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 $n$ $(> d)$ generic points $P=\Setp1,…,pn$ in a $d$ -dimensional vector space $V = K^{d}$ over a field $K$ of characteristic zero. For $X \subset P$ let $H_{X}$ denote the affine hull of $X$ . Let

A=\SetHXX⊂P,#X=d

be the set of all hyperplanes defined by $H_{X}$ for some $X \subset P$ , $# X = d$ . Here we assume that points $p_{1}, \dots, p_{n}$ are generic in the sense of Athanasiadis (1999). Then combinatorial properties of the arrangement $A$ does not depend on the points. Since in this paper we are interested only in the combinatorial properties of $A$ , we denote the arrangement by $A_{n, d}$ . We decompose the poset ideals of the intersection lattice of $A_{n, d}$ 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 $d \leq 6$ and for all $n > d$ .

By Theorem 2.2 of Falk (1994), $A_{n, d}$ is equivalent to the discriminantal arrangement $B (n, n - d - 1)$ 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 $A_{n, d}$ because we utilize the recursive structure of $A_{n, d}$ with respect to $d$ .

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 $d \leq 3$ . In Section 3, we show the fundamental structure of the intersection lattice of $A_{n, d}$ , 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 $d = 6$ and for all $n > d$ .

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 $A_{n, d}$ by

L(An,d)=\setH1∩⋯∩HkH1,…,Hk∈An,d,

where the sets are ordered by reverse inclusion. Contrary to the usual convention, here we consider that $\emptyset = ⋂_{X : # X = d} H_{X}$ belongs to $L (A_{n, d})$ , so that $L (A_{n, d})$ is not only a poset but also a lattice (cf. Proposition 2.3 of Stanley (2007)). In usual convention, this corresponds to the coning $c A_{n, d}$ of $A_{n, d}$ , except that we do not add a coordinate hyperplane. The reason for this unconventional definition is that $\emptyset \in L (A_{n, d})$ plays an essential role for recursive description of $L (A_{n, d})$ .

We now follow Athanasiadis (1999) to give an interpretation of $L (A_{n, d})$ in set theoretical terminology.

Definition 2.1.

For a finite set $X$ , we define

{codim}_{d} (X) = d + 1 - # X .

For distinct finite sets $T_{1}, \dots, T_{l}$ , we define

	$ρd(\SetT1,…,Tl)$	$= {codim}_{d} T_{1} + \dots + {codim}_{d} T_{l},$
	$Dd(\SetT1,…,Tl)$	$=codimd(T1∩⋯∩Tl)−ρd(\SetT1,…,Tl).$

We also define $ρd(∅)=ρd(\Set)=0$ .

Remark 2.2.

By definition, it follows that

(1)

D_{d}

(\SetT1,…,Tl)=−(l−1)(d+1)+#T1+⋯+#Tl−#(T1∩⋯∩Tl).

In particular for $T_{1} \neq T_{2}$ ,

(2)

Dd(\SetT1,T2)=#(T1∪T2)−(d+1).

Remark 2.3.

For $Y \subset X$ , ${codim}_{d - # Y} (X ∖ Y) = {codim}_{d} (X)$ . This implies the following fact. Let $U \subset T_{1} \cap \dots \cap T_{l}$ . Then

	$ρd−#U(\setT1∖U,…,Tl∖U)$	$=ρd(\setT1,…,Tl),$
	$Dd−#U(\setT1∖U,…,Tl∖U)$	$=Dd(\setT1,…,Tl).$

Definition 2.4.

For $d > 0$ and $n > d$ , we define $L (n, d)$ to be the set of $T⊂2\Set1,…,n$ satisfying the following two conditions:

$D_{d} (T^{'}) > 0$ for all $T^{'} \subset T$ with $# T^{'} > 1$ .
$0 \leq # T_{i} \leq d$ for all $T_{i} \in T$ .

