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Abstract

An arrangement of hyperplaneswith a modular element in its intersection lattice has a Poincarépolynomial which factors; this was proven by Stanley in the setting ofgeometric lattices. This note proves a factorization in the setting ofhyperplane arrangements under two conditions which imply a modularelement. Two well known reflection arrangements serve as motivation andtheir Poincaré polynomials are computed using the main theorem of thisnote.

Definition 1. Let \({\mathbb F}\) be a field. A hyperplane isan affine subspace of codimension one in \({\mathbb F}^\ell\). A hyperplanearrangement in \({\mathbb F}^\ell\) isa finite collection of hyperplanes in \({\mathbb F}^\ell\), written \({\mathcal A}=\{H_1,\ldots, H_n\}\). Thecardinality of \({\mathcal A}\) is\(n\) and is denoted \(|{\mathcal A}|\).

Definition 2. Let \({\mathcal A}\) be an arrangement ofhyperplanes in \(V = {\mathbbF}^\ell\). We define the partially ordered set \(L({\mathcal A})\) with objects given by\(\cap_{H \in {\mathcal B}} H\) for\({\mathcal B}\subseteq {\mathcal A}\)and \(\cap_{H\in {\mathcal B}} H \ne \emptyset\); order the objects of \(L({\mathcal A})\) opposite to inclusion.Notice \(\emptyset \subseteq {\mathcalA}\) gives \(V \in L({\mathcalA})\) with \(V \le X\) for all\(X \inL({\mathcal A})\). For \(X \inL({\mathcal A})\), we define \(\operatorname{rank}(X) := \operatorname{codim\}X\). We define \(\operatorname{rank}({\mathcal A}) :=\max_{X\in L({\mathcal A})} \operatorname{rank}(X)\).

Definition 3. Let \({\mathcal A}\) be an arrangement. If \({\mathcal B}\subseteq{\mathcal A}\) is a subset, then \({\mathcal B}\) is called a subarrangement.For \(X\in L({\mathcal A})\), we definea subarrangement \(A_X\) of \({\mathcal A}\) by \(A_X :=\{H \in {\mathcal A}\ :\ X\subsetH\}\). Define an arrangement \({\mathcal A}^X\) in \(X\) via \({\mathcal A}^X :=\{X\cap H\ :\ H\in {\mathcalA}\setminus {\mathcal A}_X \rm{\ and\ }X\cap H \ne \emptyset\}\)

Definition 4. Let \({\mathcal A}= \{H_1,...,H_n\}\) be ahyperplane arrangement in \(V={\mathbbF}^\ell\) for some field \({\mathbbF}\). We fix an order on \({\mathcalA}\); that is, for hyperplanes \(H_i\) and \(H_j\) in \({\mathcal A}\), we have \(H_i < H_j\) if and only if \(i <j\).

Let \({\mathcal K}\) be acommutative ring. Let \(E_1\) be thelinear space over \({\mathcal K}\) on\(n\) generators. Let \(E({\mathcal A}):= \Lambda (E_1)\) be theexterior algebra on \(E_1\). We have\(E({\mathcal A}) = {\bigoplus_{p \ge 0}E_p}\) is a graded algebra over \({\mathcal K}\). The standard \({\mathcal K}\)-basis for \(E_p\) is given by \[\{e_{i_1} \cdot\cdot\cdote_{i_p}:\ 1 \le i_1 < \ldots < i_p \le p\}.\] Any orderedsubset \(S =\{H_{i_1},...,H_{i_p}\}\) of \({\mathcal A}\) corresponds to an element\(e_S := e_{i_1} \cdot \cdot \cdote_{i_p}\) in \(E({\mathcalA})\).

Definition 5. We define the map \(\partial:\ E({\mathcal A})\to E({\mathcalA})\) via the usual differential. That is, \[\begin{aligned}\partial(1) &:=&0,\\\partial(e_i) &:=& 1,\quad \rm{\ and\ for\ }p\geq 2,\\\partial(e_{i_1} \cdot \cdot \cdot e_{i_p}) &:=& \sum_{k=1}^p(-1)^{k-1}e_{i_1}\cdot\cdot \cdot \hat e_{i_k} \cdot \cdot \cdot e_{i_p}.\end{aligned}\]

Definition 6. We define \(I({\mathcal A})\) to be the ideal of \(E({\mathcal A})\) which is generated by\[\{\partial(e_S):\ S \rm{\ isdependent\ }\}\cup\{e_S:\ \cap S = \emptyset \}.\]

Definition 7. The Orlik-Solomon algebra, \(A({\mathcal A})\), is defined as \[A({\mathcal A}) := E({\mathcal A})/I({\mathcalA}).\] Let \(\pi:\ E({\mathcal A}) \toA({\mathcal A})\) be the canonical projection. We write \(a_S\) to represent the image of \(e_S\) under \(\pi\).

