hyperplane arrangements generated by generic points

On intersection lattices of hyperplane arrangements generated by generic points

Hiroshi KOIZUMI Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo. Yasuhide NUMATA Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo.  and  Akimichi TAKEMURA Mathematical Informatics, Graduate School of Information Science and Technology, University of Tokyo. JST CREST
Abstract.

We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.

Key words and phrases:
discriminantal arrangements; Möbius function; characteristic polynomial; enumeration of elements
2000 Mathematics Subject Classification:
52C35;05A99
The second and the third authors are supported by JST CREST

1. Introduction

Consider a set of generic points in a -dimensional vector space over a field of characteristic zero. For let denote the affine hull of . Let

be the set of all hyperplanes defined by for some , . Here we assume that points are generic in the sense of Athanasiadis (1999). Then combinatorial properties of the arrangement does not depend on the points. Since in this paper we are interested only in the combinatorial properties of , we denote the arrangement by . We decompose the poset ideals of the intersection lattice of into direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we give an explicit description of the Möbius functions and the characteristic polynomials of the intersection lattices for and for all .

By Theorem 2.2 of Falk (1994), is equivalent to the discriminantal arrangement of Manin and Schechtman (1989). Relevant facts on the discriminantal arrangement are given in Section 5.6 of Orlik and Terao (1992), Bayer and Brandt (1997) and Athanasiadis (1999). We prefer to work with because we utilize the recursive structure of with respect to .

The organization of this paper is the following. In Section 2 we set up our definition and notation. In particular following Athanasiadis (1999) we interpret the intersection lattice of our arrangement in set theoretical terminology. We also give illustrations for . In Section 3, we show the fundamental structure of the intersection lattice of , which is the main result of this paper. Based on the main result, in Section 4 we compute the Möbius function of the intersection lattice, the number of elements of a particular type of the intersection lattice, and the characteristic polynomials of the arrangements up to and for all .

Acknowledgments.

The authors are very grateful to Hidehiko Kamiya and Hiroaki Terao for very useful comments.

2. Definition and Notation

We denote the intersection lattice of by

where the sets are ordered by reverse inclusion. Contrary to the usual convention, here we consider that belongs to , so that is not only a poset but also a lattice (cf. Proposition 2.3 of Stanley (2007)). In usual convention, this corresponds to the coning of , except that we do not add a coordinate hyperplane. The reason for this unconventional definition is that plays an essential role for recursive description of .

We now follow Athanasiadis (1999) to give an interpretation of in set theoretical terminology.

Definition 2.1.

For a finite set , we define

For distinct finite sets , we define

We also define .

Remark 2.2.

By definition, it follows that

(1)

In particular for ,

(2)
Remark 2.3.

For , . This implies the following fact. Let . Then

Definition 2.4.

For and , we define to be the set of satisfying the following two conditions:

  1. for all with .

  2. for all .

Moreover we define the partial ordering on by

(3)

Let be a collection of generic points in in the sense of Section 1. For , , define to be the affine hull . Since , there exists a subset such that and . Hence

This mapping induces a map from to , or equivalently, corresponds to . By this correspondence, is isomorphic to as lattices (Athanasiadis (1999), Falk (1994)).

Remark 2.5.

is a graded poset with the rank function . is the minimum element of with and () is the maximum element of with . In the one-to-one correspondence between and , and . In the case , the condition 2 in Definition 2.4 implies for . Hence is the poset of two elements

independent of .

Let be a nonnegative integer. We call a weakly-decreasing sequence of nonnegative integers such that a partition of . We write to say that is a partition of . We also regard a partition as a multiset of positive integers. For example, , and is the set consisting of the unique partition of zero, which is denoted by .

Definition 2.6.

Let . Without loss of generality assume . We call

the type of .

Example 2.7.

For any , and .

Definition 2.8.

For , we define to be the poset ideal generated by , i.e., .

Finally we define the Möbius function of the poset , which will be studied in Section 4. Define by

We write . The characteristic polynomial of the poset (cf. Section 3.10 of Stanley (1997)) is defined by

(4)

Note that the usual characteristic polynomial of the non-central arrangement is given as

Conversely from we can evaluate since . Equivalently

(5)

2.1. Illustration of the posets up to dimension three

We illustrate the above definitions with . For we already saw . In particular .

Let . In , in addition to the minimum and the maximum , there are rank one elements , , with . Hence . The value is relevant for .

Figure 1. Arrangement for dimension two

The case is already discussed in Section 7 of Manin and Schechtman (1989) and Section 5.6 of Orlik and Terao (1992). However we present it here from our viewpoint. As shown in Figure 1, each line (rank one element) is labeled by a pair of points, such as , which is a line connecting points and . There are two types of points (rank two elements). The first type is an element of type . Each element of type corresponds to an original point in . The second type is an element of type . Each element of type corresponds the intersection of two lines, depicted by a white circle in Figure 1. The Möbius function is evaluated as and .

Remark 2.9.

In this paper we are assuming that so that is a non-central arrangement. We usually think of as “sufficiently large” compared to . Relevant quantities are polynomials in and these polynomials are determined by sufficiently large . However our polynomials hold for all with appropriate qualifications. For example, the second type of exists if and only if . As long as , . In general, when we write , this has to exist in . Actually we are interested in the existence of some with the same type as , i.e. . The existence implies that has to be larger than or equal to some specific value, say , depending on the type of . As shown in Section 4.2, is the minimum such that the number of elements of of the type is positive.