Moreover we define the partial ordering $<$ on $L (n, d)$ by

(3)

T < T^{'}

⟺ {\begin{matrix} ρ_{d} (T) < ρ_{d} (T^{'}) & and \forall T_{i} \in T, \exists T_{j}^{'} \in T^{'} such that T_{j}^{'} \subset T_{i} . \end{matrix}

Let $P=\Setp1,…,pn$ be a collection of generic points in $V$ in the sense of Section 1. For $X⊂\Set1,…,n$ , $0 \leq # X \leq d$ , define $H_{X}$ to be the affine hull $H\Setpii∈X$ . Since $n > d$ , there exists a subset $X′⊂\Set1,…,n$ such that $X^{'} \cap X = \emptyset$ and $# (X \cup X^{'}) = d + 1$ . Hence

	$H_{X}$	$=H\Setpi1,…,pil$
		$=⋂k∈X′H\Setpii∈X∪X′∖\Setpk∈L(An,d).$

This mapping induces a map from $L (n, d)$ to $L (A_{n, d})$ , or equivalently, $T \in L (n, d)$ corresponds to $H (T) = ⋂_{T_{i} \in T} H_{T_{i}} \in L (A_{n, d})$ . By this correspondence, $L (n, d)$ is isomorphic to $L (A_{n, d})$ as lattices (Athanasiadis (1999), Falk (1994)).

Remark 2.5.

$L (n, d)$ is a graded poset with the rank function $ρ_{d}$ . $∅=\set$ is the minimum element of $L (n, d)$ with $ρ_{d} (\emptyset) = 0$ and $\set∅$ ( $∅⊂\set1,…,n$ ) is the maximum element of $L (n, d)$ with $ρd(\set∅)=d+1$ . In the one-to-one correspondence between $L (n, d)$ and $L (A_{n, d})$ , $H (\emptyset) = V = K^{d}$ and $H(\set∅)=∅$ $(\subset K^{d})$ . In the case $d = 0$ , the condition 2 in Definition 2.4 implies $# T_{i} = 0$ for $T_{i} \in T \in L (n, 0)$ . Hence $L (n, 0)$ is the poset of two elements

L(n,0)=\set∅,\set∅

independent of $n$ .

Let $d$ be a nonnegative integer. We call a weakly-decreasing sequence $δ = (δ_{1}, δ_{2}, \dots)$ of nonnegative integers such that $\sum_{i} δ_{i} = d$ a partition of $d$ . We write $δ ⊢ d$ to say that $δ$ is a partition of $d$ . We also regard a partition as a multiset of positive integers. For example, $\Setδ⊢3=\set(3),(2,1),(1,1,1)$ , and $\Setδ⊢0$ is the set consisting of the unique partition of zero, which is denoted by $(0)$ .

Definition 2.6.

Let $T=\setT1,…,Tl∈L(n,d)$ . Without loss of generality assume $# T_{1} \leq \dots \leq # T_{l}$ . We call

γ_{d} (T) = ({codim}_{d} (T_{1}), \dots, {codim}_{d} (T_{l})) ⊢ ρ_{d} (T)

the type of $T$ .

Example 2.7.

For any $d$ , $γ_{d} (\emptyset) = (0)$ and $γd(\set∅)=(d+1)$ .

Definition 2.8.

For $T \in L (n, d)$ , we define $I_{n, d} (T)$ to be the poset ideal generated by $T$ , i.e., $In,d(T)=\SetS∈L(n,d)S≤T$ .

Finally we define the Möbius function $μ_{n, d}$ of the poset $L (n, d)$ , which will be studied in Section 4. Define $μ_{n, d}$ by

μ_{n, d} (T, T) = 1, \sum S : T \leq S \leq T^{'} μ_{n, d} (T, S) = 0, T < T^{'} .

We write $μ_{n, d} (T) = μ_{n, d} (\emptyset, T)$ . The characteristic polynomial $χ_{n, d} (t)$ of the poset $L (n, d)$ (cf. Section 3.10 of Stanley (1997)) is defined by

(4)

χ_{n, d} (t) = \sum T \in L (n, d) μ_{n, d} (T) t^{d + 1 - ρ_{d} (T)} .

