aa r X i v : . [ m a t h . A C ] M a y INEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS
MICHAEL DIPASQUALE
Abstract.
We prove that, on a large cone containing the constant multiplic-ities, the only free multiplicities on the braid arrangement are those identifiedin work of Abe, Nuida, and Numata (2009). We also give a conjecture on thestructure of all free multiplicities on braid arrangements. Introduction
Let V ∼ = K ℓ +1 be a vector space over a field K of characteristic zero, V ∗ itsdual space and S = Sym( V ∗ ) ∼ = K [ x , . . . , x ℓ ]. Given a polynomial f ∈ S denoteby V ( f ) the zero-locus of f in V . The braid arrangement of type A ℓ ⊂ V is de-fined as A ℓ = ∪ ≤ i
5. We shall referto the set of multiplicities satisfying the inequalities in (2) as the balanced cone ofmultiplicities. The reason for this name will be explained in § A braid arrangement [8], which is joint work of theauthor with Francisco, Mermin, and Schweig. To state our result more concretelywe shall associate to the multi-braid arrangement ( A ℓ , m ) an edge-labeled completegraph ( K ℓ +1 , m ). The vertices of K ℓ +1 are labeled in bijection with the variables x , . . . , x ℓ ∈ S . An edge { v i , v j } corresponds to H ij = V ( x i − x j ) and is furthermore labeled by m ( H ij ) = m ij . Now suppose C is a four-cycle in K ℓ +1 which traversesthe vertices v i , v j , v s , v t in order. Define m ( C ) = | m ij − m js + m st − m it | ; sincewe take absolute value, m ( C ) is independent of orientation, depending only on thefour cycle and the multiplicity. Let C ( K ℓ +1 ) be the set of all four cycles of K ℓ +1 .Given a subset U ⊂ { v , . . . , v ℓ } of size at least four, the deviation of m over U isDV( m U ) = X C ∈ C ( K ℓ +1 ) C ⊂ U m ( C ) . Our main result is the following.
Theorem 1.1.
Suppose ( A ℓ , m ) is a multi-braid arrangement with m in the bal-anced cone of multiplicities. For a subset U ⊂ { v , . . . , v ℓ } let m U = { m ij |{ v i , v j } ⊂ U } and denote by q U the number of integers { m ij + m ik + m jk |{ v i , v j , v k } ⊂ U } that are odd. Then the following are equivalent. (1) ( A ℓ , m ) is free (2) DV ( m U ) ≤ q U ( | U | − for every subset U ⊂ { v , . . . , v ℓ } where | U | ≥ . (3) m is a free ANN multiplicity. In other words, there exist non-negativeintegers n , . . . , n ℓ and ǫ ij ∈ {− , , } (for ≤ i < j ≤ ℓ ) so that (a) m ij = n i + n j + ǫ ij (b) the signed graph G on { v , . . . , v ℓ } with E − G = {{ v i , v j } : ǫ ij < } , E + G = {{ v i , v j } : ǫ ij > } is signed-eliminable in the sense of [2] .Remark . Notice that DV( m U ) = 0 if and only if m ij − m js + m st − m it = 0 forevery four-tuple ( v i , v j , v s , v t ) of distinct vertices in U . These equations cut out thelinear space L U parametrized by m ij = n i + n j for { v i , v j } ⊂ U . Thus DV( m U ) canbe viewed as a measure of how far m U is from the linear space L U ; which in turnmeasures how far m U ‘deviates’ from being an ANN multiplicity on the sub-braidarrangement 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( m U ) in Theorem 1.1.(2) arises from studying the local and globalmixed products (introduced in [3]) of ( A ℓ , m ). The main point of our note is toshow that these ‘deviations’ not only detect freeness in the balanced cone but alsointeract well with the notion of signed-eliminable graphs.The structure of the paper is as follows. In Section 2 we give some backgroundon arrangements. The proof of Theorem 1.1 is split across sections 3 and 5. Theimplication (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 ifthe inequalities in (2) are satisfied, is Proposition 4.2. The second part, showingthat the inequalities in (2) detect when the associated signed graph is not signed-eliminable, 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 aconjecture about the structure of all free multiplicities on braid arrangements.1.1. Examples.
We provide some computations using Theorem 1.1. For the braidarrangement A ℓ , corresponding to the complete graph K ℓ +1 , we label the verticesof K ℓ +1 by v , . . . , v ℓ and, given a multiplicity m , we denote by m ij the value of m on the hyperplane H ij = V ( x i − x j ). If U ⊂ { v , . . . , v ℓ } , we denote by A U thecorresponding sub-braid arrangement of A ℓ . NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 3
Example 1.3.