We define the Orlik-Solomon algebra and a linear basis for thisalgebra, referred to as the broken circuit basis; see Chapter 3 in [3].

Let \({\mathcal A}=\{H_1,...,H_n\}\) be a hyperplane arrangement in \(V={\mathbb F}^\ell\) for some field \({\mathbb F}\). For each \(H_i\in {\mathcal A}\), we fix an affinefunctional \(\alpha_i\) with \(\operatorname{Ker}\alpha_i =H_i\). We fixan order on \({\mathcal A}\); that is,for hyperplanes \(H_i\) and \(H_j\) in \({\mathcal A}\), we have \(H_i < H_j\) if and only if \(i <j\).

Let \(I({\mathcal A})\) be the idealof \(E({\mathcal A})\) as definedpreviously, and let \(A({\mathcal A}) :=E({\mathcal A})/I({\mathcal A})\) be the Orlik-Solomon algebra.Let \(\pi:\ E({\mathcal A}) \to A({\mathcalA})\) be the canonical projection. We write \(a_S\) to represent the image of \(e_S\) under \(\pi\).

We demonstrate that \(A({\mathcalA})\) is a free graded \({\mathcalK}\)-module by defining the broken circuit basis for \(A({\mathcal A})\). By Theorem 1.9 tofollow, this is indeed a basis for \(A({\mathcal A})\).

Theorem 9. As a \({\mathcal K}\)-module, \(A({\mathcal A})\) is a free, graded module.The broken circuit basis forms a basis for \(A({\mathcal A})\).

Proof. This is proven in Theorem 3.55 in [3].◻

Example 10. Let \(\dim\ V =\ell,\) and let \({\mathcal A}\) be the braid arrangement in\(V\) given by \[{Q({\mathcal A}) =\prod_{1\leq i <j\leq \ell}\ (x_i -x_j)}.\] Let \(H_{ij}\) correspond to the hyperplane givenby \(x_i -x_j=0\). Order thehyperplanes lexicographically; that is, \(H_{ij} < H_{mn}\) if either \(i < m\) or \(i=m\) and \(j<n\). We will write \(a_{H_{ij}} =a_{ij}\) in \(A_1({\mathcalA})\).

In order to compute \(\dim\ A_p({\mathcalA})\), we need to describe the elements of the broken circuitbasis in \(A_p({\mathcal A})\). Let\(a := a_{i_1j_1}a_{i_2j_2} \cdot\cdot\cdot a_{i_pj_p}\) be an element of the broken circuit basis in\(A_p({\mathcal A})\). By definition ofthe hyperplanes, we have \(i_k <j_k\).

We first verify all the second indices of \(a\) are distinct. Suppose \(j_1 = j_2\). Without loss of generality, wemay assume \(i_1<i_2\). Then \(\{H_{i_{1}j_{1}}, H_{i_{2}j_{2}},H_{i_1i_2}\}\) is dependent with \(H_{i_1i_2}\) being minimal in the set; thiscontradicts the assumption \(a\) is inthe broken circuit basis. In a similar fashion, we have and will assume\(j_1 < j_2 < \cdot \cdot \cdot <j_p.\)

We now verify the first indices have no restriction other than \(i_k <j_k\). Suppose \(i_1 = i_2\), then \(\{H_{i_1j_1}, H_{i_2j_2}, H_{j_1,j_2}\}\)is dependent; but the minimal element of this set is \(H_{i_1j_1}\). Notice \(\{H_{i_1j_1}, H_{i_2j_2}, H_{j_1,j_2}\}\)is not basic as there are two of the second indices equal and thissituation was eliminated. Therefore, \(a\) is still an element of the brokencircuit basis as it does not contain the factor \(a_{j_1 j_2}\). Hence, there are norestrictions on \(i_k\) other than\(j_k > i_k.\)

It is now just a matter of counting the possibilities we have for\(\{i_1j_1, ..., i_pj_p\}\) with therestrictions \(j_1 <j_2 < \cdot \cdot\cdot < j_p\) and \(i_k <j_k\) for \(k = 1, ..., p\).

Fix \(j_1, ..., j_p\). There are\(\ell-j_k\) choices for \(i_k\) for each \(k= 1, ..., p.\) Thus, \[\begin{aligned}\dim\ A_p({\mathcal A}) &=&\sum_{i_p =1+i_{p-1}}^{\ell-1} \cdot \cdot \cdot\sum_{i_2=1+i_1}^{\ell-p+1} \sum_{i_1 =1}^{\ell-p}(\prod_{k=1}^p\ (\ell-j_k))\\&=& \sum_{1 \leq j_1<j_2< \cdot\cdot < j_p \leq \ell-1} j_1j_2 \cdot \cdot j_p .\end{aligned}\] As usual, if \(p=0\), then this sum is taken to be \(1\).

