s,t ). There are three four cycles: one of
these has m(C) = |2s − 2t| while the other two have m(C) = |s − t|. Hence
DV(m3s,t ) = 6(s − t)2 .
Now consider the sums mijk around three cycles. There are four such sums, two
of the form 2s + t and two of the form 2t + s. So, applying Theorem 1.1, if m3s,t
is in the balanced cone of multiplicities, it is free if and only if 6(s − t)2 ≤ 4 · 3, or
|s − t| ≤ 1. In fact, using the classification from [8], it follows that m3s,t is a free
multiplicity if and only if |s − t| ≤ 1 (regardless of whether m3s,t is in the balanced
cone or not).
Example 1.4. Next we consider a similar family of multiplicities on the A4 braid
arrangement. Let C1 be the five-cycle traversing the vertices v0 , v1 , v2 , v3 , v4 , v0 in
order and C2 be the five-cycle traversing the vertices v0 , v2 , v4 , v1 , v3 , v0 in order;
C1 and C2 are edge-disjoint and every edge of K5 is contained in either C1 or
C2 . Given positive integers s, t we define the multiplicity m4s,t by m4s,t |C1 ≡ s and
m4s,t |C2 ≡ t.
Any closed sub-arrangement of (A4 , m4s,t ) of rank three has the form (A3 , m3s,t )
considered in Example 1.3. It follows that (A4 , m4s,t ) is not free if |s − t| > 1. So we
consider the case when |s − t| ≤ 1. If |s − t| ≤ 1 then m4s,t is in the balanced cone of
multiplicities. We compute DV(m4s,t ) as follows. A four-cycle of K5 lies in a unique
complete sub-graph on four vertices and each complete sub-graph on four vertices
contains three such four-cycles. As we saw in Example 1.3, one of these satisfies
m(C) = |2s − 2t| while the other two have m(C) = |s − t|. So the contribution to
DV(m4s,t ) from each complete sub-graph on four vertices is 6(s − t)2 . As there are
five such sub-graphs, we have DV(m4s,t ) = 30(s − t)2 .
Now we consider the sums mijk around three cycles. There are ten such sums,
five of the form 2s + t and five of the form 2t + s. If |s − t| = 1, then exactly one of
s, t is odd so there are precisely five sums around three cycles that are odd. Hence
DV(m4s,t ) = 30 > 4 · 5 and m4s,t is not free by Theorem 1.1. So we conclude that
m4s,t is free if and only if s = t.
We also consider why m4s,t is not free when |s − t| = 1 using the criterion
of Abe-Nuida-Numata (which is the third statement of Theorem 1.1). Without
loss, suppose t = s + 1 and let ni = ⌈s/2⌉ for i = 0, 1, 2, 3, 4. If s is even then
mij = ni + nj = s for {i, j} ∈ C1 while mij = ni + nj + 1 = s + 1 for {i, j} ∈ C2 .
In this case the graph G on the vertices v0 , v1 , v2 , v3 , v4 is the (positive) five-cycle
given by C2 and is hence not signed-eliminable by the characterization in [2] (see
also Corollary 5.3). Similarly, if s is odd, then G is the negatively signed five-cycle
C1 .
Example 1.5. The following example shows that criterion (2) in Theorem 1.1
really does need to be checked on all proper subsets of size at least four. Consider
the A4 arrangement with the multiplicity m defined by m01 = m02 = m03 = m12 =

4

M. DIPASQUALE

m14 = 1, m04 = m13 = m23 = m24 = 2 and m34 = 3. We can check that m lies in
the balanced cone of multiplicities.
There are four odd sums around three cycles (so in the notation of Theorem 1.1,
q = 4). Also, we compute DV(m) = 16. From Theorem 1.1, we cannot conclude
that (A4 , m) is not free since qℓ = 16 also in this case. However, let us consider
the A3 sub-arrangement AU where U = {v0 , v1 , v3 , v4 }. Let mU be the restricted
multiplicity; it also lies in the balanced cone of multiplicities on AU . All sums
around three-cycles are even, and DV(mU ) = 8. Since 8 > 0, it follows from
Theorem 1.1 that (AU , mU ) is not free, hence (A4 , m) is also not free.
2. Notation and preliminaries
ℓ

Let V = K be a vector space over a field K of characteristic zero. A central
hyperplane arrangement A = ∪ni=1 Hi is a union of hyperplanes Hi ⊂ V passing
through the origin in V . In other words, if we let {x1 , . . . , xℓ } be a basis for the
dual space V ∗ and S = Sym(V ∗ ) ∼
= K[x1 , . . . , xl ], then Hi = V (αHi ) for some
choice of linear form αHi ∈ V ∗ , unique up to scaling. We will use the language
of graphic arrangements for referring to the braid arrangement Aℓ and its subarrangements. Namely, suppose G = (VG , EG ) is a graph with vertices ordered
as VG = {v0 , . . . , vℓ }, and let S = K[x0 , . . . , xℓ ]. If {vi , vj } is an edge in EG
then let Hij = V (xi − xj ). The graphic arrangement associated to G is AG =
∪{i,j}⊂E(G) Hij . Clearly AG is a sub-arrangement of the full braid arrangement
Aℓ , which may be identified with the graphic arrangement corresponding to the
complete graph Kℓ+1 on (ℓ + 1) vertices.
A multi-arrangement is a pair (A, m) of a central arrangement A = ∪ki=1 Hi and
a map m : {H1 , . . . , Hk } → Z≥0 , called a multiplicity. If m ≡ 1, then (A, m) is
denoted A and is called a simple arrangement. If AG is a graphic arrangement then
the multi-arrangement (AG , m) is equivalent to the information of the edge-labeled
graph (G, m), where {vi , vj } is labeled by m(Hij ) = mij . We will frequently move
back and forth between these notations. We will always assume that a graph G
comes with some ordering VG = {v0 , . . . , vℓ } of its vertices and may refer to the
vertices simply by their integer labels {0, . . . , ℓ}.
Lℓ
The module of derivations on S is defined by DerK (S) =
i=1 S∂xi , the free
S-module with basis ∂xi = ∂/∂xi for i = 1, . . . , ℓ. The module DerK (S) acts on S
by partial differentiation. Given a multi-arrangement (A, m), our main object of
study is the module D(A, m) of logarithmic derivations of (A, m):
m(H)

