THE NON-VANISHING COHOMOLOGY OF ORLIK-SOLOMON ALGEBRAS

YUKIHITO KAWAHARA

Abstract. The cohomology on the complement of hyperplanes with the coefficients in the rank one local system associated to a generic weight vanishes except in the highest dimension. In this paper, we construct matroids or arrangements and its weights with non-vanishing cohomology of Orlik-Solomon algebras, using decomposable relations arising from Latin hypercubes.

1. Introduction

Let R be a commutative ring with 1. Write [n] := {1, 2, …, n}. Let E = E_{R} denote the graded exterior algebra over R generated by 1 and degree-one elements e_{i} for i ∈ [n]. Define a R-linear map ∂ : E^{p} → E^{p−1} by ∂1 = 0, ∂e_{i} = 1 for i ∈ [n], and

∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{p}}) = ∑_{k=1}^{p}(−1)^{k−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{k}} ∧ ⋯ ∧ e_{i_{p}}

for p ≥ 2 and i_{j} ∈ [n]. Let M be a loopless matroid on [n] with rank ℓ + 1.

Definition 1.1. The Orlik-Solomon algebra of M is the quotient A(M) of E by the ideal ⟨∂M⟩ generated by ∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{s}}) for every circuit c = (i_{1}, …, i_{s}) of M.

If 1 and 2 are parallel, that is, {1, 2} is a circuit, then e_{1} = e_{2}. So the Orlik-Solomon algebra of the simple matroid associated with M is equal to that of M. The Orlik-Solomon algebra A(M) has the natural grading. The linear map ∂ on E induces the linear map ∂_{M} on A(M). Let e_{λ} = λ_{1}e_{1} + ⋯ + λ_{n}e_{n} ∈ E^{1}. We call λ = (λ_{1}, …, λ_{n}) a weight of M. The left multiplication e_{λ}∧ : A^{p}(M) → A^{p+1}(M) induces the complex (A(M), e_{λ}). Let H(A(M), e_{λ}) denote the cohomology of this complex. If λ = 0 then H(A(M), e_{λ}) is just A(M), otherwise we have H^{0}(A(M), e_{λ}) = 0. If ∑_{j=1}^{n} λ ≠ 0 then we have H^{p}(A(M), e_{λ}) = 0 for all p (see [15]). If ∂e_{λ} = ∑_{j=1}^{n} λ_{j} = 0 then e_{λ} induces the complex (∂_{M}(A(M)), e_{λ}) and the cohomology H(∂_{M}(A(M)), e_{λ}), where ∂_{M}(A(M)) is the image of ∂_{M}. It is known that

H^{p+1}(A(M), e_{λ}) = H^{p+1}(∂_{M}(A(M)), e_{λ}) ⊕ H^{p}(∂_{M}(A(M)), e_{λ}).

For a generic weight λ, Yuzvinsky [15] showed the vanishing theorem:

H^{k}(∂_{M}(A(M)), e_{λ}) = 0 for k ≠ ℓ

and hence we have

H^{k}(A(M), e_{λ}) = 0 for k ≠ ℓ, ℓ + 1.

Date: February 1, 2005.

2000 Mathematics Subject Classification. Primary 52C35; Secondary 32S22, 14F99.

Key words and phrases. matroid, hyperplane arrangement, Orlik-Solomon algebra, local system, Latin square, Latin hypercube.

Research at MSRI is supported in part by NSF grant DMS-9810361.

Partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

An arrangement A of hyperplanes in ℙ^{ℓ} has the underlying matroid M(A) = M with rank ℓ + 1 as a combinatorial structure. The cohomology of the complement of A is isomorphic to ∂_{M}(A(M)) (see [10] and [7]). If a weight λ = (λ_{i})_{i∈A} satisfies some generic condition, then the cohomology of the complement of A with the coefficients in the rank one local system associated to λ is isomorphic to H(∂_{M}(A(M)), e_{λ}) (see [5, 14]). The local system cohomologys is an important subject in the multivariable theory of hypergeometric functions [2, 11]. By the vanishing theorem [15], for a generic weight λ, the local system cohomology vanishes in all but the top dimension. In this paper, our purpose is to construct matroids and arrangements with non-vanishing cohomology of Orlik-Solomon algebras, or rather H^{ℓ−1}(A(M), e_{λ}) ≠ 0.

In particular, the case of ℓ = 2 was studied in [6, 9]. Falk [6] defined the resonance variety, that is, the space of weights with non-vanishing cohomology. The resonance variety is deeply related to the cohomology support loci [1] and the characteristic variety [8, 3]. Libgober and Yuzvinsky [9] showed that, under some condition, weights with non-vanishing first cohomology are parametrized by Latin squares.

In this paper, we prove that, in general, matroids associated to Latin hypercubes have weights with non-vanishing cohomology, by using decomposable relations arising from Latin hypercubes. This decomposable relation is the generalization of the relation discovered by Rybnikov (see [6]). Moreover, in the case of ℓ = 2, we study well, using terms of Latin squares. In the last section, we shall give examples of realizations including the higher case. Some of them appear in the classical projective geometry (see Figure 1, 2 and 3).

We shall use the following notation and terminology. A k-set is a set with cardinality k. Denote the family of all k-subset of a set S by \binom{S}{k}. Often, we regard a p-tuple (i_{1}, …, i_{p}) as a p-set {i_{1}, …, i_{p}}. We refer to [12] for terminology of the matroid theory.