The dimensions of \(A_1({\mathcalA})\) and \(A_2({\mathcal A})\)can be easily simplified. Obviously, we have \(\dim A_1({\mathcal A}) = {\ell \choose2}\). For the dimension of \(A_2({\mathcal A})\), consider minimallydependent sets of three hyperplanes. Any such set must be of the form\(\{H_{ij}, H_{ik},H_{jk}:\ \ i<j<k\}\). There are \({\ell \choose 3}\) of these sets. Hence,\(\dim\ A_2({\mathcal A}) = \dim\ E_2 - {\ell\choose3}\). Using the fact \(n = {\ell\choose 2},\) we arrive at \(\dim\A_2({\mathcal A}) = \displaystyle{{\ell(\ell-1)(\ell-2)(3\ell-1)} \over{24}}\).

Definition 11. Let \({\mathcal A}\) be an arrangement. Let \(H_0 \in {\mathcal A}\). We define thearrangements given by deletion and restriction \[{\mathcal A}' =\{H:\ H\in{\mathcal A}\setminus H_0\}, \rm{and}\] \[{\mathcal A}''=\{H_0 \cap H:\ H \in{\mathcal A}\rm{\ and\ } H \cap H_0 \neq \emptyset\}.\]

Definition 12. Let \(\pi(A({\mathcal A}), t)\) be the Poincarépolynomial of the free graded \({\mathcalK}\)-module \(A({\mathcal A})\);that is, \(\pi(A({\mathcal A}), t) =\sum_{p=0}^\ell {\operatorname{rank}}(A_p({\mathcal A}))t^p\).

Theorem 13. Let \(({\mathcal A}, {\mathcal A}', {\mathcalA}'')\) be a triple given by deletion and restriction.Then \(\pi(A({\mathcal A}), t) =\pi(A({\mathcal A}'), t) + t\pi(A({\mathcal A}''),t)\).

Proof. This is Corollary 3.67 in <[3].◻

We end this section by furnishing two additional definitions whichare needed in the subsequent section.

Definition 14. An element \(X\in L({\mathcal A})\) is said to bemodular if for any \(Y\in L({\mathcalA})\) and any \(Z\in L({\mathcalA})\) with \(Z\le Y\), we have\[Z\vee (X\wedge Y)=(Z\vee X))\wedgeY.\]

Definition 15. Let \({\mathcal A}\) be an arrangement. We say\({\mathcal A}\) is supersolvable if\(L({\mathcal A})\) has a maximal chainof modular elements \[V=X_0<X_1<\cdots<X_\ell=\cap_{H\in{\mathcal A}} H,\] while \(\operatorname{rank}({\mathcalA})=\ell\).

Main Theorem

Factorization of the Poincaré polynomial has been studiedextensively. Stanley showed that supersolvable arrangements havePoincare polynomials that factor into linear factors [4]. A generalization ofsupersolvable arrangements gave a factorization into linear factors bylooking at nice partitions [5]. Other generalizations of supersolvablearrangements are given in [1]and [2]. In this section, weshow a factorization of the Poincaré polynomial when the arrangement hasa special subarrangement which implies the existence of a modularelement in \(L({\mathcal A})\).

In the above definition, \(Z=\cap_{H \in{\mathcal H}} H\) is a modular element of \(L({\mathcal A})\). See Stanley [4]. However, let \({\mathcal A}\) be the arrangement given bythe hyperplanes \(\{x, y, z, x+y-z\}\)with \({\mathcal H}\) given by \(\{z\}\). Then the hyperplane given by \(\{z=0\}\) is a modular element but does notsatisfy condition (B). Hence, modularity of \(Z=\cap_{H \in {\mathcal H}} H\) does notimply that conditions (A) and (B) are satisfied.

Theorem 17. Suppose \({\mathcal H}\subset {\mathcal A}\)satisfies condition (A). There exists an ordering of the hyperplanes sothat the broken circuit basis contains no elements \(a_{\vec \nu}\) where \(\vec \nu\) contains two indicescorresponding to hyperplanes in \({\mathcalH}\).

Proof. Order the hyperplanes so that for any \(H_i \in{\mathcal H}\) and any \(H_k \in{\mathcal A}\setminus{\mathcal H}\), we have \(i>k\). Let \(a_{\vec \nu}\) be a basic element of \(A({\mathcal A})\) and suppose \(\vec \nu\) contains two indicescorresponding to hyperplanes in \({\mathcalH}\), say \(H_\alpha\) and \(H_\beta\). Since \({\mathcal H}\) satisfies condition (A),there exists \(H_\gamma \in{\mathcal A}\setminus{\mathcal H}\) with \(H_\alpha \cap H_\beta \subset H_\gamma\).By our choice of ordering, \(\gamma<\alpha,\beta\) and hence \(a_{\vec\nu}\) is not basic.◻

Suppose \({\mathcal H}\subseteq {\mathcalA}\) satisfies condition (A). Let \(X\in L({\mathcal A})\) have rank greaterthan or equal to 2. Since \({\mathcalH}\) satisfies condition (A), we must have some hyperplanescontaining \(X\) that are in \({\mathcal A}\setminus{\mathcal H}\). Let\(X'\in L({\mathcal A})\) representthe intersection of the hyperplanes containing \(X\) that are in \({\mathcal A}\setminus {\mathcal H}\).

