On intersection lattices of hyperplane arrangements generated by generic points
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
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 .
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.
For a finite set , we define
For distinct finite sets , we define
We also define .
By definition, it follows that
In particular for ,
For , . This implies the following fact. Let . Then
For and , we define to be the set of satisfying the following two conditions:
for all with .
for all .
Moreover we define the partial ordering on by
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
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 .
Let . Without loss of generality assume . We call
the type of .
For any , and .
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
Note that the usual characteristic polynomial of the non-central arrangement is given as
Conversely from we can evaluate since . Equivalently
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 .
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 .
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 .
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).
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.
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.
and for all other , .
We need the numbers of elements of to evaluate . These are tabulated in Table 1.
3. Main result
In this section we show the following main theorem.
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.
For , is isomorphic to as posets.
To prove the first part of Theorem 3.1, we show one proposition and three lemmas.
Let . For each , there uniquely exists such that .
It suffices to show the uniqueness. Let and , , satisfy . This means . Since , by (2), . This conflicts with . ∎
Let and . If for all , then .
Let . Then by (1)
Hence . ∎
Let , , , …, , and . Assume for all . If and for all , then .
Let . Then
Since , it follows from Lemma 3.4 that
Let , and . Then and are isomorphic as posets.
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. ∎
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 .
For , ,
Note that for we have . Also, as discussed at the beginning of Section 2.1, if .
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.
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.
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