2. Non-vanishing Theorem

A Latin hypercube of dimension ℓ and order m is an m^{ℓ}-array such that, if ℓ − 1 coordinates are fixed, the m positions so determined contain a permutation of m symbols. Let K = [k(i_{1}, …, i_{ℓ})]_{1≤i_{1},...,i_{ℓ}≤m} be a Latin ℓ-dimensional hypercube on [m], that is, an m^{ℓ}-matrix satisfying the condition

{k(i′_{1}, i_{2}, …, i_{ℓ}) : i′_{1} ∈ [m]} = {k(i_{1}, i′_{2}, …, i_{ℓ}) : i′_{2} ∈ [m]} = ⋯
⋯ = {k(i_{1}, i_{2}, …, i′_{ℓ}) : i′_{ℓ} ∈ [m]} = [m],

for 1 ≤ i_{1}, …, i_{ℓ} ≤ m. Define the family of (ℓ + 1)-subsets in [n] associated to K by

C[K] = [(i_{1}, m + i_{2}, 2m + i_{3}, …, (ℓ − 1)m + i_{ℓ}, ℓm + k(i_{1}, …, i_{ℓ}))]_{1≤i_{1},...,i_{ℓ}≤m}.

On the other hand, a matroid is said to be ℓ-generic if it has no i-circuits for i ≤ ℓ. Note that an 1-generic matroid is just a loopless matroid and a 2-generic matroid is just a simple matroid. The uniform matroid U_{m,n} of rank m is m-generic. So we can mention the main theorem as follows.

Theorem 2.1. Let m ≥ 2, ℓ ≥ 2 and n = (ℓ+1)m. Let K be a Latin ℓ-dimensional hypercube on [m]. Then there exists a unique ℓ-generic matroid M[K] on [n] with rank ℓ + 1, for which the family of all (ℓ + 1)-circuits is equal to C[K]. This matroid has weights with non-vanishing cohomology, in fact,

H^{k}(A(M[K]), e_{λ}) = 0  for k ≤ ℓ − 2,
H^{ℓ−1}(A(M[K]), e_{λ}) ≠ 0,

for a non-zero weight

λ = (\underbrace{λ_{1}, …, λ_{1}}_{m}, \underbrace{λ_{2}, …, λ_{2}}_{m}, · · · · · · , \underbrace{λ_{ℓ+1}, …, λ_{ℓ+1}}_{m}); ∑_{j=1}^{ℓ+1} λ_{j} = 0.

In the rest of this section, we will prove this theorem. First of all, we prove some lemmas.

Lemma 2.2. A family C of (ℓ + 1)-subsets in [n] satisfies the condition (C_{ℓ+1}) if C_{1}, C_{2} ∈ C and |C_{1} ∪ C_{2}| = ℓ + 2 then every (ℓ + 1)-subset C_{3} of C_{1} ∪ C_{2}

is a member of C, if and only if, there exists an ℓ-generic matroid on [n] for which the family of all (ℓ + 1)-circuits is equal to C.

Proof. It is clear when n < ℓ + 1. Assume that n ≥ ℓ + 1. Let C be a family of (ℓ + 1)-subsets in [n] satisfying (C_{ℓ+1}). Let I be a ℓ-subset of [n]. Define X_{I} = I ∪ \{e ∈ [n] : I ∪ e ∈ C\}, \binom{X_{I}}{ℓ+1} = \{all (ℓ + 1)-subsets of X_{I}.\}, and \binom{X_{I}}{ℓ+1}_{s} =

\{S ∈ \binom{X_{I}}{ℓ+1} : |S \ I| = s\}. Note that \binom{X_{I}}{ℓ+1} = ∪_{s=1}^{ℓ+1}\binom{X_{I}}{ℓ+1}_{s}. First of all, we show that

\binom{X_{I}}{ℓ+1}_{s} is a subfamily of C by induction on s. For s = 1, since \binom{X_{I}}{ℓ+1}_{1} = \{I ∪ e ∈ C\}, it is clear. Let assume that \binom{X_{I}}{ℓ+1}_{s} ⊂ C for s ≥ 1. Take a member S of \binom{X_{I}}{ℓ+1}_{s+1}. Let T := S \ I and I′ := S ∩ I. Note that S = I′ ∪ T, I′ ⊂ I, T ⊂ X_{I} \ I, |I′| = ℓ − s and |T | = s + 1. Now we can choose e ∈ I \ I′ and f_{1}, f_{2} ∈ T with f_{1} ≠ f_{2}. By the inductive assumption, C_{1} := I′ ∪ e ∪ (T \ {f_{1}}) and C_{2} := I′ ∪ e ∪ (T \ {f_{2}}) are in \binom{X_{I}}{ℓ+1}_{s} ⊂ C. We can check C_{1} and C_{2} satisfy the condition in (C_{ℓ+1}), and S is a (ℓ + 1)-subset of C_{1} ∪ C_{2}. So we have S ∈ C. Therefore, we have \binom{X_{I}}{ℓ+1}_{s} ⊂ C and hence \binom{X_{I}}{ℓ+1} ⊂ C.

Assume that C is not the family of all (ℓ + 1)-subsets of [n]. We shall show that

I = {I ⊂ [n] : |I| ≤ ℓ + 1, I ∉ C}

is a matroid complex (see [12]). Note that I have all i-subsets of [n] for i < ℓ + 1. Since ∅ ∈ I and if I′ ⊂ I ∈ I then I′ ∈ I, we should prove the independence augmentation axiom for I, that is, for I_{1}, I_{2} ∈ I with |I_{2}| = |I_{1}| + 1, there exists e ∈ I_{2} \ I_{1} such that I_{1} ∪ {e} ∈ I. If |I_{1}| < ℓ, it is clear. Let |I_{1}| = ℓ. Suppose that I_{1} ∪ {e} ∉ I for all e ∈ I_{2} \ I_{1}. Then we have I_{2} ⊂ X_{I_{1}}. By the above claim, we have \binom{X_{I_{1}}}{ℓ+1} ⊂ C and hence we have I_{2} ∈ C, this is a contradiction. Therefore, I defines the matroid of rank ℓ + 1. The converse is easy by the circuit elimination axiom of the matroids (see [12, 1.1.4]). □

Remark 2.3. (1) When C = ∅, the uniform matroid U_{m,n} of rank m with m ≥

ℓ + 1 is one of matroids in the above lemma.