D(A, m) := {θ ∈ DerK (S) : θ(αH ) ∈ hαH
m(H)

m(H)

i for all H ∈ A},

where hαH i ⊂ S is the ideal generated by αH . If D(A, m) is a free S-module,
then we say (A, m) is free or m is a free multiplicity of the simple arrangement A.
For a simple arrangement, D(A, m) is denoted D(A); if D(A) is free we say A is
free.
The intersection lattice of A is the ranked poset L = L(A) consisting of all intersections of hyperplanes of A ordered with respect to reverse inclusion (the vector
space V is included as the ‘empty’ intersection). We denote by Lk the intersections
of rank k, where the rank of an intersection is its codimension. If X ∈ Lk , AX denotes the sub-arrangement consisting of hyperplanes which contain X, LX denotes
the lattice of AX , and mX denotes the multiplicity function restricted to hyperplanes containing X. If AG is a graphic arrangement with lattice L and H ⊂ G

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

5

is a connected induced sub-graph of G on (k + 1) vertices, then H corresponds
to an intersection X(H) ∈ Lk , and the graphic arrangement AH is the same as
(AG )X(H) . In this setting, if m is a multiplicity on AG , we denote by mH the
restriction of m to the sub-arrangement AH .
Proposition 2.1. [4, Proposition 1.7] If (A, m) is a free multi-arrangement, then
so is (AX , mX ).
If D(A, m) is free then it has ℓ minimal generators as an S-module whose
degrees are an invariant of D(A, m). These degrees are called the exponents
of (A, m) and we will list them as a non-increasing sequence (d1 , . . . , dℓ ). Put
P
P
|m| = H∈L1 m(H). Then ℓi=1 di = |m| (this follows for instance by an extension of Saito’s criterion to multi-arrangements [17]). For a free multi-arrangement,
define the kth global mixed product by
X
GMP(k) =
di1 di2 · · · dik ,
where the sum runs across all k-tuples satisfying 1 ≤ i1 < · · · < ik ≤ ℓ. Furthermore, define the kth local mixed product by
X
X
X
LMP(k) =
dX
1 d2 · · · dk ,
X∈Lk

k
dX
1 , . . . , dX

where
are the (non-zero) exponents of the rank k sub-arrangement AX .
We make use of the following result for k = 2.
Theorem 2.2. [3, Corollary 4.6] If (A, m) is free then GMP(k) = LMP(k) for
every 2 ≤ k ≤ ℓ.
3. Deviations and mixed products in the balanced cone
In this section we study the local and global mixed products of multiplicities
in the balanced cone. In particular, we prove the implication (1) =⇒ (2) of
Theorem 1.1. Recall the balanced cone of multiplicities on a braid arrangement
Aℓ is the set of multiplicities satisfying the three inequalities mij + mjk + 1 ≥
mik , mij +mik +1 ≥ mjk , and mik +mjk +1 ≥ mij for every triple 0 ≤ i < j < k ≤ ℓ.
The following proposition (due to Wakamiko) explains why we call this the balanced
cone; it is because the exponents of every sub-A2 arrangement are as balanced as
possible.
Proposition 3.1. [15] Suppose H is a three-cycle on the vertices i, j, k of the
edge-labeled complete graph (Kℓ+1 , m). Put mijk = mij + mik + mjk . If m is in
the balanced cone of multiplicities then the (non-zero) exponents of (AH , mH ) are
(⌊mijk /2⌋, ⌈mijk /2⌉).
If {i, j, k} are vertices of Kℓ+1 so that mij + mik + mjk is odd then we will call
{i, j, k} an odd three-cycle.
Proposition 3.2. Let (Aℓ , m) be a multi-braid
arrangement so that m is in the
P
balanced cone of multiplicities. Set |m| = ij mij and mijk = mij + mjk + mik . If
q is the number of odd three cycles of m, then
X
X
mij mst − q/4
LMP(2) =
(mijk /2)2 +
0≤i<j<k≤ℓ

{i,j}∩{s,t}=∅

6

M. DIPASQUALE

and
 
ℓ |m|2
.
GMP(2) ≤
2 ℓ2
Proof. We prove the formula for LMP(2) first. If X ∈ L2 , then either (1) : X =
Hij ∩Hst for a pair of non-adjacent edges {i, j} and {s, t} or (2) : X = Hij ∩Hjk ∩Hik
corresponds to a triangle. In the first case the arrangement is boolean with (nonzero) exponents (mij , mst ), contributing mij mst to LMP(2). In the second case the
arrangement is an A2 braid arrangement with exponents (mijk /2, mijk /2) if mijk
is even and ((mijk − 1)/2, (mijk + 1)/2) if mijk is odd from Proposition 3.1. The
former contributes m2ijk /4 to LMP(2) while the latter contributes m2ijk /4 − 1/4.
P
X
This yields the expression for LMP(2) = X∈L2 dX
1 d2 .
Now consider the inequality for GMP(2). Supposing (Aℓ , m) is free, let (d1 , . . . , dℓ )
be its (non-zero) exponents, ordered so that d1 ≥ · · · ≥ dℓ . By an extension of
Pℓ
Saito’s criterion to multi-arrangements [17],
i=1 di = |m|. Now, following remarks
just
after
[3,
Corollary
4.6],
we
say
that
(b
. . . , bℓ ) with b1 ≥P· · · ≥ bℓ and
1, P
P
bi = |m| is ‘more balanced’ than (d1 , . . . , dℓ ) if (bi+1 − bi ) ≤ (di+1 − di ).
Then
X
X
di1 · · · dik = GMP(k).
bi1 · · · bik ≥
Let |m| = kℓ + p be the result of dividing |m| by ℓ, so k is a positive integer and
0 ≤ p < ℓ. The ‘most balanced’ distribution of exponents occurs when