Note that the usual characteristic polynomial $χ (A_{n, d}, t)$ of the non-central arrangement $A_{n, d}$ is given as

χ(An,d,t)=∑T∈L(n,d),T≠\set∅μn,d(T)td−ρd(T)=χn,d(t)−μn,d(\set∅)t.

Conversely from $χ (A_{n, d}, t)$ we can evaluate $μn,d(\set∅)=−χ(An,d,1)$ since $χ_{n, d} (1) = 0$ . Equivalently

(5)

μn,d(\set∅)=−∑T∈L(n,d),T≠\set∅μn,d(T).

2.1. Illustration of the posets up to dimension three

We illustrate the above definitions with $d = 0, \dots, 3$ . For $d = 0$ we already saw $L(n,0)=\set∅,\set∅$ . In particular $μn,0(\set∅)=−1$ .

Let $d = 1$ . In $L (n, 1)$ , in addition to the minimum $\emptyset$ and the maximum $\set∅$ , there are $n$ rank one elements $\Set\seti$ , $i = 1, \dots, n$ , with $μn,1(\Set\seti)=−1$ . Hence $χ (A_{n, 1}, t) = t - n$ . The value $μn,1(\set∅)=n−1$ is relevant for $d > 1$ .

The case $d = 2$ 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 $T=\set\seti,j$ , which is a line connecting points $p_{i}$ and $p_{j}$ . There are two types of points (rank two elements). The first type is an element of type $(2) ⊢ 2$ . Each element $\set\seti$ of type $(2) ⊢ 2$ corresponds to an original point in $P$ . The second type is an element of type $(1, 1) ⊢ 2$ . Each element $T=\set\seti,j,\setk,l$ of type $(1, 1)$ corresponds the intersection of two lines, depicted by a white circle in Figure 1. The Möbius function is evaluated as $μn,2(\set\seti)=n−2$ and $μn,2(\set\seti,j,\setk,l)=1$ .

Remark 2.9.

In this paper we are assuming that $n > d$ so that $A_{n, d}$ is a non-central arrangement. We usually think of $n$ as “sufficiently large” compared to $d$ . Relevant quantities are polynomials in $n$ and these polynomials are determined by sufficiently large $n$ . However our polynomials hold for all $n > d$ with appropriate qualifications. For example, the second type $\set\seti,j,\setk,l$ of $L (n, 2)$ exists if and only if $n \geq 4$ . As long as $n \geq 4$ , $μn,2(\set\seti,j,\setk,l)=1$ . In general, when we write $T \in L (n, d)$ , this $T$ has to exist in $L (n, d)$ . Actually we are interested in the existence of some $T^{'}$ with the same type as $T$ , i.e. $γ_{d} (T^{'}) = γ_{d} (T)$ . The existence implies that $n$ has to be larger than or equal to some specific value, say $n_{γ_{d} (T)}$ , depending on the type of $T$ . As shown in Section 4.2, $n_{γ_{d} (T)}$ is the minimum $n$ such that the number of elements of $L (n, d)$ of the type $γ_{d} (T)$ is positive.

We now count the number of elements of $L (n, 2)$ . This is also needed to evaluate $μn,2(\set∅)$ . There are $(\frac{n}{2})$ lines. There are $n$ points of the first type and

\frac{1}{2} (\frac{n}{2}) (\frac{n - 2}{2}) = 3 (\frac{n}{4})

points of the second type. As discussed in Remark 2.9, this $3 (\frac{n}{4})$ is positive if and only if $n \geq 4$ .

Therefore for $n \geq 3$

	$χ (A_{n, 2}, t)$	$= t^{2} - (\frac{n}{2}) t + 3 (\frac{n}{4}) + n (n - 2)$
		$= t^{2} - (\frac{n}{2}) t + 3 (\frac{n}{4}) + 2 (\frac{n}{2}) - n,$
(6)		$μn,2(\set∅)$	$= - 3 (\frac{n}{4}) - (\frac{n}{2}) + n - 1.$

These quantities are polynomials in $n$ and we prefer to write these polynomials as integer combinations of binomial coefficients $(\frac{n}{k})$ . Note that, in view of Remark 2.9, $(\frac{n}{k}) = 0$ for integer $k > n$ .