(2) If C consists of all (ℓ + 1)-subsets of [n], the uniform matroid U_{ℓ,n} of rank ℓ is only one ℓ-generic matroid in the above lemma. Otherwise, the rank of such a matroid is greater than ℓ, and there exists uniquely such an ℓ-generic matroid with rank ℓ + 1.

Lemma 2.4. Let n = (ℓ+1)m. Let a_{s} = e_{(s−1)m+1}+⋯+e_{sm} for 1 ≤ s ≤ ℓ+1. For a Latin ℓ-dimensional hypercube K on [m], we obtain the following decomposable relation

∂(a_{1} ∧ a_{2} ∧ ⋯ ∧ a_{ℓ+1}) = −(a_{1} − a_{2}) ∧ (∂(a_{2} ∧ ⋯ ∧ a_{ℓ+1}))

= (−1)^{ℓ}m(a_{1} − a_{2}) ∧ (a_{2} − a_{3}) ∧ ⋯ ∧ (a_{ℓ} − a_{ℓ+1})

= m ∑_{S∈C[K]} ∂(e_{S}),

where e_{S} = e_{i_{1}} ∧ ⋯ ∧ e_{i_{p}} for a p-tuple (i_{1}, …, i_{p}).

Proof. The first and second equations are obtained by

∂(a_{1} ∧ a_{2} ∧ ⋯ ∧ a_{ℓ+1}) = ∂((a_{1} − a_{2}) ∧ a_{2} ∧ ⋯ ∧ a_{ℓ+1})
= ∂(a_{1} − a_{2}) ∧ a_{2} ∧ ⋯ ∧ a_{ℓ+1} − (a_{1} − a_{2}) ∧ (∂(a_{2} ∧ ⋯ ∧ a_{ℓ+1}))
= −(a_{1} − a_{2}) ∧ (∂(a_{2} ∧ ⋯ ∧ a_{ℓ+1})) = ⋯
= (−1)^{ℓ}(a_{1} − a_{2}) ∧ (a_{2} − a_{3}) ∧ ⋯ ∧ (a_{ℓ} − a_{ℓ+1}) ∧ ∂(a_{ℓ+1})
= (−1)^{ℓ}m(a_{1} − a_{2}) ∧ (a_{2} − a_{3}) ∧ ⋯ ∧ (a_{ℓ} − a_{ℓ+1}).

Let E_{s} = {(s − 1)m + 1, (s − 1)m + 2, …, sm} for 1 ≤ s ≤ ℓ + 1. Note that E_{1} ∪⋯∪E_{ℓ+1} = [n]. We regard K as a Latin hypercube K̃ = (k̃(i_{1}, …, i_{ℓ})) with saxis indexed by E_{s} for 1 ≤ s ≤ ℓ and symbol set E_{ℓ+1}. We note that k̃(i_{1}, …, i_{ℓ}) = ℓm + k(i_{1}, …, i_{ℓ}) ∈ E_{ℓ+1}. Since ∂(e_{1} ∧ ⋯ ∧ e_{k} ∧ e_{k+1}) = ∂(e_{1} ∧ ⋯ ∧ e_{k}) ∧ e_{k+1} + (−1)^{k}e_{1} ∧ ⋯ ∧ e_{k}, we have

(−1)^{k}e_{1} ∧ e_{2} ∧ ⋯ ∧ e_{k} = −∂(e_{1} ∧ ⋯ ∧ e_{k}) ∧ e_{k+1} + ∂(e_{1} ∧ ⋯ ∧ e_{k} ∧ e_{k+1}).

Hence, we can get

(−1)^{ℓ}m · a_{1} ∧ ⋯ ∧ a_{ℓ} = m ∑_{i_{1}∈E_{1},...,i_{ℓ}∈E_{ℓ}}(−1)^{ℓ}e_{i_{1}} ∧ ⋯ ∧ e_{i_{ℓ}} = m×

∑_{i_{1}∈E_{1},...,i_{ℓ}∈E_{ℓ}} {−∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ e_{k̃(i_{1},...,i_{ℓ})} + ∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{ℓ}} ∧ e_{k̃(i_{1},...,i_{ℓ})})} .

The second term is

∑_{i_{1}∈E_{1},...,i_{ℓ}∈E_{ℓ}} ∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{ℓ}} ∧ e_{k̃(i_{1},...,i_{ℓ})}) = ∑_{S∈C[K]} ∂(e_{S}).

On the other hand, since K is a Latin hypercube, we have

∑_{i_{1}∈E_{1},...,i_{ℓ}∈E_{ℓ}} ∂(e_{i_{1}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ e_{k̃(i_{1},...,i_{ℓ})}

= ∑_{i_{1},...,i_{ℓ}} (∑_{p=1}^{ℓ}(−1)^{p−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{p}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ e_{k̃(i_{1},...,i_{ℓ})}

= ∑_{p=1}^{ℓ} ∑_{i_{1},...,i_{ℓ}} ((−1)^{p−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{p}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ e_{k̃(i_{1},...,i_{ℓ})}

= ∑_{p=1}^{ℓ} ∑_{i_{1},...,î_{p},...,i_{ℓ}} ((−1)^{p−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{p}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ ∑_{i_{p}} e_{k̃(i_{1},...,i_{ℓ})}

= ∑_{p=1}^{ℓ} ∑_{i_{1},...,î_{p},...,i_{ℓ}} ((−1)^{p−1}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{p}} ∧ ⋯ ∧ e_{i_{ℓ}}) ∧ a_{ℓ+1},

and

∂(a_{1} ∧ ⋯ ∧ a_{ℓ}) = ∂(a_{p}) ∑_{p=1}^{ℓ}(−1)^{p−1}a_{1} ∧ ⋯ ∧ â_{p} ∧ ⋯ ∧ a_{ℓ}

= m ∑_{p=1}^{ℓ}(−1)^{p−1} ∑_{i_{1},...,î_{p},...,i_{ℓ}} e_{i_{1}} ∧ ⋯ ∧ ê_{i_{p}} ∧ ⋯ ∧ e_{i_{ℓ}}.