b1 = · · · = bp = k + 1
and
bp+1 = . . . = bℓ = k,
P
so (bi+1 − bi ) = bp−1 − bp is zero if p = 0 and one if p > 0. Some algebra yields
that, for this choice of exponents,
 
 
X
p(ℓ − p)
ℓ |m|2
ℓ |m|2
−
,
≤
bi bj =
2
2ℓ
2 ℓ2
2 ℓ
1≤i<j≤ℓ

which gives the result.



Definition 3.3. Fix a multiplicity m on the braid arrangement Aℓ and a subset
U ⊂ {v0 , . . . , vℓ }. Denote by C4 (Kℓ+1 ) the set of four-cycles in Kℓ+1 and by C4 (U )
the set of four-cycles in Kℓ+1 whose vertices are contained in U . If C ∈ C4 (Kℓ+1 )
traverses the vertices i, j, s, t in order, set m(C) = |mij − mjs + mst − mit |. The
deviation of m over U is the sum of squares
X
m(C)2 .
DV(mU ) =
C∈C4 (U)

If U consists of all vertices of Kℓ+1 , then we write DV(m) instead of DV(mU ).
Theorem 3.4. Suppose (Aℓ , m) is a multi-braid arrangement, m is in the balanced
cone of multplicities, and q is the number of odd three cycles of m. If DV(m) > qℓ,
then m is not free.
Remark 3.5. The inequality DV(m) > qℓ in Theorem 3.4 can be strengthened to
DV(m) > qℓ − 2p(ℓ − p), where p is the remainder of |m| on division by ℓ. We will
see that the simpler inequality DV(m) > qℓ suffices to detect non-freeness.

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

7

Proof. By Proposition 3.2, we know that
LMP(2) − GMP(2) ≥
X

X

2

(mijk /2) +

0≤i<j<k≤ℓ

{i,j}∩{s,t}

Our primary claim is

X
(1)
4ℓ 
(mijk /2)2 +

X

0≤i<j<k≤n

{i,j}∩{s,t}

 
ℓ |m|2
mij mst −
− q/4.
2 ℓ2


 
2
ℓ |m| 
= DV(m).
mij mst −
2 ℓ2

Once Equation (1) is proved, notice that
4ℓ(LMP(2) − GMP(2)) ≥ DV(m) − qℓ.
Then Theorem 2.2 immediately yields Theorem 3.4. So we prove Equation (1). We
first consider the right hand
 side, namely the sum DV(m). Since every edge of
ℓ−1
Kℓ+1 is contained in 2 2 four-cycles, every pair of disjoint edges is contained in
two four-cycles, and every pair of adjacent edges is contained in (ℓ − 2) four-cycles,


ℓ−1
(2) DV(m) = 2
2

X

X

m2ij + 4

0≤i<j≤ℓ

mij mst

{i,j}∩{s,t}=∅

X

− 2(ℓ − 2)

(mij mik + mij mjk + mik mjk ).

0≤i<j<k≤ℓ

Now consider the left hand side of Equation (1). Distributing 4ℓ yields
X
X
mij mst − 2(ℓ − 1)|m|2 .
ℓ
(mijk )2 + 4ℓ
0≤i<j<k≤ℓ

{i,j}∩{s,t}

This expression can be re-written in the form of Equation (2) using the following
two expressions and simplifying:
P
P
m2ij + 2
mij mst
|m|2 =
0≤i<j≤ℓ
{i,j}∩{s,t}=∅
P
+2
(mij mik + mij mjk + mik mjk )
0≤i<j<k≤ℓ

P

0≤i<j<k≤ℓ

m2ijk =

(ℓ − 1)

P

0≤i<j≤ℓ

+2

P

m2ij

(mij mik + mij mjk + mik mjk ). 

0≤i<j<k≤ℓ

As an immediate corollary we obtain (1) =⇒ (2) in Theorem 1.1.
Corollary 3.6. If (Aℓ , m) is free then DV(mU ) ≤ qU (|U | − 1) for every subset
U ⊂ {v0 , . . . , vℓ }.
Proof. By Proposition 2.1, freeness of (Aℓ , m) implies freeness of (AU , mU ) for every
subset U ⊂ {v0 , . . . , vℓ }. By Theorem 3.4, we must have DV(mU ) ≤ qU (|U | − 1)
for every subset U ⊂ {v0 , . . . , vℓ } as well.