We now discuss the case of $d = 3$ .

We first look at rank two elements (lines) of $A_{n, 3}$ . There are two types of elements. The first type is an element of type $(2) ⊢ 2$ . Each element of type $(2)$ , such as $T=\set\set1,2$ , corresponds to the line connecting two points as in the leftmost picture of Figure 2. $\set\set1,2$ is understood as the intersection of all hyperplanes $\set\set1,2,i$ , $i = 3, \dots, n$ . The second type is an element of type $(1, 1) ⊢ 2$ . Each elements of type $(1, 1)$ corresponds to an intersection of two hyperplanes, such as $H(\set\set1,2,3)∩H(\set\set4,5,6)$ . As shown in the rightmost picture in Figure 2, two points ( $p_{3}$ and $p_{4}$ in the picture) may overlap in this case without violating 1 of Definition 2.4. This type of element exists for $n \geq 5$ (cf. Remark 2.9).

Figure 2. Rank two elements for dimension three

Finally we look at rank three elements (points) of $A_{n, 3}$ . We will not repeat remarks on existence of these elements of $L (n, 3)$ . There are three types of rank three elements, corresponding to three partitions of $3$ . Each element $\set\seti$ of the first type $(3) ⊢ 3$ , corresponds to an original point in $P$ . Each element the second type $(2, 1) ⊢ 3$ corresponds to an intersection of a line of type $(2) ⊢ 2$ and a hyperplane, e.g. $H(\set\set1,2)∩H(\set\set3,4,5)$ as shown in Figure 3.

Figure 3. Rank three element for dimension three of type $(2, 1) ⊢ 3$

Figure 4. Rank three elements for dimension three of type $(1, 1, 1) ⊢ 3$

The third type is $(1, 1, 1) ⊢ 3$ , 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)		$μn,3(\set\set1,2)$	$=−μn,3(\set\set1,2,\set3,4,5)=μn−2,1(\set∅)$
		$= n - 3,$
	$μn,3(\set\set1)$	$=μn−1,2(\set∅)$
		$= - 3 (\frac{n - 1}{4}) - (\frac{n - 1}{2}) + n - 2,$

and $μ_{n, 3} (T) = (- 1)^{ρ_{3} (T)}$ for all other $T$ , $T≠\set∅$ .

We need the numbers of elements of $L (n, 3)$ to evaluate $μn,3(\set∅)$ . These are tabulated in Table 1.

(1)	(2)	(1,1)	(3)	(2,1)	(1,1,1)
$(\frac{n}{3})$	$(\frac{n}{2})$	$10 (\frac{n}{6}) + 15 (\frac{n}{5})$	$n$	$10 (\frac{n}{5})$	$280 (\frac{n}{9}) + 840 (\frac{n}{8}) + 630 (\frac{n}{7}) + 120 (\frac{n}{6})$

Table 1. Number of elements for

d = 3

An element of a particular type exists if and only if the number of elements is positive in Table 1. For example, $T$ of type $(1, 1) ⊢ 2$ exists if and only if $10 (\frac{n}{6}) + 15 (\frac{n}{5}) > 0$ , i.e. $n \geq 5$ . From Table 1 and (7) we obtain (for $n \geq 4$ )

	$χ (A_{n, 3}, t)$	$= t^{3} - (\frac{n}{3}) t^{2} + [- (\frac{n}{2}) + 3 (\frac{n}{3}) + 15 (\frac{n}{5}) + 10 (\frac{n}{6})] t$
		$- [n - 2 (\frac{n}{2}) + 3 (\frac{n}{3}) + 35 (\frac{n}{5}) + 180 (\frac{n}{6}) + 630 (\frac{n}{7})$
		$+ 840 (\frac{n}{8}) + 280 (\frac{n}{9})],$
	$μn,3(\set∅)$	$= - 1 + n - (\frac{n}{2}) + (\frac{n}{3}) + 20 (\frac{n}{5}) + 170 (\frac{n}{6}) + 630 (\frac{n}{7})$
		$+ 840 (\frac{n}{8}) + 280 (\frac{n}{9}) .$

3. Main result

In this section we show the following main theorem.