First we consider a family of multiplicities on the A arrangement.Given positive integers s, t , define the multiplicity m s,t by m = m = m = s and m = m = m = t (this assigns different multiplicities along two edge-disjoint paths of length three).The multiplicity m s,t is in the balanced cone of multiplicities if and only if s ≤ t + 1 and t ≤ s + 1. Assuming m s,t is in the balanced cone of multiplicities,we now compute the deviation DV( m s,t ). There are three four cycles: one ofthese has m ( C ) = | s − t | while the other two have m ( C ) = | s − t | . HenceDV( m s,t ) = 6( s − t ) .Now consider the sums m ijk around three cycles. There are four such sums, twoof the form 2 s + t and two of the form 2 t + s . So, applying Theorem 1.1, if m s,t is in the balanced cone of multiplicities, it is free if and only if 6( s − t ) ≤ · , or | s − t | ≤
1. In fact, using the classification from [8], it follows that m s,t is a freemultiplicity if and only if | s − t | ≤ m s,t is in the balancedcone or not). Example 1.4.
Next we consider a similar family of multiplicities on the A braidarrangement. Let C be the five-cycle traversing the vertices v , v , v , v , v , v inorder and C be the five-cycle traversing the vertices v , v , v , v , v , v in order; C and C are edge-disjoint and every edge of K is contained in either C or C . Given positive integers s, t we define the multiplicity m s,t by m s,t | C ≡ s and m s,t | C ≡ t .Any closed sub-arrangement of ( A , m s,t ) of rank three has the form ( A , m s,t )considered in Example 1.3. It follows that ( A , m s,t ) is not free if | s − t | >
1. So weconsider the case when | s − t | ≤
1. If | s − t | ≤ m s,t is in the balanced cone ofmultiplicities. We compute DV( m s,t ) as follows. A four-cycle of K lies in a uniquecomplete sub-graph on four vertices and each complete sub-graph on four verticescontains three such four-cycles. As we saw in Example 1.3, one of these satisfies m ( C ) = | s − t | while the other two have m ( C ) = | s − t | . So the contribution toDV( m s,t ) from each complete sub-graph on four vertices is 6( s − t ) . As there arefive such sub-graphs, we have DV( m s,t ) = 30( s − t ) .Now we consider the sums m ijk around three cycles. There are ten such sums,five of the form 2 s + t and five of the form 2 t + 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. HenceDV( m s,t ) = 30 > · m s,t is not free by Theorem 1.1. So we conclude that m s,t is free if and only if s = t .We also consider why m s,t is not free when | s − t | = 1 using the criterionof Abe-Nuida-Numata (which is the third statement of Theorem 1.1). Withoutloss, suppose t = s + 1 and let n i = ⌈ s/ ⌉ for i = 0 , , , ,
4. If s is even then m ij = n i + n j = s for { i, j } ∈ C while m ij = n i + n j + 1 = s + 1 for { i, j } ∈ C .In this case the graph G on the vertices v , v , v , v , v is the (positive) five-cyclegiven by C and is hence not signed-eliminable by the characterization in [2] (seealso Corollary 5.3). Similarly, if s is odd, then G is the negatively signed five-cycle C . Example 1.5.
The following example shows that criterion (2) in Theorem 1.1really does need to be checked on all proper subsets of size at least four. Considerthe A arrangement with the multiplicity m defined by m = m = m = m = M. DIPASQUALE m = 1, m = m = m = m = 2 and m = 3. We can check that m lies inthe 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 concludethat ( A , m ) is not free since qℓ = 16 also in this case. However, let us considerthe A sub-arrangement A U where U = { v , v , v , v } . Let m U be the restrictedmultiplicity; it also lies in the balanced cone of multiplicities on A U . All sumsaround three-cycles are even, and DV( m U ) = 8. Since 8 >
0, it follows fromTheorem 1.1 that ( A U , m U ) is not free, hence ( A , m ) is also not free.2. Notation and preliminaries
Let V = K ℓ be a vector space over a field K of characteristic zero. A centralhyperplane arrangement A = ∪ ni =1 H i is a union of hyperplanes H i ⊂ V passingthrough the origin in V . In other words, if we let { x , . . . , x ℓ } be a basis for thedual space V ∗ and S = Sym( V ∗ ) ∼ = K [ x , . . . , x l ], then H i = V ( α H i ) for somechoice of linear form α H i ∈ V ∗ , unique up to scaling. We will use the languageof graphic arrangements for referring to the braid arrangement A ℓ and its sub-arrangements. Namely, suppose G = ( V G , E G ) is a graph with vertices orderedas V G = { v , . . . , v ℓ } , and let S = K [ x , . . . , x ℓ ]. If { v i , v j } is an edge in E G then let H ij = V ( x i − x j ). The graphic arrangement associated to G is A G = ∪ { i,j }⊂ E ( G ) H ij . Clearly A G is a sub-arrangement of the full braid arrangement A ℓ , which may be identified with the graphic arrangement corresponding to thecomplete graph K ℓ +1 on ( ℓ + 1) vertices.A multi-arrangement is a pair ( A , m ) of a central arrangement A = ∪ ki =1 H i anda map m : { H , . . . , H k } → Z ≥ , called a multiplicity. If m ≡