8

M. DIPASQUALE

4. From deviations to ANN multiplicities
In this section we prove the first part of the implication (2) =⇒ (3) in Theorem 1.1. Namely, we prove that for a multiplicity m in the central cone, the
inequalities DV(m) ≤ qU (|U | − 1) on deviations are enough to guarantee that m is
an ANN multiplicity. In fact, we show that it is enough to have these inequalities
on subsets of size four.
Recall from the introduction that we call m an ANN multiplicity on Aℓ if m is in
the balanced cone of multiplicities and there exist non-negative integers n0 , . . . , nℓ
and ǫij ∈ {−1, 0, 1} so that mij = ni + nj + ǫij for 0 ≤ i < j ≤ ℓ.
Lemma 4.1. Suppose m is a multiplicity on A3 and DV(m) ≤ 3q. Then m(C) ≤ 2
for each four-cycle C in K4 .
Proof. There are three four-cycles. Set
T1 =
T2 =
T3 =

m01 − m12 + m23 − m03
m13 − m01 + m02 − m23
m13 − m12 + m02 − m03 .

Notice T1 + T2 = T3 , and DV(m) = T12 + T22 + T32 . Now, suppose without loss that
|T3 | ≥ 3. Then either |T1 | ≥ 2 or |T2 | ≥ 2. But then P (m) ≥ 13, contradicting that
DV(m) ≤ 3q ≤ 12 (since q ≤ 4).

Proposition 4.2. Let (Aℓ , m) be a multi-braid arrangement so that m is in the
balanced cone of multiplicities and DV(mU ) ≤ 3qU for every subset U ⊂ {v0 , . . . , vℓ }
with |U | = 4. Then m is an ANN multiplicity.
Proof. We need only show that there exist non-negative integers ni for i = 0, . . . , ℓ
and integers ǫij ∈ {−1, 0, 1} for 0 ≤ i < j ≤ ℓ so that mij = ni + nj + ǫij . By
Lemma 4.1, we must have m(C) ≤ 2 for every four-cycle C ∈ C4 (Kℓ+1 ). We use
this condition to provide
producing
the integers


 n0 , . . . , nℓ .
 an inductive algorithm
m01 + m12 − m02
m01 + m02 − m12
, n1 =
, and n2 =
If ℓ = 2, set n0 =
2
2


m02 + m12 − m01
. Since m is in the balanced cone, ni ≥ 0 for i = 0, 1, 2.
2
Moreover, mij = ni + nj + ǫij , where ǫij ∈ {−1, 0}.
Now assume ℓ > 2. We make an initial guess at what the non-negative integers
n0 , . . . , nℓ and ǫij should be, and then adjust as necessary. By induction on ℓ,
there exist non-negative integers ñ0 , . . . , ñℓ−1 and ǫ̃ij ∈ {−1, 0, 1} such that mij =
ñi + ñj + ǫ̃ij for 0 ≤ i < j ≤ ℓ − 1. Let ñℓ be a non-negative integer satisfying
ñℓ + ñi ≥ miℓ − 1 and set ǫ̃iℓ = miℓ − (ñi + ñℓ ) for every i < ℓ, so miℓ = ñi + ñℓ + ǫ̃iℓ .
By the choice of ñℓ , we have ǫ̃iℓ ≤ 1 for all i < ℓ.
Now suppose there is an index 0 ≤ j < ℓ so that ǫ̃jℓ ≤ −2. Our goal is to
decrease either ñℓ or ñj by one, thereby increasing ǫ̃jℓ , without disturbing any of
the hypotheses made so far, namely
ñi ≥ 0 for all 0 ≤ i ≤ ℓ,
(⋆)

ǫ̃iℓ ≤ 1 for all i < ℓ,
ǫ̃st ∈ {−1, 0, 1} for all 0 ≤ s < t ≤ ℓ − 1.

First we assume ñℓ > 0 and try to decrease ñℓ by one. We can do this without
disturbing assumptions (⋆) provided there is no index s so that ǫsℓ = 1. So, assume

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

9

that there is an index 0 ≤ s < ℓ so that ǫsℓ = 1. We claim that in this situation,
ǫst ≥ 0 for every t 6= s. Suppose to the contrary that there is an index t so that
ǫst = −1 and consider the four-cycle C : ℓ → s → t → j → ℓ. Then
m(C) = |ǫ̃sℓ − ǫ̃jℓ + ǫ̃jt − ǫ̃st |
≥ 1 + 2 + ǫ̃jt + 1
≥ 3,
since ǫ̃jt ∈ {−1, 0, 1} by the inductive hypothesis. This contradicts our assumption
that m(C) ≤ 2. So it follows that ǫ̃st ∈ {0, 1} for all t. Thus we may increase
ñs by one, thereby decreasing ǫ̃st by one for every t 6= s, without disturbing the
hypothesis that ǫ̃st ∈ {−1, 0, 1}. Since we can apply this argument at every index
s so that ǫ̃sℓ = 1, we may assume ǫ̃sℓ ≤ 0 for every 0 ≤ s < ℓ. Hence, if ñℓ > 0, it
is now clear that we can decrease ñℓ by one without disturbing assumptions (⋆).
Now assume that ñℓ = 0. Then, for any s < ℓ,
msℓ + mjℓ − mjs

= (ñs + ǫ̃sℓ ) + (ñj + ǫ̃jℓ ) − (ñj + ñs + ǫ̃js )
= ǫ̃sℓ + ǫ̃jℓ − ǫ̃js
≤ 0 − 2 − ǫ̃js
≤ −1,

since ǫ̃js ∈ {−1, 0, 1} by the inductive hypothesis. Since m is in the balanced cone,
we must have equality for all of these, so ǫjs = −1 for every s 6= j, s < ℓ. If
ñj = 0 as well, then mjℓ = ñj + ñℓ + ǫjℓ ≤ −2, contradicting that mjℓ is nonnegative. Hence ñj > 0 and we can decrease ñj by one without disturbing any of
assumptions (⋆).
In either case, we have shown how to increase ǫ̃jℓ if ǫ̃jℓ ≤ −2 without disturbing
assumptions (⋆). So we iterate the above arguments until ǫ̃jℓ ≥ −1 for every j < ℓ,
then set ni = ñi for 0 ≤ i ≤ ℓ and ǫ̃ij = ǫij for 0 ≤ i < j ≤ ℓ. This completes the
algorithm and the proof.