Theorem 3.1.

Let $T \in L (n, d)$ , $T \neq \emptyset$ . Then the ideal $I_{n, d} (T)$ is isomorphic to the direct product $\prod_{T_{i} \in T} I_{n, d} ({T_{i}})$ as posets. They are also isomorphic to $\prod_{T_{i} \in T} L (n - # T_{i}, d - # T_{i})$ .

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

Lemma 3.2.

For $\setT1∈L(n,d)$ , $I_{n, d} ({T_{1}})$ is isomorphic to $L (n - # T_{1}, d - # T_{1})$ as posets.

Proof.

Suppose that $S=\setS1,…,Sl∈In,d(\setT1)$ . Then $\setS1,…,Sl≤\setT1$ , so $S_{i} \supset T_{1}$ , $\forall i$ , by (3). Hence $\setS1∖T1,…,Sl∖T1∈L(n−#T1,d−#T1)$ by Remark 2.3. Therefore we have a map

In,d(\setT1)∋\setS1,…,Sl↦\setS1∖T1,…,Sl∖T1∈L(n−#T1,d−#T1),

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 $T < T^{'} \in L (n, d)$ . For each $T_{i} \in T$ , there uniquely exists $T_{j}^{'} \in T^{'}$ such that $T_{j}^{'} \subset T_{i}$ .

Proof.

It suffices to show the uniqueness. Let $T_{i} \in T$ and $T_{j}^{'}, T_{k}^{'} \in T^{'}$ , $j \neq k$ , satisfy $T_{j}^{'}, T_{k}^{'} \subset T_{i}$ . This means $T_{j}^{'} \cup T_{k}^{'} \subset T_{i}$ . Since $Dd(\SetT′j,T′k)>0$ , by (2), $# T_{i} \geq # (T_{j}^{'} \cup T_{k}^{'}) > d + 1$ . This conflicts with $# T_{i} \leq d$ . ∎

Lemma 3.4.

Let $T=\SetT1,…,Tl$ and $S=\SetS1,…,Sl$ . If $T_{i} \subset S_{i}$ for all $i$ , then $D_{d} (S) \geq D_{d} (T)$ .

Proof.

Let $S_{i}^{'} = S_{i} ∖ T_{i}$ . Then by (1)

	$D_{d} (S) - D_{d} (T)$	$= l \sum i = 1 (# S_{i} - # T_{i}) + # l ⋂ i = 1 T_{i} - # l ⋂ i = 1 S_{i}$
		$= l \sum i = 1 # S_{i}^{'} + # l ⋂ i = 1 T_{i} - # l ⋂ i = 1 S_{i} .$

Since $⋂_{i} S_{i} = ⋂_{i} (S_{i}^{'} \cup T_{i}) = (⋂_{i} T_{i}) \cup (S_{1}^{'} \cap ⋂_{j} S_{j}) \cup (S_{2}^{'} \cap ⋂_{j} S_{j}) \cup \dots \cup (S_{l}^{'} \cap ⋂_{j} S_{j})$ ,

# l ⋂ i = 1 S_{i} \leq # l ⋂ i = 1 T_{i} + # (S_{1}^{'} \cap l ⋂ j = 1 S_{j}) + \dots + # (S_{l}^{'} \cap l ⋂ j = 1 S_{j}) .

This implies

	$D_{d} (S) - D_{d} (T)$	$\geq l \sum i = 1 # S_{i}^{'} + # l ⋂ i = 1 T_{i} - # l ⋂ i = 1 T_{i} - l \sum i = 1 # (S_{i}^{'} \cap l ⋂ j = 1 S_{j})$
		$= l \sum i = 1 # S_{i}^{'} - l \sum i = 1 # (S_{i}^{'} \cap l ⋂ j = 1 S_{j})$
		$= l \sum i = 1 # (S_{i}^{'} ∖ l ⋂ j = 1 S_{j}) .$

Hence $D_{d} (S) - D_{d} (T) \geq 0$ . ∎

Lemma 3.5.