1, then ( A , m ) isdenoted A and is called a simple arrangement. If A G is a graphic arrangement thenthe multi-arrangement ( A G , m ) is equivalent to the information of the edge-labeledgraph ( G, m ), where { v i , v j } is labeled by m ( H ij ) = m ij . We will frequently moveback and forth between these notations. We will always assume that a graph G comes with some ordering V G = { v , . . . , v ℓ } of its vertices and may refer to thevertices simply by their integer labels { , . . . , ℓ } .The module of derivations on S is defined by Der K ( S ) = L ℓi =1 S∂ x i , the free S -module with basis ∂ x i = ∂/∂x i for i = 1 , . . . , ℓ . The module Der K ( S ) acts on S by partial differentiation. Given a multi-arrangement ( A , m ), our main object ofstudy is the module D ( A , m ) of logarithmic derivations of ( A , m ): D ( A , m ) := { θ ∈ Der K ( S ) : θ ( α H ) ∈ h α m ( H ) H i for all H ∈ A} , where h α m ( H ) H i ⊂ S is the ideal generated by α m ( H ) H . 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 isfree.The intersection lattice of A is the ranked poset L = L ( A ) consisting of all inter-sections of hyperplanes of A ordered with respect to reverse inclusion (the vectorspace V is included as the ‘empty’ intersection). We denote by L k the intersectionsof rank k , where the rank of an intersection is its codimension. If X ∈ L k , A X de-notes the sub-arrangement consisting of hyperplanes which contain X , L X denotesthe lattice of A X , and m X denotes the multiplicity function restricted to hyper-planes containing X . If A G is a graphic arrangement with lattice L and H ⊂ G NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 5 is a connected induced sub-graph of G on ( k + 1) vertices, then H correspondsto an intersection X ( H ) ∈ L k , and the graphic arrangement A H is the same as( A G ) X ( H ) . In this setting, if m is a multiplicity on A G , we denote by m H therestriction of m to the sub-arrangement A H . Proposition 2.1. [4, Proposition 1.7] If ( A , m ) is a free multi-arrangement, thenso is ( A X , m X ) . If D ( A , m ) is free then it has ℓ minimal generators as an S -module whosedegrees 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 ( d , . . . , d ℓ ). Put | m | = P H ∈ L m ( H ). Then P ℓi =1 d i = | m | (this follows for instance by an exten-sion of Saito’s criterion to multi-arrangements [17]). For a free multi-arrangement,define the kth global mixed product byGMP( k ) = X d i d i · · · d i k , where the sum runs across all k -tuples satisfying 1 ≤ i < · · · < i k ≤ ℓ . Further-more, define the k th local mixed product byLMP( k ) = X X ∈ L k d X d X · · · d Xk , where d X , . . . , d kX are the (non-zero) exponents of the rank k sub-arrangement A X .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 ) forevery ≤ k ≤ ℓ . Deviations and mixed products in the balanced cone
In this section we study the local and global mixed products of multiplicitiesin the balanced cone. In particular, we prove the implication (1) = ⇒ (2) ofTheorem 1.1. Recall the balanced cone of multiplicities on a braid arrangement A ℓ is the set of multiplicities satisfying the three inequalities m ij + m jk + 1 ≥ m ik , m ij + m ik +1 ≥ m jk , and m ik + m jk +1 ≥ m ij 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- A arrangement are as balanced aspossible. Proposition 3.1. [15]
Suppose H is a three-cycle on the vertices i, j, k of theedge-labeled complete graph ( K ℓ +1 , m ) . Put m ijk = m ij + m ik + m jk . If m is inthe balanced cone of multiplicities then the (non-zero) exponents of ( A H , m H ) are ( ⌊ m ijk / ⌋ , ⌈ m ijk / ⌉ ) . If { i, j, k } are vertices of K ℓ +1 so that m ij + m ik + m jk is odd then we will call { i, j, k } an odd three-cycle . Proposition 3.2.