With Proposition 4.2, we now prove (1) ⇐⇒ (3) in Theorem 1.1. Most of the
heavy lifting is done by Abe-Nuida-Numata in [2].
Corollary 4.3. Suppose m is in the balanced cone of multiplicities on the Aℓ braid
arrangement. Then (Aℓ , m) is free if and only if m is an ANN multiplicity and
+
−
the signed graph with EG
= {{vi , vj } : ǫij = −1} and EG
= {{vi , vj } : ǫij = 1} is
signed-eliminable.
Proof. If (Aℓ , m) is free and m is in the balanced cone, then DV(mU ) ≤ 3qU for
every subset U ⊂ {v0 , . . . , vℓ } of size four by Corollary 3.6. By Proposition 4.2, m is
+
an ANN multiplicity. By [2, Theorem 0.3], the signed graph with EG
= {{vi , vj } :
−
ǫij = −1} and EG = {{vi , vj : ǫij = 1} is signed-eliminable. For the converse, if
m is an ANN multiplicity associated to a signed-eliminable graph, then (Aℓ , m) is
free by [2, Theorem 0.3].

Remark 4.4. In the result [2, Theorem 0.3], Abe-Nuida-Numata do not have the
condition that m is in the balanced cone. However, this turns out to be a necessary
condition for their arguments [1]. Furthermore their arguments, using additiondeletion techniques for multi-arrangements from [4], work for any ANN multiplicity
as we have defined it [1].

10

M. DIPASQUALE

v0
v1

v2

v3

···

v0
vℓ−2 vℓ−1 vℓ

v2

v3

v4

v1
···

vℓ−2 vℓ−1 vℓ

Figure 1. σ-mountain (at left) and σ-hill (at right)
5. Detecting signed-eliminable graphs
In this section we finish the proof of Theorem 1.1. We have already shown in
Proposition 4.2 that if the inequalities of Theorem 1.1.(2) are satisfied then m is
an ANN multiplicity. Now we show that these inequalities also detect when the
associated signed graph is not signed-eliminable. We follow the presentation of
signed-eliminable graphs from [11, 2].
Let G be a signed graph on ℓ+1 vertices. That is, each edge of G is assigned either
+
−
a + or a −, and so the edge set EG decomposes as a disjoint union EG = EG
∪ EG
.
Define

+
 1 {i, j} ∈ EG
−
mG (ij) =
−1 {i, j} ∈ EG

0 otherwise.

The graph G is signed-eliminable with signed-elimination ordering ν : V (G) →
{0, . . . , ℓ} if ν is bijective and, for every three vertices vi , vj , vk ∈ V (G) with
ν(vi ), ν(vj ) < ν(vk ), the induced sub-graph G|vi ,vj ,vk satisfies the following conditions.
σ
σ
• For σ ∈ {+, −}, if {vi , vk } and {vj , vk } are edges in EG
then {vi , vj } ∈ EG
−σ
σ
• For σ ∈ {+, −}, if {vk , vi } ∈ EG and {vi , vj } ∈ EG then {vk , vj } ∈ EG
These two conditions generalize the notion of a graph possessing an elimination
ordering, which is equivalent to the graph being chordal. A graph is chordal if and
only if it has no induced sub-graph which is a cycle of length at least four. In [11],
Nuida establishes a similar characterization for signed-eliminable graphs, to which
we now turn.

Definition 5.1.
(1) A graph with (ℓ + 1) vertices v0 , v1 , . . . , vℓ with ℓ ≥ 3 is
σ
a σ-mountain, where σ ∈ {+, −}, if {v0 , vi } ∈ EG
for i = 2, . . . , ℓ − 1,
−σ
{vi , vi+1 } ∈ EG for i = 1, . . . , ℓ − 1, and no other pair of vertices is joined
by an edge. (See Figure 1 - edges of sign σ are denoted by a single edge
and edges of sign −σ are denoted by a doubled edge.)
(2) A graph with (ℓ + 1) vertices v0 , v1 , v2 , . . . , vℓ with ℓ ≥ 3 is a σ-hill, where
σ
σ
σ
σ ∈ {+, −}, if {v0 , v1 } ∈ EG
, {v0 , vi } ∈ EG
for i = 2, . . . , ℓ−1, {v1 , vi } ∈ EG
−σ
for i = 3, . . . , ℓ, {vi , vi+1 } ∈ EG for i = 2, . . . , ℓ − 1, and no other pair of
vertices is connected by an edge. (See Figure 1.)
(3) A graph with (ℓ + 1) vertices v0 , . . . , vℓ with ℓ ≥ 2 is a σ-cycle if {vi , vi+1 } ∈
σ
σ
EG
for i = 0, . . . , ℓ − 1, {v0 , vℓ } ∈ EG
, and no other pair of vertices is
connected by an edge.
Theorem 5.2. [11, Theorem 5.1] Let G be a signed graph. Then G is signedeliminable if and only if the following three conditions are satisfied.
(C1) Both G+ and G− are chordal.