Therefore we obtain

(−1)^{ℓ}m · a_{1} ∧ ⋯ ∧ a_{ℓ} = −∂(a_{1} ∧ ⋯ ∧ a_{ℓ}) ∧ a_{ℓ+1} + m ∑_{S∈C[K]} ∂(e_{S})

and hence we have

∂(a_{1} ∧ ⋯ ∧ a_{ℓ} ∧ a_{ℓ+1}) = ∂(a_{1} ∧ ⋯ ∧ a_{ℓ}) ∧ a_{ℓ+1} + (−1)^{ℓ}m · a_{1} ∧ ⋯ ∧ a_{ℓ}

= m ∑_{S∈C[K]} ∂(e_{S}).

□

Proof of Theorem 2.1. Let K be a Latin ℓ-dimensional hypercube on [m]. By the construction of C[K], for C_{1}, C_{2} ∈ C[K] with C_{1} ≠ C_{2}, we have |C_{1} ∩ C_{2}| = ℓ − 1 and |C_{1} ∪ C_{2}| = ℓ + 3. Hence, due to Lemma 2.2 and its remark, there exists a unique ℓ-generic matroid M[K] with rank ℓ+1. In general, for an ℓ-generic matroid M and a non-zero weight λ of M, we have H^{k}(M, e_{λ}) = 0 for k ≤ ℓ − 2. Thus, we shall prove H^{ℓ−1}(A(M[K]), e_{λ}) ≠ 0. Let λ be a weight given in the statement, and assume without loss of generality that λ_{1} ≠ 0. Since ∑_{j=1}^{ℓ+1} λ_{j} = 0, we have

e_{λ} = λ_{1}(e_{1} + ⋯ + e_{m}) + ⋯ + λ_{ℓ+1}(e_{ℓm+1} + ⋯ + e_{(ℓ+1)m})
= λ_{1}a_{1} + λ_{2}a_{2} + ⋯ + λ_{ℓ+1}a_{ℓ+1}
= λ_{1}(a_{1} − a_{2}) + (λ_{1} + λ_{2})(a_{2} − a_{3}) + ⋯ + (λ_{1} + ⋯ + λ_{ℓ})(a_{ℓ} − a_{ℓ+1}),

where a_{j} is defined in Lemma 2.4. Define a (ℓ − 1)-form

b := ∂(a_{2} ∧ a_{3} ∧ ⋯ ∧ a_{ℓ+1})
= (−1)^{ℓ−1}m(a_{2} − a_{3}) ∧ (a_{3} − a_{4}) ∧ ⋯ ∧ (a_{ℓ} − a_{ℓ+1}).

By Lemma 2.4, we have

e_{λ} ∧ b = λ_{1}(a_{1} − a_{2}) ∧ b ∈ ⟨∂M[K]⟩,

that is, e_{λ} ∧ b vanishes in the Orlik-Solomon algebra A(M[K]). Since M[K] is ℓ-generic, the (ℓ − 1)-form b is not in ⟨∂M[K]⟩. Finally, we shall check that b is a non-vanishing cohomology class in H^{ℓ−1}(M[K], e_{λ}).

For a finite set {e_{1}, …, e_{n}}, denote E(e_{1}, …, e_{n}) the graded exterior algebra over R generated by 1 and degree-one elements e_{1}, …, e_{n}. Note that E(e_{2}, …, e_{n}) is a subalgebra of E(e_{1}, …, e_{n}). Let e_{λ} = λ_{1}e_{1} +⋯+λ_{n}e_{n} with λ_{i} ∈ R and λ_{1} ≠ 0. Then we have E(e_{1}, …, e_{n}) = E(e_{λ}, e_{2}, …, e_{n}). It is easy to see the following: if ω ∈ E(e_{2}, …, e_{n}) with ω ≠ 0, then ω is not belong to the ideal of E(e_{1}, …, e_{n}) generated by e_{λ}.

By the above, since b is in E(e_{m+1}, …, e_{n}) and λ_{1} ≠ 0, b is not in the ideal of E(e_{1}, …, e_{n}) generated by e_{λ}, that is, there exists no (ℓ−2)-form η with e_{λ} ∧η = b. This completes the proof. □

3. The case of ℓ = 2

We refer to [4] for the Latin squares. A Latin square of order m is a Latin hypercube of dimension 2 and order m, that is, an m × m matrix with entries in an m-set (we call the symbol set.) such that each element occurs exactly once in each row and exactly once in each column. The two Latin squares K and K′ are isotopic if K′ is obtained by permutations of rows, permutations of columns, and a bijection from the symbol set of K. Let E_{1}, E_{2} and E_{3} be three m-sets and let K be a Latin square with rows indexed by E_{1}, columns by E_{2}, and symbols by E_{3}. Define T (K) = {{x_{1}, x_{2}, x_{3}} : x_{i} ∈ E_{i}(i = 1, 2, 3), k_{x_{1},x_{2}} = x_{3}}. For any permutation σ of {1, 2, 3}, the σ-conjugate of L is the Latin square K_{σ} with rows indexed by E_{σ1}, columns by E_{σ2}, and symbols by E_{σ3}, defined by T (K) = T (K_{σ}).

The two Latin squares K and K′ are main class isotopic if K′ is isotopic to any conjugate of K.