We now count the number of elements of . This is also needed to evaluate . There are lines. There are points of the first type and

points of the second type. As discussed in Remark 2.9, this is positive if and only if .

Therefore for

(6)

These quantities are polynomials in and we prefer to write these polynomials as integer combinations of binomial coefficients . Note that, in view of Remark 2.9, for integer .

We now discuss the case of .

We first look at rank two elements (lines) of . There are two types of elements. The first type is an element of type . Each element of type , such as , corresponds to the line connecting two points as in the leftmost picture of Figure 2. is understood as the intersection of all hyperplanes , . The second type is an element of type . Each elements of type corresponds to an intersection of two hyperplanes, such as . As shown in the rightmost picture in Figure 2, two points ( and in the picture) may overlap in this case without violating 1 of Definition 2.4. This type of element exists for (cf. Remark 2.9).

Figure 2. Rank two elements for dimension three

Finally we look at rank three elements (points) of . We will not repeat remarks on existence of these elements of . There are three types of rank three elements, corresponding to three partitions of . Each element of the first type , corresponds to an original point in . Each element the second type corresponds to an intersection of a line of type and a hyperplane, e.g.  as shown in Figure 3.

Figure 3. Rank three element for dimension three of type
Figure 4. Rank three elements for dimension three of type

The third type is , corresponding to an intersection of three hyperplanes as depicted by a white circle in Figure 4. Without violating 1 of Definition 2.4, there are four patterns of overlaps of points.

As will be proved in Section 4, the Möbius function depends only on the above types (i.e. the overlaps of points do not affect the Möbius function) and it is given as follows.

(7)

and for all other , .

We need the numbers of elements of to evaluate . These are tabulated in Table 1.

(1) (2) (1,1) (3) (2,1) (1,1,1)
Table 1. Number of elements for

An element of a particular type exists if and only if the number of elements is positive in Table 1. For example, of type exists if and only if , i.e. . From Table 1 and (7) we obtain (for )

3. Main result

In this section we show the following main theorem.

Theorem 3.1.

Let , . Then the ideal is isomorphic to the direct product as posets. They are also isomorphic to .

The second part of this theorem is a consequence of the following lemma.

Lemma 3.2.

For , is isomorphic to as posets.

Proof.

Suppose that . Then , so , , by (3). Hence by Remark 2.3. Therefore we have a map

which is seen to be one-to-one and onto, and preserves the partial order. ∎

To prove the first part of Theorem 3.1, we show one proposition and three lemmas.

Proposition 3.3.

Let . For each , there uniquely exists such that .

Proof.

It suffices to show the uniqueness. Let and , , satisfy . This means . Since , by (2), . This conflicts with . ∎

Lemma 3.4.

Let and . If for all , then .

Proof.

Let . Then by (1)

Since ,

This implies

Hence . ∎

Lemma 3.5.

Let , , , …, , and . Assume for all . If and for all , then .

Proof.

Let . Then

Hence

Since ,

Since , it follows from Lemma 3.4 that

Lemma 3.6.

Let , and . Then and are isomorphic as posets.

Proof.

Let , and . For , let us define . Then by Lemma 3.5. By definition . Hence is a map from to . Moreover, if and satisfy and , then . On the other hand, we can define the following map from to :

which is the inverse map of . Hence and are isomorphic as posets. ∎

Applying Lemma 3.6 recursively, we have Theorem 3.1.

4. Computation of Möbius function and the characteristic polynomial

In this section we apply Theorem 3.1 to compute the Möbius function and the characteristic polynomial of the intersection lattice for . This section is divided into four subsections.

In Section 4.1 we derive an explicit formula for the value of the Möbius function of and show that it only depends on the type of . Next in Section 4.2 we derive a formula for the number of elements of the same type as . Then in Section 4.3 we derive some identities for these numbers, which are useful for checking the results of computations by computer. Finally in Section 4.4 we present lists of the numbers of elements and the characteristic polynomials for .

4.1. Möbius function of the intersection lattice

We first obtain the value of Möbius function of .

Proposition 4.1.

For , ,

Note that for we have . Also, as discussed at the beginning of Section 2.1, if .

Proposition 4.1 is an immediate consequence of Theorem 3.1 and the following well-known lemma.

Lemma 4.2 (Proposition 3.8.2 of Stanley (1997)).

Let and be posets, and the direct product of posets and . Then for and , where denotes the Möbius function for each poset.

Proposition 4.1 shows that the Möbius function of is completely determined by the values of , . In particular for , is a product of for smaller than . As seen in the examples of Section 2.1, is a polynomial in . Hence , , can be immediately obtained from for . Therefore for the recursion on , the essential step is to compute by (5), which will be discussed in the next subsection.

As a corollary to Proposition 4.1 we have the following result.

Corollary 4.3.

Let and satisfy for each . Define , . Then

In this sense the value of the Möbius function depends only on the multiset of codimensions, i.e., the type of . Therefore from now on we denote if .

4.2. Number of elements of the intersection lattice

The results of the previous subsection implies that the terms of the summations in (4) and (5) can be grouped into different types. Then the question is how to obtain the number of elements of the same type in , denoted by below. In this subsection we give an explicit expression for in Proposition 4.7.

Let

denote the number of of type . Then (4) and (5) are written as follows.

(8)
(9)

For stating Proposition 4.7 we need some more definitions. For a partition of a nonnegative integer let

denote the stabilizer of the symmetric group fixing . Then we have

where denotes the multiplicity of in .

Denote the elements of as