Let ( A ℓ , m ) be a multi-braid arrangement so that m is in thebalanced cone of multiplicities. Set | m | = P ij m ij and m ijk = m ij + m jk + m ik . If q is the number of odd three cycles of m , thenLMP (2) = X ≤ i We prove the formula for LMP(2) first. If X ∈ L , then either (1) : X = H ij ∩ H st for a pair of non-adjacent edges { i, j } and { s, t } or (2) : X = H ij ∩ H jk ∩ H ik corresponds to a triangle. In the first case the arrangement is boolean with (non-zero) exponents ( m ij , m st ), contributing m ij m st to LMP(2). In the second case thearrangement is an A braid arrangement with exponents ( m ijk / , m ijk / 2) if m ijk is even and (( m ijk − / , ( m ijk + 1) / 2) if m ijk is odd from Proposition 3.1. Theformer contributes m ijk / m ijk / − / P X ∈ L d X d X .Now consider the inequality for GMP(2). Supposing ( A ℓ , m ) is free, let ( d , . . . , d ℓ )be its (non-zero) exponents, ordered so that d ≥ · · · ≥ d ℓ . By an extension ofSaito’s criterion to multi-arrangements [17], P ℓi =1 d i = | m | . Now, following re-marks just after [3, Corollary 4.6], we say that ( b , . . . , b ℓ ) with b ≥ · · · ≥ b ℓ and P b i = | m | is ‘more balanced’ than ( d , . . . , d ℓ ) if P ( b i +1 − b i ) ≤ P ( d i +1 − d i ).Then X b i · · · b i k ≥ X d i · · · d i k = GMP( k ) . Let | m | = kℓ + p be the result of dividing | m | by ℓ , so k is a positive integer and0 ≤ p < ℓ . The ‘most balanced’ distribution of exponents occurs when b = · · · = b p = k + 1 and b p +1 = . . . = b ℓ = k, so P ( b i +1 − b i ) = b p − − b p is zero if p = 0 and one if p > 0. Some algebra yieldsthat, for this choice of exponents, X ≤ i Fix a multiplicity m on the braid arrangement A ℓ and a subset U ⊂ { v , . . . , v ℓ } . Denote by C ( K ℓ +1 ) the set of four-cycles in K ℓ +1 and by C ( U )the set of four-cycles in K ℓ +1 whose vertices are contained in U . If C ∈ C ( K ℓ +1 )traverses the vertices i, j, s, t in order, set m ( C ) = | m ij − m js + m st − m it | . The deviation of m over U is the sum of squaresDV( m U ) = X C ∈ C ( U ) m ( C ) . If U consists of all vertices of K ℓ +1 , then we write DV( m ) instead of DV( m U ). Theorem 3.4. Suppose ( A ℓ , m ) is a multi-braid arrangement, m is in the balancedcone of multplicities, and q is the number of odd three cycles of m . If DV ( m ) > qℓ ,then m is not free.Remark . The inequality DV( m ) > qℓ in Theorem 3.4 can be strengthened toDV( m ) > qℓ − p ( ℓ − p ), where p is the remainder of | m | on division by ℓ . We willsee that the simpler inequality DV( m ) > qℓ suffices to detect non-freeness. NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 7 Proof. By Proposition 3.2, we know thatLMP(2) − GMP(2) ≥ X ≤ i 2) four-cycles,(2) DV( m ) = 2 (cid:18) ℓ − (cid:19) X ≤ i In this section we prove the first part of the implication (2) = ⇒ (3) in The-orem 1.1. Namely, we prove that for a multiplicity m in the central cone, theinequalities DV( m ) ≤ q U ( | U | − 1) on deviations are enough to guarantee that m isan ANN multiplicity. In fact, we show that it is enough to have these inequalitieson subsets of size four.Recall from the introduction that we call m an ANN multiplicity on A ℓ if m is inthe balanced cone of multiplicities and there exist non-negative integers n , . . . , n ℓ and ǫ ij ∈ {− , , } so that m ij = n i + n j + ǫ ij for 0 ≤ i < j ≤ ℓ . Lemma 4.1. Suppose m is a multiplicity on A and DV ( m ) ≤ q . Then m ( C ) ≤ for each four-cycle C in K .Proof. There are three four-cycles. Set T = m − m + m − m T = m − m + m − m T = m − m + m − m . Notice T + T = T , and DV( m ) = T + T + T . Now, suppose without loss that | T | ≥ 3. Then either | T | ≥ | T | ≥ 2. But then P ( m ) ≥ 13, contradicting thatDV( m ) ≤ q ≤ 12 (since q ≤ (cid:3) Proposition 4.2. Let ( A ℓ , m ) be a multi-braid arrangement so that m is in thebalanced cone of multiplicities and DV ( m U ) ≤ q U for every subset U ⊂ { v , . . . , v ℓ } with | U | = 4 . Then m is an ANN multiplicity.Proof. We need only show that there exist non-negative integers n i for i = 0 , . . . , ℓ and integers ǫ ij ∈ {− , , } for 0 ≤ i < j ≤ ℓ so that m ij = n i + n j + ǫ ij . ByLemma 4.1, we must have m ( C ) ≤ C ∈ C ( K ℓ +1 ). We usethis condition to provide an inductive algorithm producing the integers n , . . . , n ℓ .If ℓ = 2, set n = (cid:24) m + m − m (cid:25) , n = (cid:24) m + m − m (cid:25) , and n = (cid:24) m + m − m (cid:25) . Since m is in the balanced cone, n i ≥ i = 0 , , m ij = n i + n j + ǫ ij , where ǫ ij ∈ {− , } .Now assume ℓ > 2. We make an initial guess at what the non-negative integers n , . . . , n ℓ and ǫ ij should be, and then adjust