Let K = (k_{i,j}) be a a Latin square on [m], that is, an m × m-matrix satisfying the condition {k_{i,1}, k_{i,2}, …, k_{i,m}} = {k_{1,j}, k_{2,j}, …, k_{m,j}} = [m] for 1 ≤ i, j ≤ m. In the previous section, we define C[K] by the family

[\begin{matrix}(1, m + 1, 2m + k_{1,1}) & (1, m + 2, 2m + k_{1,2}) & ⋯ & (1, 2m, 2m + k_{1,m}) \\ (2, m + 1, 2m + k_{2,1}) & (2, m + 2, 2m + k_{2,2}) & ⋯ & (1, 2m, 2m + k_{2,m}) \\ ⋮ & & & ⋮ \\ (m, m + 1, 2m + k_{m,1}) & (m, m + 2, 2m + k_{m,2}) & ⋯ & (1, 2m, 2m + k_{m,m}) \end{matrix}] .

We can see K as a Latin square K̃ with rows indexed by {1, 2, …, m}, columns by {m + 1, m + 2, …, 2m}, and symbols by {2m + 1, 2m + 2, …, 3m}. So we can consider C[K] = T (K̃). By Theorem 2.1, there exists a unique simple matroid M[K] on [n] with rank 3, for which the family of all 3-circuits is equal to C[K]. The simple matroid M[K] has weights with non-vanishing first cohomology.

Proposition 3.1. Let m ≥ 2. If K_{1} and K_{2} are main class isotopic Latin squares then matroids M[K_{1}] and M[K_{2}] are isomorphic. If a Latin square K_{1} is not main class isotopic to K_{2} then matroids M[K_{1}] is not isomorphic to M[K_{2}].

Proof. It is clear by the definition of main class isotopic Latin squares. □

Remark 3.2. The number of main class isotopic Latin squares of order m ≤ 8 is known (see [4]).

m =  1 2 3 4 5 6 7  8
main classes 1 1 1 2 2 12 147 283, 657

The two Latin squares K = (k_{i,j}) and K′ = (k′_{i,j}) of same order are orthogonal if all pairs (k_{i,j}, k′_{i,j}) are distinct. A set of Latin squares of order m is mutually orthogonal if any two distinct squares are orthogonal.

Theorem 3.3. Let m ≥ 1, s ≥ 1 and n = (s + 2)m. Let K_{1}, …, K_{s} be mutually orthogonal Latin squares on [m]. Then there exists a simple matroid M[K_{1}, …, K_{s}] on [n] satisfying

dim H^{1}(A(M[K_{1}, …, K_{s}]), e_{λ}) = s

for a non-zero weight

λ = (\underbrace{λ_{1}, …, λ_{1}}_{m}, \underbrace{λ_{2}, …, λ_{2}}_{m}, · · · · · · , \underbrace{λ_{s+2}, …, λ_{s+2}}_{m}); ∑_{j=1}^{s+2} λ_{j} = 0.

Proof. By Lemma 2.2 in the case of ℓ = 2, a family C of 3-subsets in [n] satisfies the condition

(C_{3}) if {i, j, k} and {i, j, l} are members of C then {i, k, l} and {j, k, l} are members of C,

if and only if, there exists a simple matroid on [n] for which the family of all 3circuits is equal to C. Recall that the set of flats of a matroid is a geometric lattice. The closure of C ∈ C is the set ∪{C′ ∈ C : |C′ ∩ C| ≥ 2}, that is a flat of rank 2. A 2-subset contained in no C ∈ C is a flat of rank 2.

Construction of M[K_{1}, …, K_{s}]: Let K_{1}, …, K_{s} be mutually orthogonal Latin squares on [m]. A sift Latin square K̃_{p} = (k̃_{i,j}^{p}) associated to K_{p} = (k_{i,j}^{p}) is given by a Latin square with row indexed by {1, 2, …, m}, column by {m+1, m+2, …, 2m}, and symbols by {(p + 1)m + 1, (p + 1)m + 2, …, (p + 2)m}, given by k̃_{i,j}^{p} = (p + 1)m + k_{i,j}^{p} for 1 ≤ i ≤ m and m + 1 ≤ j ≤ 2m. We define

C[K_{1}, …, K_{s}] := T (K̃_{1}) ∪ ⋯ ∪ T (K̃_{s}),

X_{i,j} := {i, j, k̃_{i,j}^{1}, …, k̃_{i,j}^{s}} for 1 ≤ i ≤ m, m + 1 ≤ j ≤ 2m, and

C := C[K_{1}, …, K_{s}] ∪ ( ⋃_{1≤i≤m,m+1≤j≤2m} \binom{X_{i,j}}{3}) .

By mutually orthogonality, we have |C ∩ X_{i,j}| = 1 for any C ∈ C[K_{1}, …, K_{s}] not contained in X_{i,j}, and |X_{i,j} ∩ X_{k,l}| = 1 for (i, j) ≠ (k, l). This implies that C satisfies (C_{3}). If m ≥ 2 then we obtain a simple matroid M[K_{1}, …, K_{s}] on [n] with rank 3 such that C is the family of all 3-circuits. If m = 1 then C gives the uniform matroid U_{2,n}.