Lemma 18. Supposes \({\mathcal A}\) has the trio separationproperty under \({\mathcal H}\). Let\(X\in L({\mathcal A})\) have rankgreater than or equal to 2. Fix \(H_0\in{\mathcal H}\). Then \(X' = (X \cap H_0)'\). Moreover, if\((X\cap H_0)'=(Y\cap H_0)'\)for any \(X,Y\in L({\mathcal A})\setminus \{V,H_0\}\), then \(X\cap H_0 = Y\capH_0\).

Lemma 19. Supposes \({\mathcal A}\) has the trio separationproperty under \({\mathcal H}\). Fix\(H_0\in {\mathcal H}\). Then \(L({\mathcal A}^{H_0})\cong L({\mathcal A}\setminus{\mathcal H})\).

We are now ready to state and prove the following:

Theorem 20. Suppose \({\mathcal A}\) has the trio separationproperty under \({\mathcal H}\). ThePoincaré polynomial of \({\mathcal A}\)is computed via \[\pi(A({\mathcal A}), t) =(1 +|{\mathcal H}|\cdot t)\pi(A({\mathcal A}\setminus{\mathcal H}),t).\]

Definition 21. Let \({\mathcal A}_\ell\) be the braidarrangement defined by \[Q({\mathcal A}_\ell)= \prod_{1 \le i <j\le \ell}(x_i - x_j).\]

Lemma 22. Let \(A_{\ell}\) denote the braid arrangement.Let \(H_{i,j}\) be the hyperplanedetermined by \(x_i - x_j\) for \(1 \le i <j\le \ell\). Then for any \(2\le \beta\le\ell\), we have: \[({\mathcal A}_{\ell}\setminus\{H_{1,\ell},\ldots, H_{\beta,\ell}\})^{H_{\beta+1,\ell}}\cong {\mathcal A}_{\ell-1}.\]

Proof. Let \({\mathcalH}=\{H_{1,\ell},\ldots, H_{\beta,\ell}\}\). Then \({\mathcal H}\) satisfies conditions (A) and(B). By Lemma 2.4, the result is immediate.◻

Theorem 23. Let \(A_{\ell}\) denote the braid arrangement.Then \[\pi({\mathcal A}_\ell) =(1+(\ell-1)\cdot t) \pi({\mathcal A}_{\ell-1}).\]

Proof. Let \({\mathcalH}=\{H_{1,\ell},\ldots, H_{\beta,\ell+1}\}\). Then \({\mathcal A}\) has the trio separationproperty under \({\mathcal H}\). ByTheorem 2.5 and Lemma 3.2, the result is immediate.◻

Definition 24. Let \(V\) be an \(\ell\)-dimensional vector space over thefinite field of \(q\) elements, \({\mathbb F}_q\). Let \({\mathcal A}_\ell\) be the centralarrangement of all hyperplanes through the origin.

Lemma 25. Let \(A_{\ell}\) denote the arrangement definedin Definition 3.4. Let \(\vec c =\{c_1,\ldots, c_{\ell-1}\}\) for \(c_i \in{\BbbF}_q\). Denote \(H_{\vec c,\ell}\) by the hyperplane determined by \(x_\ell + \sum_{1 \lei\le\ell-1}c_ix_i\). Let \({\mathcalH}\) be the collection of hyperplanes \(H_{\vecc, \ell}\). For any \(U\subset {\mathcal H}\) with \(H_{\vecc, \ell} \not\inU\), we have \[({\mathcal A}_{\ell}\setminus U)^{H_{1, \vec c}}\cong{\mathcal A}_{\ell-1}.\]

Proof. Since \({\mathcalA}\) has the trio separation property under \({\mathcal H}\), the result is immediate byLemma 2.4.◻

Theorem 26. Let \(A_{\ell}\) denote the arrangement ofDefinition 3.4. Then \[\pi({\mathcal A}_\ell)= (1+q^{\ell-1}\cdot t)\pi({\mathcal A}_{\ell-1}).\]

Proof. Take \({\mathcalH}\) to be the collection of hyperplanes \(H_{\vecc, \ell}\) as defined in Lemma 3.5. By Theorem 2.5 and Lemma 3.5,the result is immediate.◻

Competing interest

The authors declare that they have nocompeting interests.

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