as necessary. By induction on ℓ ,there exist non-negative integers ˜ n , . . . , ˜ n ℓ − and ˜ ǫ ij ∈ {− , , } such that m ij =˜ n i + ˜ n j + ˜ ǫ ij for 0 ≤ i < j ≤ ℓ − 1. Let ˜ n ℓ be a non-negative integer satisfying˜ n ℓ + ˜ n i ≥ m iℓ − ǫ iℓ = m iℓ − (˜ n i + ˜ n ℓ ) for every i < ℓ , so m iℓ = ˜ n i + ˜ n ℓ + ˜ ǫ iℓ .By the choice of ˜ n ℓ , we have ˜ ǫ iℓ ≤ i < ℓ .Now suppose there is an index 0 ≤ j < ℓ so that ˜ ǫ jℓ ≤ − 2. Our goal is todecrease either ˜ n ℓ or ˜ n j by one, thereby increasing ˜ ǫ jℓ , without disturbing any ofthe hypotheses made so far, namely˜ n i ≥ ≤ i ≤ ℓ, ˜ ǫ iℓ ≤ i < ℓ, ( ⋆ ) ˜ ǫ st ∈ {− , , } for all 0 ≤ s < t ≤ ℓ − . First we assume ˜ n ℓ > n ℓ by one. We can do this withoutdisturbing assumptions ( ⋆ ) provided there is no index s so that ǫ sℓ = 1. So, assume NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 9 that there is an index 0 ≤ s < ℓ so that ǫ sℓ = 1. We claim that in this situation, ǫ st ≥ t = s . Suppose to the contrary that there is an index t so that ǫ st = − C : ℓ → s → t → j → ℓ . Then m ( C ) = | ˜ ǫ sℓ − ˜ ǫ jℓ + ˜ ǫ jt − ˜ ǫ st |≥ ǫ jt + 1 ≥ , since ˜ ǫ jt ∈ {− , , } by the inductive hypothesis. This contradicts our assumptionthat m ( C ) ≤ 2. So it follows that ˜ ǫ st ∈ { , } for all t . Thus we may increase˜ n s by one, thereby decreasing ˜ ǫ st by one for every t = s , without disturbing thehypothesis that ˜ ǫ st ∈ {− , , } . Since we can apply this argument at every index s so that ˜ ǫ sℓ = 1, we may assume ˜ ǫ sℓ ≤ ≤ s < ℓ . Hence, if ˜ n ℓ > 0, itis now clear that we can decrease ˜ n ℓ by one without disturbing assumptions ( ⋆ ).Now assume that ˜ n ℓ = 0. Then, for any s < ℓ , m sℓ + m jℓ − m js = (˜ n s + ˜ ǫ sℓ ) + (˜ n j + ˜ ǫ jℓ ) − (˜ n j + ˜ n s + ˜ ǫ js )= ˜ ǫ sℓ + ˜ ǫ jℓ − ˜ ǫ js ≤ − − ˜ ǫ js ≤ − , since ˜ ǫ js ∈ {− , , } by the inductive hypothesis. Since m is in the balanced cone,we must have equality for all of these, so ǫ js = − s = j , s < ℓ . If˜ n j = 0 as well, then m jℓ = ˜ n j + ˜ n ℓ + ǫ jℓ ≤ − 2, contradicting that m jℓ is non-negative. Hence ˜ n j > n j by one without disturbing any ofassumptions ( ⋆ ).In either case, we have shown how to increase ˜ ǫ jℓ if ˜ ǫ jℓ ≤ − ⋆ ). So we iterate the above arguments until ˜ ǫ jℓ ≥ − j < ℓ ,then set n i = ˜ n i for 0 ≤ i ≤ ℓ and ˜ ǫ ij = ǫ ij for 0 ≤ i < j ≤ ℓ . This completes thealgorithm and the proof. (cid:3) With Proposition 4.2, we now prove (1) ⇐⇒ (3) in Theorem 1.1. Most of theheavy lifting is done by Abe-Nuida-Numata in [2]. Corollary 4.3. Suppose m is in the balanced cone of multiplicities on the A ℓ braidarrangement. Then ( A ℓ , m ) is free if and only if m is an ANN multiplicity andthe signed graph with E + G = {{ v i , v j } : ǫ ij = − } and E − G = {{ v i , v j } : ǫ ij = 1 } issigned-eliminable.Proof. If ( A ℓ , m ) is free and m is in the balanced cone, then DV( m U ) ≤ q U forevery subset U ⊂ { v , . . . , v ℓ } of size four by Corollary 3.6. By Proposition 4.2, m isan ANN multiplicity. By [2, Theorem 0.3], the signed graph with E + G = {{ v i , v j } : ǫ ij = − } and E − G = {{ v i , v j : ǫ ij = 1 } is signed-eliminable. For the converse, if m is an ANN multiplicity associated to a signed-eliminable graph, then ( A ℓ , m ) isfree by [2, Theorem 0.3]. (cid:3) Remark . In the result [2, Theorem 0.3], Abe-Nuida-Numata do not have thecondition that m is in the balanced cone. However, this turns out to be a necessarycondition for their arguments [1]. Furthermore their arguments, using addition-deletion techniques for multi-arrangements from [4], work for any ANN multiplicityas we have defined it [1]. · · · v v v v ℓ − v ℓ − v ℓ v · · · v v v v ℓ − v ℓ − v ℓ v v 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 inProposition 4.2 that if the inequalities of Theorem 1.1.(2) are satisfied then m isan ANN multiplicity. Now we show that these inequalities also detect when theassociated signed graph is not signed-eliminable. We follow the presentation ofsigned-eliminable graphs from [11, 2].Let G be a signed graph on ℓ +1 vertices. That is, each edge of G is assigned eithera + or a − , and so the edge set E G decomposes as a disjoint union E G = E + G ∪ E − G .Define m G ( ij ) = { i, j } ∈ E + G − { i, j } ∈ E − G . The graph G is signed-eliminable with signed-elimination ordering ν : V ( G ) →{ , . . . , ℓ } if ν is bijective and, for every three vertices v i , v j , v k ∈ V ( G ) with ν ( v i ) , ν ( v j ) < ν ( v k ), the induced sub-graph G | v