Non-vanishing: Let a_{i} = e_{(i−1)m}+e_{(i−1)m+1}+⋯+e_{(i−1)m} for i = 1, 2, …, s+2. By Lemma 2.4, we have

(a_{1} − a_{i}) ∧ (a_{2} − a_{i}) ∈ ⟨∂M[K_{1}, …, K_{s}]⟩

for 3 ≤ i ≤ s + 2. We take two one-forms

e_{λ^{t}} = λ_{1}^{t}a_{1} + λ_{2}^{t}a_{2} + ⋯ + λ_{s+2}^{t}a_{s+2}

with ∑_{j=1}^{s+2} λ_{j}^{t} = 0 for t = 1, 2. Since e_{λ^{1}} = λ_{2}^{1}(a_{2} − a_{1}) + ⋯ + λ_{s+2}^{1}(a_{s+2} − a_{1}) and e_{λ^{2}} = λ_{1}^{2}(a_{1} − a_{2}) + ⋯ + λ_{s+2}^{2}(a_{s+2} − a_{2}), we have e_{λ^{1}} ∧ e_{λ^{2}} ∈ ⟨∂M[K_{1}, …, K_{s}]⟩. This implies dim H^{1}(A(M[K_{1}, …, K_{s}]), e_{λ}) = s. □

Remark 3.4. When m = 1, the matroid in this theorem is the uniform matroid U_{2,n} with rank 2. When m ≥ 2, the matroid M[K_{1}, …, K_{s}] has rank 3.

Remark 3.5. There exists a Latin square of order m for m ≥ 1. Let N(m) be the maximum number of mutually orthogonal Latin squares of order m. The following is known (see [4]).

• N(0) = N(1) = ∞ and 1 ≤ N(m) ≤ m − 1 for every m > 1.

• If m is a prime power then N(m) = m − 1.

• If m ≢ 2 mod 4, then N(m) ≥ 2.

• N(p × q) ≥ min {N(p), N(q)}.

• N(2) = 1, N(3) = 2, N(4) = 3, N(5) = 4, N(6) = 1, N(7) = 6, N(8) = 7.

Remark 3.6. In the case of s = 1, we have dim H^{1}(A(M[K]), e_{λ}) = 1 for non-zero one-form

e_{λ} = λ_{1}(e_{1} + ⋯ + e_{m}) + λ_{2}(e_{m+1} + ⋯ + e_{2m}) + λ_{3}(e_{2m+1} + ⋯ + e_{3m})

with λ_{1} + λ_{2} + λ_{3} = 0.

Let M and M′ be loopless matroids M on [n] of rank 3. We call M′ a degeneration of M if the family of 3-circuits of M′ contains that of M. Mostly, degenerations of M[K_{1}, …, K_{s}] have weights with non-vanishing first cohomology. The uniform matroid U_{2,n} of rank 2 is its degeneration. Next, without U_{2,n}, we shall construct its degeneration with non-vanishing first cohomology.

Proposition 3.7. Let m ≥ 2, s ≥ 1 and n = (s + 2)m. Let K_{1}, …, K_{s} be mutually orthogonal Latin squares on [m]. Let M_{i} be a simple matroid on I_{i} := {(i − 1)m + 1, (i − 1)m + 2, …, im} for i = 1, 2, …, s + 2. There exists a simple matroid M[K_{1}, …, K_{s} : M_{1}, …, M_{s+2}] with rank 3 such that it is a degeneration of M[K_{1}, …, K_{s}] and its restriction on I_{i} is M_{i} for i = 1, 2, …, s + 2. Then we have

dim H^{1}(A(M[K_{1}, …, K_{s} : M_{1}, …, M_{s+2}]), e_{λ}) = s

for a weight λ given in Theorem 3.3.

Proof. Let C_{3}(M_{1}, …, M_{s+2}) be the union of families of 3-curcuits of M_{i}; i = 1, …, s + 2. For a 3-curcuit C_{i} of M_{i} and C ∈ C[K_{1}, …, K_{s}], we have C_{i} ∩ C_{j} = ∅ for i ≠ j and |C_{i} ∩ C| = 1. Thus C[K_{1}, …, K_{s}] ∪ C_{3}(M_{1}, …, M_{s+2}) satisfies (C_{3}) and it yields a simple matroid M[K_{1}, …, K_{s} : M_{1}, …, M_{s+2}] in this statement. By the same argument as that in the proof of Theorem 3.3, we can prove the proposition. □

Remark 3.8. A realization of M[K_{1}, …, K_{s} : M_{1}, …, M_{s+2}] is a (s + 2, m)-net in ℙ^{2} defined in [17]. Therefore, there is no (k, m)-net for k > N(m)+2. In particular, there is no (k, 6)-net for k > 3.

In a Latin square K, a s × s-matrix obtained by s rows and s columns is called a Latin s-subsquare of K if it forms a Latin square of order s. Let K be a Latin square on [m] and J be a subsquare of K. We treat J̃ as a subsquare of K̃. J̃ has row index set I_{1}(J), column index set I_{2}(J) and symbol set I_{3}(J) where I_{1}(J) ⊂ I_{1}, I_{2}(J) ⊂ I_{2}, I_{3}(J) ⊂ I_{3} and |I_{1}(J)| = |I_{2}(J)| = |I_{3}(J)|. We define X(J) = I_{1}(J) ∪ I_{2}(J) ∪ I_{3}(J).

Proposition 3.9. Let J be a subsquare of a Latin square K on [m]. There exists a simple degeneration M[K; J] of M[K], whose restriction on X(J) is the uniform matroid of rank 2. Then we have

dim H^{1}(A(M[K; J]), e_{λ}) = 1

for a weight λ given in Remark 3.6.

Proof. Let C = C[K] ∪ \binom{X(J)}{3}. Since J is a subsquare of K, for C ∈ C[K] \ \binom{X(J)}{3}, we have |C ∩ X(J)| = 1. This leads to (C_{3}) for C. By the same way of Proposition 3.7, we can show this. □

Remark 3.10. The following is known (see [4]).

• There exists a Latin square of order m with a proper k-subsquare if and only if k ≤ [\frac{m}{2}].

• There exists a Latin square of order m with no proper subsquares if m ≠ 2^{a}3^{b} or if m = 3,9,12,16,18,27,81 or 243.