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

11

Table 1. Graphs on four vertices which are not signed-eliminable

(C2) Every induced sub-graph on four vertices is signed eliminable.
(C3) No induced sub-graph of G is a σ-mountain or a σ-hill.
All signed-eliminable graphs on four vertices are listed (with an elimination ordering) in [2, Example 2.1], along with those which are not signed-eliminable. For
use in the proof of Corollary 5.3, we also list those graphs which are not signedeliminable in Table 1. The property of being signed-eliminable is preserved under
interchanging + and −. Consequently, we list these graphs in Table 1 up to automorphism with the convention that a single edge takes one of the signs +, −, while
a double edge takes the other sign.
Corollary 5.3. Let G be a signed graph. Then G is signed-eliminable if and only
if the following three conditions are satisfied.
(C1′ ) No induced sub-graph of G is a σ-cycle of length > 3.
(C2) Every induced sub-graph on four vertices is signed eliminable.
(C3) No induced sub-graph of G is a σ-mountain or a σ-hill.
Proof. Clearly (C1) from Theorem 5.2 implies (C1′ ). We show that (C1′ ) and (C2)
σ
imply condition (C1). Assume for contradiction that EG
, σ ∈ {−, +}, is a cycle of
σ
length ℓ + 1 > 3 and VG = {v0 , . . . , vℓ } where {vi , vi+1 } ∈ EG
for i = 0, . . . , ℓ − 1
−σ
σ
and {v0 , vℓ } ∈ EG . If EG = ∅ then G is a σ-cycle which is forbidden by (C1′ ),
−σ
so we assume EG
6= ∅. Let m be the maximal integer so that there is a sequence
of consecutive vertices vi , vi+1 . . . , vi+m−1 so that the induced sub-graph on these
−σ
σ
6= ∅, m < ℓ + 1.
consecutive vertices consists only of edges in EG
. Since EG
Relabel the vertices so that v0 , . . . , vm−1 are the vertices of a maximal induced
σ
sub-graph with edges only in EG
. If m = 2 or m = 3, then the induced sub-graph
on v0 , v1 , v2 , v3 consists of the three σ edges {v0 , v1 }, {v1 , v2 }, {v2 , v3 } along with
at least one −σ edge. No such graph is signed eliminable (see Table 1). So m ≥ 4.
Now consider the induced sub-graph H on v0 , v1 , . . . , vm−1 , vm . By definition of m,
H has exactly one −σ edge, namely {v0 , vm }. But then the induced sub-graph on
v0 , v1 , vm−1 , vm consists of the two σ edges {v0 , v1 }, {vm−1 , vm } and the −σ edge
σ
{v0 , vm }, which is not signed-eliminable. It follows that EG
cannot have a cycle of
σ
length > 3, so EG is chordal.

The following proposition proves the implication (2) =⇒ (3) in Theorem 1.1,
thereby completing the proof of Theorem 1.1.
Proposition 5.4. Suppose n0 , . . . , nℓ are non-negative integers, G is a signed graph
on v0 , . . . , vℓ , and let m be the multiplicity on Aℓ given by mij = ni +nj +mG (ij). If
G is not signed-eliminable, then there is a subset U ⊂ {0, . . . , ℓ} so that DV(mU ) >
qU · (|U | − 1).

12

M. DIPASQUALE

Proof. Notice that, for a four-cycle traversing i, j, s, t in order,
m(C) = |mij − mjs + mst − mit | = |mG (ij) − mG (js) + mG (st) − mG (it)|.
Furthermore, for a three-cycle {i, j, k},
mij + mik + mjk = 2(ni + nj + nk ) + mG (ij) + mG (ik) + mG (jk).
P
It follows that the values of DV(m) =
m(C)2 and qℓ = (# odd three cycles) · ℓ
from Theorem 3.4 may be determined after replacing mij by mG (ij), which takes
values only in {−1, 0, 1}. Hereafter we write DV(G) for DV(m) and qG for q to
emphasize their dependence only on the signed graph G. If U ⊂ {v0 , . . . , vℓ }, we
let DV(GU ) represent DV(mU ) to emphasize dependence only on G and the subset
U . As usualy, qU denotes the number of odd three cycles contained in U .
Now, if G is not signed eliminable then by Corollary 5.3 G contains an induced
sub-graph H which is
• a signed graph on four vertices which is not signed-eliminable,
• a σ-cycle of length > 3,
• a σ-hill,
• or a σ-mountain.
We assume G = H and show that DV (G) > qG ℓ in each of these cases, where ℓ
is one less than the number of vertices of G. The inequality DV (G) > 3qG can
easily be verified by hand for each of the twelve graphs on four vertices which are
not signed-eliminable (see Table 1); this is also done in [8, Corollary 6.2]. If G is a
σ-cycle, σ-mountain, or σ-hill on (ℓ + 1) vertices we will show that DV(G) and qG
are given by the formulas:
(3)

DV(G) =ℓ3 − 2ℓ2 − ℓ + 2

(4)

qG =ℓ2 − 2ℓ − 3.

Given these formulas, note that DV(G) = qℓ + 2(ℓ + 1) > qℓ, thus proving the
result. We prove Equations (3) and (4) for the σ-cycle directly, relying on the two
additional formulas:
X
(5)
DV(GU )
DV(G) =
U⊂VG ,|U|=4

(6)

qG = (

X

qU )/(ℓ − 2).