i ,v j ,v k satisfies the following con-ditions. • For σ ∈ { + , −} , if { v i , v k } and { v j , v k } are edges in E σG then { v i , v j } ∈ E σG • For σ ∈ { + , −} , if { v k , v i } ∈ E σG and { v i , v j } ∈ E − σG then { v k , v j } ∈ E G These two conditions generalize the notion of a graph possessing an eliminationordering, which is equivalent to the graph being chordal. A graph is chordal if andonly 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 whichwe now turn. Definition 5.1. (1) A graph with ( ℓ + 1) vertices v , v , . . . , v ℓ with ℓ ≥ σ -mountain, where σ ∈ { + , −} , if { v , v i } ∈ E σG for i = 2 , . . . , ℓ − { v i , v i +1 } ∈ E − σG for i = 1 , . . . , ℓ − 1, and no other pair of vertices is joinedby an edge. (See Figure 1 - edges of sign σ are denoted by a single edgeand edges of sign − σ are denoted by a doubled edge.)(2) A graph with ( ℓ + 1) vertices v , v , v , . . . , v ℓ with ℓ ≥ σ -hill, where σ ∈ { + , −} , if { v , v } ∈ E σG , { v , v i } ∈ E σG for i = 2 , . . . , ℓ − { v , v i } ∈ E σG for i = 3 , . . . , ℓ , { v i , v i +1 } ∈ E − σG for i = 2 , . . . , ℓ − 1, and no other pair ofvertices is connected by an edge. (See Figure 1.)(3) A graph with ( ℓ + 1) vertices v , . . . , v ℓ with ℓ ≥ σ -cycle if { v i , v i +1 } ∈ E σG for i = 0 , . . . , ℓ − { v , v ℓ } ∈ E σG , and no other pair of vertices isconnected by an edge. Theorem 5.2. [11, Theorem 5.1] Let G be a signed graph. Then G is signed-eliminable if and only if the following three conditions are satisfied. ( C Both G + and G − are chordal. NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 11 Table 1. Graphs on four vertices which are not signed-eliminable( C Every induced sub-graph on four vertices is signed eliminable. ( C 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 or-dering) in [2, Example 2.1], along with those which are not signed-eliminable. Foruse in the proof of Corollary 5.3, we also list those graphs which are not signed-eliminable in Table 1. The property of being signed-eliminable is preserved underinterchanging + and − . Consequently, we list these graphs in Table 1 up to auto-morphism with the convention that a single edge takes one of the signs + , − , whilea double edge takes the other sign. Corollary 5.3. Let G be a signed graph. Then G is signed-eliminable if and onlyif the following three conditions are satisfied. ( C ′ ) No induced sub-graph of G is a σ -cycle of length > . ( C Every induced sub-graph on four vertices is signed eliminable. ( C No induced sub-graph of G is a σ -mountain or a σ -hill.Proof. Clearly ( C 1) from Theorem 5.2 implies ( C ′ ). We show that ( C ′ ) and ( C C E σG , σ ∈ {− , + } , is a cycle oflength ℓ + 1 > V G = { v , . . . , v ℓ } where { v i , v i +1 } ∈ E σG for i = 0 , . . . , ℓ − { v , v ℓ } ∈ E σG . If E − σG = ∅ then G is a σ -cycle which is forbidden by ( C ′ ),so we assume E − σG = ∅ . Let m be the maximal integer so that there is a sequenceof consecutive vertices v i , v i +1 . . . , v i + m − so that the induced sub-graph on theseconsecutive vertices consists only of edges in E σG . Since E − σG = ∅ , m < ℓ + 1.Relabel the vertices so that v , . . . , v m − are the vertices of a maximal inducedsub-graph with edges only in E σG . If m = 2 or m = 3, then the induced sub-graphon v , v , v , v consists of the three σ edges { v , v } , { v , v } , { v , v } along withat least one − σ edge. No such graph is signed eliminable (see Table 1). So m ≥ H on v , v , . . . , v m − , v m . By definition of m , H has exactly one − σ edge, namely { v , v m } . But then the induced sub-graph on v , v , v m − , v m consists of the two σ edges { v , v } , { v m − , v m } and the − σ edge { v , v m } , which is not signed-eliminable. It follows that E σG cannot have a cycle oflength > 3, so E σG is chordal. (cid:3) The following proposition proves the implication (2) = ⇒ (3) in Theorem 1.1,thereby completing the proof of Theorem 1.1. Proposition 5.4. Suppose n , . . . , n ℓ are non-negative integers, G is a signed graphon v , . . . , v ℓ , and let m be the multiplicity on A ℓ given by m ij = n i + n j + m G ( ij ) . If G is not signed-eliminable, then there is a subset U ⊂ { , . . . , ℓ } so that DV ( m U ) >q U · ( | U | − . Proof. Notice that, for a four-cycle traversing i, j, s, t in order, m ( C ) = | m ij − m js + m st − m it | = | m G ( ij ) − m G ( js ) + m G ( st ) − m G ( it ) | . Furthermore, for a three-cycle { i, j, k } , m ij + m ik + m jk = 2( n i + n j + n k ) + m G ( ij ) + m G ( ik ) + m G ( jk ) . It follows that the values of DV( m ) = P m ( C ) and qℓ = ( · ℓ from Theorem 3.4 may be determined after replacing m ij by m G ( ij ), which takesvalues only in {− , , } . Hereafter we write DV( G ) for DV( m ) and q G for q toemphasize their dependence only on the signed graph G . If U ⊂ { v , . . . , v ℓ } , welet DV( G U ) represent DV( m U ) to emphasize dependence only on G and the subset U . As usualy, q U 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 inducedsub-graph H which is • a signed graph on four vertices which is not signed-eliminable, • a σ -cycle of length > • a σ -hill, • or a σ -mountain.We assume G = H and show that DV ( G ) > q G ℓ in each of these cases, where ℓ is one less than the number of vertices of G . The inequality DV ( G ) > q G caneasily be verified by hand for each of the twelve graphs on four vertices which arenot 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 q G are given by the formulas: DV( G ) = ℓ − ℓ − ℓ + 2(3) q G = ℓ − ℓ − . (4)Given these formulas, note that DV( G ) = qℓ + 2( ℓ + 1) > qℓ , thus proving theresult. We prove Equations (3) and (4) for the σ -cycle directly, relying on the twoadditional formulas: DV( G ) = X U ⊂ V G , | U | =4 DV( G U )(5) q G = ( X U ⊂ V G , | U | =4 q U ) / ( ℓ − . (6)Equation (5) follows since each four-cycle is contained in a unique induced sub-graph 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 identifyall possible types of induced sub-graphs of the σ -cycle on four vertices, how manyof each type there are, and compute DV( G U ) and q U for each of these. Thenwe use Equation (5) to compute DV( G ) and Equation (6) to compute q G . Thelist 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 inthe second column, while the third and fourth columns record q U and DV( G U ),respectively, for each type of sub-graph. The final row records the total numberof sub-graphs on four vertices, the number of odd three-cycles, and the deviationof m , DV( m ) = P C ∈ C ( K ℓ +1 ) m ( C ) . We find that DV( m ) = ℓ − ℓ − ℓ + 2and q = ℓ − ℓ − 3, proving Equations (3) and (4) for the σ -cycle. The same NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 13 σ -cycle of length ( ℓ + 1)Type of sub-graph Count q U DV( G U ) (cid:18) ℓ − (cid:19) + (cid:18) ℓ − (cid:19) ℓ + 1) (cid:18) ℓ − (cid:19) ℓ + 1)( ℓ − ℓ + 1)( ℓ − 4) 2 2 ℓ + 1 2 6Total (cid:18) ℓ + 14 (cid:19) q = ℓ − ℓ − ℓ − ℓ − ℓ + 2 Table 2. Computing DV( G ) where G is a σ -cyclecomputations can be done to prove Equations (3) and (4) for the σ -hill and σ -mountain; for the convenience of the reader we collect these in Appendix A. (cid:3) Free vertices and a conjecture In this final section we discuss free vertices of a multiplicity on a graphic ar-rangement and present a conjecture on the structure of free multiplicities on braidarrangements. Definition 6.1. Suppose G is a graph. A vertex v i ∈ V G is a simplicial vertex if the sub-graph of G induced by v i and its neighbors is a complete graph. Givena multi-arrangement ( A G , m ) and the corresponding edge-labeled graph ( G, m ), avertex v i is a free vertex of ( G, m ) if it is a simplicial vertex and for every trianglewith vertices v i , v j , v k we have m ij + m ik ≤ m jk + 1. Theorem 6.2. Suppose G is a graph, v i is a free vertex of ( G, m ) , and G ′ is theinduced sub-graph on the vertex set V G \ { v i } . Then ( A G , m ) is free if and only if ( A G ′ , m G ′ ) is free. Proof of Theorem 6.2. We use a result whose proof we omit since it is virtuallyidentical 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 = K ℓ . Theorem 6.3. Suppose ( A , m ) is a central multi-arrangement of rank ℓ ≥ and X is a modular flat of rank ℓ − . Suppose ( A X , m X ) is free with exponents ( d , . . . , d ℓ − , and for all H ∈ A \ A X and H ′ ∈ A X , set Y := H ∩ H ′ . Ifone of the following two conditions is satisfied: (1) A Y = H ∪ H ′ or (2) m ( H ′ ) ≥ P H ∈A\A ′ m ( H ) − .Then ( A , m ) is free with exponents ( d , . . . , d ℓ − , | m | − | m ′ | ) . Now suppose G is a graph on ℓ + 1 vertices { v , . . . , v ℓ } and A G is the asso-ciated graphic arrangement. Further suppose that v i is a free vertex of ( G, m ),and G ′ is the induced sub-graph on the vertex set V G \ { v i } . Set m ′ = m | G ′ . ByProposition 2.1, if ( A G ′ , m ′ ) is not free, then neither is ( A G , m ).Suppose now that ( A G ′ , m ′ ) is free. We show that ( A G , m ) is free using The-orem 6.3. Write H ij = V ( x i − x j ). Since v i is a simplicial vertex of G , the flat X = ∩ v j ,v k = v i H jk is modular and has rank ℓ − 1. The sub-arrangement ( A G ) X isthe graphic arrangement A G ′ . Suppose H = H ij ∈ A G \ A G ′ , H ′ = H st ∈ A G ′ , andset Y = H ij ∩ H st . If { s, t } ∩ { i, j } = ∅ , then A Y = H ij ∪ H st . Otherwise, suppose s = j . Since v i is a simplicial vertex, { i, t } ∈ E G , so A Y = H ij ∪ H it ∪ H jt . Since v i is a free vertex, m ij + m it ≤ m jt + 1, which is condition (2) from Theorem 6.3.Hence ( A G , m ) is free by Theorem 6.3. (cid:3) Remark . 