We note that there are degenerations of matroids associated to Latin square with non-vanishing cohomology, except for those of Proposition 3.7 and 3.9. Especially, it is not necessary to be simple, for example, see Section 4.5.

4. Arrangements

For a matroid M, an arrangement over a field F with underlying matroid M is called a F -realization or representation of M. A matroid is said to be realizable or representable over F if M has a F -realization. We shall find realizations of matroids obtained in the previous section. In this section, we will know the following:

Proposition 4.1. If 1 ≤ m ≤ 4 then the matroid M[K] associated to a Latin square K on [m] is realizable over real.

In addition, these realizations are arrangements appearing in the classical projective geometry (Figure 1, 2 and 3). Besides, we shall give many other examples including the higher case.

4.1. m = 1. Lemma 2.4 implies (e_{1} −e_{3})∧(e_{2} −e_{3}) = ∂(e_{1} ∧ e_{2} ∧ e_{3}). The matroid M[K] is realized by the arrangement in ℙ^{2} consisting of three lines through one point.

4.2. m = 2 (Falk [6]). We have only one main class isotopic Latin square K =

(\begin{matrix}1 & 2 \\ 2 & 1 \end{matrix}). The decomposable relation is (e_{1} + e_{2} − e_{5} − e_{6}) ∧ (e_{3} + e_{4} − e_{5} − e_{6}) =

∂(e_{1} ∧ e_{3} ∧ e_{5}) + ∂(e_{1} ∧ e_{4} ∧ e_{6}) + ∂(e_{2} ∧ e_{3} ∧ e_{6}) + ∂(e_{2} ∧ e_{4} ∧ e_{5}). The matroid M[K] is realized by the arrangement in ℙ^{2} arising from the Ceva Theorem (the left side in Figure 1).

4.3. m = 3. We have only one main class isotopic Latin square, which is given by

K = (\begin{matrix}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{matrix}) , C[K] = [\begin{matrix}(1, 4, 7) & (1, 5, 8) & (1, 6, 9) \\ (2, 4, 9) & (2, 5, 7) & (2, 6, 8) \\ (3, 4, 8) & (3, 5, 9) & (3, 6, 7) \end{matrix}] .

The realization is given by the arrangement of 9 lines in ℙ^{2} arising from the Pappus Theorem (the right side in Figure 1).

Figure 1. The Ceva Theorem and the Pappus Theorem

4.4. m = 4. There are two main class isotopic Latin squares, that we can give by

K_{1} = (\begin{matrix}1 & 2 & 3 & 4 \\ 4 & 1 & 2 & 3 \\ 3 & 4 & 1 & 2 \\ 2 & 3 & 4 & 1 \end{matrix}) , C[K_{1}] = [\begin{matrix}(1, 5, 9) & (1, 6, 10) & (1, 7, 11) & (1, 8, 12) \\ (2, 5, 12) & (2, 6, 9) & (2, 7, 10) & (2, 8, 11) \\ (3, 5, 11) & (3, 6, 12) & (3, 7, 9) & (3, 8, 10) \\ (4, 5, 10) & (4, 6, 11) & (4, 7, 12) & (4, 8, 9) \end{matrix}] ,

K_{2} = (\begin{matrix}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \\ 3 & 4 & 1 & 2 \\ 4 & 3 & 2 & 1 \end{matrix}) , C[K_{2}] = [\begin{matrix}(1, 5, 9) & (1, 6, 10) & (1, 7, 11) & (1, 8, 12) \\ (2, 5, 10) & (2, 6, 9) & (2, 7, 12) & (2, 8, 11) \\ (3, 5, 11) & (3, 6, 12) & (3, 7, 9) & (3, 8, 10) \\ (4, 5, 12) & (4, 6, 11) & (4, 7, 10) & (4, 8, 9) \end{matrix}] .

The matroid M[K_{1}] or M[K_{2}] is realized by the arrangement of 12 lines in ℙ^{2} defined by Figure 2 or 3, which is arising from the Kirkman Theorem or the Steiner Theorem, respectively (see [13, Chapter 16]).

4.5. Degenerations. Let K_{1} and K_{2} be in the preceding section.

Let J be a subsquare of K_{1} given by

J = ( \begin{matrix}2 & 4 \\ 4 & 2 \end{matrix}) .

By Proposition 3.9, we obtain X(J) = {1, 3, 6, 8, 10, 12} and the matroid M[K_{1}; J]. Let M_{1} be a simple matroid on [4] for which the family of 3-circuits is {(1, 2, 4)}. By Proposition 3.7, we have the matroid M[K_{1}; M_{1}]. Furthermore, the family C[K_{1}] ∪ \binom{X(J)}{3} ∪ C_{3}(M_{1}) satisfies (C_{3}) and then yields the matroid M[K_{1} : M_{1}; J] with non-vanishing first cohomology. This matroid M[K_{1} : M_{1}; J] is realized by the arrangement of 11 lines in ℂ^{2} with the infinite line 1 in Figure 4.

Figure 2. The Kirkman Theorem

Figure 3. The Steiner Theorem

The degeneration of M[K_{2}] such that 1 and 2 are parallel, that is, {1, 2} is a circuit, has a realization defined by the left one in Figure 5. Moreover, the degeneration of M[K_{2}] such that {1, 2}, {5, 6} and {11, 12} are circuits, is realizable. This realization is the B_{3}-arrangement (the right one in Figure 5). Therefore, these two arrangements have weights with non-vanishing first cohomology in the same way of Remark 3.6.