U⊂VG ,|U|=4

Equation (5) follows since each four-cycle is contained in a unique induced subgraph on four vertices and Equation (6) follows since each three-cycle appears in
(ℓ − 2) sub-graphs on four vertices. Using these equations, it suffices to identify
all possible types of induced sub-graphs of the σ-cycle on four vertices, how many
of each type there are, and compute DV(GU ) and qU for each of these. Then
we use Equation (5) to compute DV(G) and Equation (6) to compute qG . The
list of all possible induced sub-graphs with four vertices of a σ-cycle on (ℓ + 1)
vertices are listed in Table 2. The number of sub-graphs of each type is listed in
the second column, while the third and fourth columns record qU and DV(GU ),
respectively, for each type of sub-graph. The final row records the total number
of sub-graphs on four
of odd three-cycles, and the deviation
P vertices, the number
2
m(C)
.
We
find that DV(m) = ℓ3 − 2ℓ2 − ℓ + 2
of m, DV(m) =
C∈C4 (Kℓ+1 )
2
and q = ℓ − 2ℓ − 3, proving Equations (3) and (4) for the σ-cycle. The same

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

13

σ-cycle of length (ℓ + 1)

Type of sub-graph

Total

Count

qU

DV(GU )


 

ℓ−4
ℓ−3
+
2
2

0

0



ℓ−4
(ℓ + 1)
2

2

2

(ℓ + 1)(ℓ − 4)
2

4

8

(ℓ + 1)(ℓ − 4)

2

2

ℓ+1

2

6



ℓ+1
q = ℓ2 − 2ℓ − 3 DV = ℓ3 − 2ℓ2 − ℓ + 2
4
Table 2. Computing DV(G) where G is a σ-cycle

computations can be done to prove Equations (3) and (4) for the σ-hill and σmountain; for the convenience of the reader we collect these in Appendix A.

6. Free vertices and a conjecture
In this final section we discuss free vertices of a multiplicity on a graphic arrangement and present a conjecture on the structure of free multiplicities on braid
arrangements.
Definition 6.1. Suppose G is a graph. A vertex vi ∈ VG is a simplicial vertex
if the sub-graph of G induced by vi and its neighbors is a complete graph. Given
a multi-arrangement (AG , m) and the corresponding edge-labeled graph (G, m), a
vertex vi is a free vertex of (G, m) if it is a simplicial vertex and for every triangle
with vertices vi , vj , vk we have mij + mik ≤ mjk + 1.
Theorem 6.2. Suppose G is a graph, vi is a free vertex of (G, m), and G′ is the
induced sub-graph on the vertex set VG \ {vi }. Then (AG , m) is free if and only if
(AG′ , mG′ ) is free.

14

M. DIPASQUALE

Proof of Theorem 6.2. We use a result whose proof we omit since it is virtually
identical to the proof of [4, Theorem 5.10]. Recall that a flat X ∈ L(A) is called
modular if X + Y ∈ L(A) for every Y ∈ L(A), where X + Y is the linear span of
X, Y considered as linear sub-spaces of V = Kℓ .
Theorem 6.3. Suppose (A, m) is a central multi-arrangement of rank ℓ ≥ 3 and
X is a modular flat of rank ℓ − 1. Suppose (AX , mX ) is free with exponents
(d1 , . . . , dℓ−1 , 0) and for all H ∈ A \ AX and H ′ ∈ AX , set Y := H ∩ H ′ . If
one of the following two conditions is satisfied:
′
(1) AY = H ∪ HP
or
′
(2) m(H ) ≥
m(H) − 1.
H∈A\A′

Then (A, m) is free with exponents (d1 , . . . , dℓ−1 , |m| − |m′ |).

Now suppose G is a graph on ℓ + 1 vertices {v0 , . . . , vℓ } and AG is the associated graphic arrangement. Further suppose that vi is a free vertex of (G, m),
and G′ is the induced sub-graph on the vertex set VG \ {vi }. Set m′ = m|G′ . By
Proposition 2.1, if (AG′ , m′ ) is not free, then neither is (AG , m).
Suppose now that (AG′ , m′ ) is free. We show that (AG , m) is free using Theorem 6.3. Write Hij = V (xi − xj ). Since vi is a simplicial vertex of G, the flat
X = ∩vj ,vk 6=vi Hjk is modular and has rank ℓ − 1. The sub-arrangement (AG )X is
the graphic arrangement AG′ . Suppose H = Hij ∈ AG \ AG′ , H ′ = Hst ∈ AG′ , and
set Y = Hij ∩ Hst . If {s, t} ∩ {i, j} = ∅, then AY = Hij ∪ Hst . Otherwise, suppose
s = j. Since vi is a simplicial vertex, {i, t} ∈ EG , so AY = Hij ∪ Hit ∪ Hjt . Since
vi is a free vertex, mij + mit ≤ mjt + 1, which is condition (2) from Theorem 6.3.
Hence (AG , m) is free by Theorem 6.3.

Remark 6.4. Theorem 6.2 can also be proved using homological techniques from [7].
We use Theorem 6.2 to inductively construct two types of free multiplicities.
Given a graph G, an elimination ordering is an ordering v0 , . . . , vℓ of the vertices
VG so that vi is a simplicial vertex of the induced sub-graph on v0 , . . . , vi for every
i = 1, . . . , ℓ. It is known that VG admits an elimination ordering if and only if G is
chordal [9].
Corollary 6.5. Suppose (G, m) is an edge-labeled chordal graph with elimination
ordering v0 , . . . , vℓ satisfying that vi is a free vertex of the induced sub-graph on
{v0 , . . . , vi } for every i ≥ 2. Then (AG , m) is free.
Corollary 6.6. Let (Aℓ , m) be a multi-braid arrangement corresponding to the
complete graph Kℓ+1 on (ℓ + 1) vertices. Suppose that Kℓ+1 admits an ordering
{v0 , . . . , vℓ } so that:
(1) For some integer 0 ≤ k ≤ ℓ, the induced sub-graph G′ on {v0 , . . . , vk }
satisfies that mG′ is a free ANN multiplicity.
(2) For k + 1 ≤ i ≤ ℓ, vi is a free vertex of the induced graph on {v0 , . . . , vi }.
Then (Aℓ , m) is free.
We conjecture that all free multi-braid arrangements take the form of Corollary 6.6.
Conjecture 6.7. The multi-braid arrangement (Aℓ , m) is free if and only if it is
one of the multi-braid arrangements constructed in Corollary 6.6. Equivalently, by