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 v , . . . , v ℓ of the vertices V G so that v i is a simplicial vertex of the induced sub-graph on v , . . . , v i for every i = 1 , . . . , ℓ . It is known that V G admits an elimination ordering if and only if G ischordal [9]. Corollary 6.5. Suppose ( G, m ) is an edge-labeled chordal graph with eliminationordering v , . . . , v ℓ satisfying that v i is a free vertex of the induced sub-graph on { v , . . . , v i } for every i ≥ . Then ( A G , m ) is free. Corollary 6.6. Let ( A ℓ , m ) be a multi-braid arrangement corresponding to thecomplete graph K ℓ +1 on ( ℓ + 1) vertices. Suppose that K ℓ +1 admits an ordering { v , . . . , v ℓ } so that: (1) For some integer ≤ k ≤ ℓ , the induced sub-graph G ′ on { v , . . . , v k } satisfies that m G ′ is a free ANN multiplicity. (2) For k + 1 ≤ i ≤ ℓ , v i is a free vertex of the induced graph on { v , . . . , v i } .Then ( A ℓ , m ) is free. We conjecture that all free multi-braid arrangements take the form of Corol-lary 6.6. Conjecture 6.7. The multi-braid arrangement ( A ℓ , m ) is free if and only if it isone of the multi-braid arrangements constructed in Corollary 6.6. Equivalently, by NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 15 Theorem 6.2, if ( A ℓ , m ) is free then either m is a free ANN multiplicity or m hasa 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 . Conjecture 6.7 is proved for the A braid arrangement in [8]. UsingMacaulay2 [10], we have verified Conjecture 6.7 for many multiplicities on the A arrangement. 7. Acknowledgements I would especially like to thank Jeff Mermin, Chris Francisco, and Jay Schweigfor their collaboration on the analysis of free multiplicities on the A braid ar-rangement. The current work would not be possible without their help. I am verygrateful 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 multi-plicities 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 multiar-rangement. Adv. Math. , 215(2):825–838, 2007.[4] Takuro Abe, Hiroaki Terao, and Max Wakefield. The Euler multiplicity and addition-deletiontheorems for multiarrangements. J. Lond. Math. Soc. (2) , 77(2):335–348, 2008.[5] Takuro Abe and Masahiko Yoshinaga. Coxeter multiarrangements with quasi-constant mul-tiplicities. J. Algebra , 322(8):2839–2847, 2009.[6] Christos A. Athanasiadis. Deformations of Coxeter hyperplane arrangements and their char-acteristic 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 Multiplicitieson the A Arrangement. ArXiv e-prints , September 2016.[9] G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 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] G¨unter 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. Appendix A. Computations for mountains and hills σ -mountain on ( ℓ + 1) verticesType of subgraph Count q U DV( G U ) (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) ℓ − (cid:18) ℓ − (cid:19) ℓ − 3) 2 6 ℓ − (cid:18) ℓ − (cid:19) ℓ − 4) 4 82( ℓ − 4) 2 2 NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 17 (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) ℓ − (cid:18) ℓ + 14 (cid:19) q G = ℓ − ℓ − ℓ − ℓ − ℓ + 2Table 3: Computing DV( G ) where G is a σ -mountain σ -hill on ( ℓ + 1) verticesType of subgraph Count q U DV( G U ) (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) ℓ − 11 + (cid:18) ℓ − (cid:19) ℓ − (cid:18) ℓ − (cid:19) ℓ − 4) 4 82( ℓ − 4) 2 22 2 62 (cid:18) ℓ − (cid:19) (cid:18) ℓ − (cid:19) ℓ − 4) 2 21 2 62( ℓ − 4) 2 22 2 6 (cid:18) ℓ − (cid:19) NEQUALITIES FOR FREE MULTI-BRAID ARRANGEMENTS 19 ℓ − (cid:18) ℓ + 14 (cid:19) q G = ℓ − ℓ − ℓ − ℓ − ℓ + 2Table 4: Computing DV( G ) where G is a σ -hill Michael DiPasquale, Department of Mathematics, Oklahoma State University, Still-water, OK 74078-1058, USA E-mail address : [email protected] URL ::