4.6. m = 3 and s = 2 (Libgober [8]). Two Latin squares

K_{1} = (\begin{matrix}1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{matrix}) ,  and  K_{2} = (\begin{matrix}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{matrix})

Figure 4. Degeneration of Kirkman’s arrangement

Figure 5. Degenerations of Steiner’s arrangement

are mutually orthogonal. We have

C[K_{1}] = [\begin{matrix}(1, 4, 7) & (1, 5, 8) & (1, 6, 9) \\ (2, 4, 9) & (2, 5, 7) & (2, 6, 8) \\ (3, 4, 8) & (3, 5, 9) & (3, 6, 7) \end{matrix}] , C[K_{2}] = [\begin{matrix}(1, 4, 10) & (1, 5, 11) & (1, 6, 12) \\ (2, 4, 11) & (2, 5, 12) & (2, 6, 10) \\ (3, 4, 12) & (3, 5, 10) & (3, 6, 11) \end{matrix}] .

The matroid M[K_{1}, K_{2}] is AG(2, 3) (see [12]) and realized as the Hessian configuration. The Hessian configuration is the arrangement of 12 projective lines passing through the nine inflection points of a nonsingular cubic in ℙ^{2}(ℂ) [10, Example 6.30], which we can define by lines

H_{1} = {x = 0}, H_{2} = {y = 0}, H_{3} = {z = 0},

H_{4} = {x + y + z = 0}, H_{5} = {x + ω^{2}y + ωz = 0}, H_{6} = {x + ωy + ω^{2}z = 0},

H_{7} = {x + ωy + ωz = 0}, H_{8} = {x + y + ω^{2}z = 0}, H_{9} = {x + ω^{2}y + z = 0},

H_{10} = {x + ω^{2}y + ω^{2}z = 0}, H_{11} = {x + ωy + z = 0}, H_{12} = {x + y + ωz = 0}, where ω = e^{2πi/3}. Underlying matroids of arrangements

{H_{1}, …, H_{6}, H_{7}, H_{8}, H_{9}} and {H_{1}, …, H_{6}, H_{10}, H_{11}, H_{12}}

are M[K_{1}] and M[K_{2}], respectively. The Hessian configuration {H_{1}, …, H_{12}} has the underlying matroid M[K_{1}, K_{2}] and we have dim H^{1}(A(M[K_{1}, K_{2}]), e_{λ}) = 2 for a non-zero one-form

e_{λ} = λ_{1}(e_{1} + e_{2} + e_{3}) + λ_{2}(e_{4} + e_{5} + e_{6}) + λ_{3}(e_{7} + e_{8} + e_{9}) + λ_{4}(e_{10} + e_{11} + e_{12})

with ∑_{j=1}^{4} λ_{j} = 0.

4.7. Monomial arrangements (Cohen and Suciu [3]). Let K be the Latin square of order m defined by the addition table for ℤ_{m} × ℤ_{m} for m ≥ 2. The monomial arrangement A_{m,m,3} in ℂ^{3} is given by the defining polynomial

Q(A_{m,m,3}) = (x_{1}^{m} − x_{2}^{m})(x_{1}^{m} − x_{3}^{m})(x_{2}^{m} − x_{3}^{m}).

Set ζ = exp (2πi/m). Define

A_{ij} = {H_{i,j}^{k} = Ker (x_{i} − ζ^{k}x_{j}) : 1 ≤ k ≤ m}

for 1 ≤ i < j ≤ 3. So we have A_{m,m,3} = A_{12} ∪ A_{23} ∪ A_{13}. Since ∩_{k=1}^{m}H_{i,j} has rank two, the underlying matroid M(A_{ij}) of A_{ij} is isomorphic to the uniform matroid U_{2,m} of rank two. Another rank two intersections are H_{1,2}^{p} ∩H_{2,3}^{q} ∩H_{1,3}^{r} for p+q ≡ r mod m. Hence, K can be consider as the Latin square with rows indexed by A_{12}, columns by A_{23}, and symbols by A_{13} . The underlying matroid of A_{m,m,3} is the matroid M[K; M(A_{12}), M(A_{23}), M(A_{13})]. By Proposition 3.7, A_{m,m,3} has weights with non-vanishing first cohomology.

4.8. Higher case (ℓ = 3). Let K be a Latin 3-dimensional hypercube on [2] defined by Figure 6. The matroid M[K] is the matroid of type L_{8} in [12, p.510].

[figure: 1; 2; 1; 2; 2; 1; 2; 1; 1357; 2358; 2457; 1458; 1368; 2367; 2468; 1467]

Figure 6. K and C[K]

Let A be an 4-arrangement defined by the defining polynomial

x_{1}x_{2}x_{3}x_{4}(x_{1}+x_{2}+x_{3}+x_{4})(x_{1}+bcx_{2}+bx_{3}+cx_{4})(x_{1}+cx_{2}+x_{3}+cx_{4})(x_{1}+bx_{2}+bx_{3}+x_{4}),

where 0, 1, b, c, bc are distinct each other. By the simple computation, A is a realization of M[K]. Therefore, A has weights with non-vanishing second cohomology. Let B be an 4-arrangement defined by the defining polynomial

(x_{1} − x_{2})(x_{1} + x_{2})(x_{2} − x_{3})(x_{2} + x_{3})(x_{3} − x_{4})(x_{3} + x_{4})(x_{4} − x_{1})(x_{4} + x_{1}).

By the simple computation, we can check that B has no 3-circuits and the family of 4-circuits is

C[K] ∪ {(1, 2, 3, 4), (1, 2, 7, 8), (3, 4, 5, 6), (5, 6, 7, 8)}.

Therefore, B has weights with non-vanishing second cohomology.

Acknowledgments. A main part of this work was done while the author was visiting Mathematical Sciences Research Institute in August-November, 2004. The author would like to thank MSRI for its hospitality. The author is grateful to Professor Sergey Yuzvinsky and Professor Michael Falk for many helpful suggestions and valuable discussions and the author would also like to thank Professor Hiroaki Terao for his help.

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Department of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

   E-mail address: ykawa@comp.metro-u.ac.jp