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

15

Theorem 6.2, if (Aℓ , m) is free then either m is a free ANN multiplicity or m has
a free vertex. Using Theorem 1.1, this is equivalent to the following statement: if
m is a free multiplicity which is not in the balanced cone of multiplicities, then m
has a free vertex.
Remark 6.8. Conjecture 6.7 is proved for the A3 braid arrangement in [8]. Using
Macaulay2 [10], we have verified Conjecture 6.7 for many multiplicities on the A4
arrangement.
7. Acknowledgements
I would especially like to thank Jeff Mermin, Chris Francisco, and Jay Schweig
for their collaboration on the analysis of free multiplicities on the A3 braid arrangement. The current work would not be possible without their help. I am very
grateful to Takuro Abe for freely corresponding and offering many suggestions.
Computations in Macaulay2 [10] were indispensable for this project.
References
[1] Takuro Abe. personal communication.
[2] Takuro Abe, Koji Nuida, and Yasuhide Numata. Signed-eliminable graphs and free multiplicities on the braid arrangement. J. Lond. Math. Soc. (2), 80(1):121–134, 2009.
[3] Takuro Abe, Hiroaki Terao, and Max Wakefield. The characteristic polynomial of a multiarrangement. Adv. Math., 215(2):825–838, 2007.
[4] Takuro Abe, Hiroaki Terao, and Max Wakefield. The Euler multiplicity and addition-deletion
theorems for multiarrangements. J. Lond. Math. Soc. (2), 77(2):335–348, 2008.
[5] Takuro Abe and Masahiko Yoshinaga. Coxeter multiarrangements with quasi-constant multiplicities. J. Algebra, 322(8):2839–2847, 2009.
[6] Christos A. Athanasiadis. Deformations of Coxeter hyperplane arrangements and their characteristic polynomials. In Arrangements—Tokyo 1998, volume 27 of Adv. Stud. Pure Math.,
pages 1–26. Kinokuniya, Tokyo, 2000.
[7] M. DiPasquale. Generalized Splines and Graphic Arrangements. J. Algebraic Combin., 2016.
doi:10.1007/s10801-016-0704-8.
[8] M. DiPasquale, C. A. Francisco, J. Mermin, and J. Schweig. Free and Non-free Multiplicities
on the A3 Arrangement. ArXiv e-prints, September 2016.
[9] G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 25:71–76, 1961.
[10] Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in
algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[11] Koji Nuida. A characterization of signed graphs with generalized perfect elimination orderings. Discrete Math., 310(4):819–831, 2010.
[12] Kyoji Saito. On the uniformization of complements of discriminant loci. In Conference Notes.
Amer. Math. Soc. Summer Institute, Williamstown, 1975.
[13] Louis Solomon and Hiroaki Terao. The double Coxeter arrangement. Comment. Math. Helv.,
73(2):237–258, 1998.
[14] Hiroaki Terao. Multiderivations of Coxeter arrangements. Invent. Math., 148(3):659–674,
2002.
[15] Atsushi Wakamiko. On the exponents of 2-multiarrangements. Tokyo J. Math., 30(1):99–116,
2007.
[16] Masahiko Yoshinaga. The primitive derivation and freeness of multi-Coxeter arrangements.
Proc. Japan Acad. Ser. A Math. Sci., 78(7):116–119, 2002.
[17] Günter M. Ziegler. Multiarrangements of hyperplanes and their freeness. In Singularities
(Iowa City, IA, 1986), volume 90 of Contemp. Math., pages 345–359. Amer. Math. Soc.,
Providence, RI, 1989.

16

M. DIPASQUALE

Appendix A. Computations for mountains and hills

σ-mountain on (ℓ + 1) vertices

Type of subgraph

Count

qU

DV(GU )



ℓ−3
4

0

0



ℓ−3
3
3

2

2



ℓ−3
2

2

2



ℓ−5
2ℓ − 9 +
2

4

8

(ℓ − 3)

2

6

ℓ−4

2

2

2

2

6



ℓ−4
2
2

2

2

2(ℓ − 4)

4

8

2(ℓ − 4)

2

2

2

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

2

2

6



ℓ−4
3

0

0



ℓ−4
2

2

2

ℓ−4

2

2



ℓ+1
4

qG = ℓ2 − 2ℓ − 3

DV = ℓ3 − 2ℓ2 − ℓ + 2

2

Total

17

Table 3: Computing DV(G) where G is a σ-mountain

σ-hill on (ℓ + 1) vertices

Type of subgraph

Count

qU

DV(GU )



ℓ−4
4

0

0



ℓ−4
3

2

2



ℓ−4
2
2

2

2

4

8

3

2ℓ − 11 +



ℓ−6
2

18

M. DIPASQUALE

ℓ−4

2

6



ℓ−4
2
2

2

2

2(ℓ − 4)

4

8

2(ℓ − 4)

2

2

2

2

6



ℓ−4
3

0

0



ℓ−4
4
2

2

2

2(ℓ − 4)

2

2

1

2

6

2(ℓ − 4)

2

2

2

2

6



ℓ−4
2

2

2

2

INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS

Total

19

ℓ−4

4

8



ℓ+1
4

qG = ℓ2 − 2ℓ − 3

DV = ℓ3 − 2ℓ2 − ℓ + 2

Table 4: Computing DV(G) where G is a σ-hill

Michael DiPasquale, Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, USA
E-mail address: mdipasq@okstate.edu
URL: http://math.okstate.edu/people/mdipasq/