Let $T=\SetT1,…,Tl$ , $S(1)=\SetS(1)1,…,S(1)m1$ , $S(2)=\SetS(2)1,…,S(2)m2$ , …, $S(l)=\SetS(l)1,…,S(l)ml$ , and $S = S^{(1)} \cup \dots \cup S^{(l)}$ . Assume $T_{i} \subset S_{j}^{(i)}$ for all $i, j$ . If $D_{d} (T) > 0$ and $D_{d} (S^{(i)}) > 0$ for all $i$ , then $D_{d} (S) > 0$ .

Proof.

Let $m = \sum_{i = 1}^{l} m_{i}$ . Then

- (m - 1) (d + 1) = - (l - 1) (d + 1) - l \sum i = 1 (m_{i} - 1) (d + 1) .

Hence

	$D_{d} (S)$
	$= - (m - 1) (d + 1) + l \sum i = 1 m_{i} \sum j = 1 # S_{j}^{(i)} - # l ⋂ i = 1 m_{i} ⋂ j = 1 S_{j}^{(i)},$
	$= - (l - 1) (d + 1) - l \sum i = 1 (m_{i} - 1) (d + 1) + l \sum i = 1 m_{i} \sum j = 1 # S_{j}^{(i)} - # l ⋂ i = 1 m_{i} ⋂ j = 1 S_{j}^{(i)}$
	$= l \sum i = 1 (- (m_{i} - 1) (d + 1) + m_{i} \sum j = 1 # S_{j}^{(i)}) - (l - 1) (d + 1) - # l ⋂ i = 1 m_{i} ⋂ j = 1 S_{j}^{(i)} .$

Since $D_{d} (S^{(i)}) + # ⋂_{j = 1}^{m_{i}} S_{j}^{(i)} = - (m_{i} - 1) (d + 1) + \sum_{j = 1}^{m_{i}} # S_{j}^{(i)}$ ,

	$D_{d} (S)$	$= l \sum i = 1 D_{d} (S^{(i)}) - (l - 1) (d + 1) + l \sum i = 1 # m_{i} ⋂ j = 1 S_{j}^{(i)} - # l ⋂ i = 1 m_{i} ⋂ j = 1 S_{j}^{(i)}$
		$=l∑i=1Dd(S(i))+Dd(\Setm1⋂j=1S(1)j,…,ml⋂j=1S(l)j).$

Since $T_{i} \subset ⋂_{j = 1}^{m_{i}} S_{j}^{(i)}$ , it follows from Lemma 3.4 that

D_{d} (S)

\geq l \sum i = 1 D_{d} (S^{(i)}) + D_{d} (T) > 0.

∎

Lemma 3.6.

Let $T \in L (n, d)$ , $T_{1} \in T$ and $T′=T∖\SetT1$ . Then $In,d(\SetT1)×In,d(T′)$ and $I_{n, d} (T)$ are isomorphic as posets.

Proof.

Let $T \in L (n, d)$ , $T_{1} \in T$ and $T′=T∖\SetT1$ . For $(S,S′)∈In,d(\SetT1)×In,d(T′)$ , let us define $φ (S, S^{'}) = S \cup S^{'}$ . Then $φ (S, S^{'}) \in L (n, d)$ by Lemma 3.5. By definition $φ (S, S^{'}) \leq T$ . Hence $φ$ is a map from $In,d(\SetT1)×In,d(T′)$ to $I_{n, d} (T)$ . Moreover, if $(S, S^{'})$ and $(S^{''}, S^{'''})$ satisfy $S \leq S^{''}$ and $S^{'} \leq S^{'''}$ , then $φ (S, S^{'}) \leq φ (S^{''}, S^{'''})$ . On the other hand, we can define the following map $ψ$ from $I_{n, d} (T)$ to $In,d(\SetT1)×In,d(T′)$ :

ψ(S)=(\SetSiT1⊂Si,\SetSiT1⊄Si),

which is the inverse map of $φ$ . Hence $In,d(\SetT1)×In,d(T′)$ and $I_{n, d} (T)$ 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 $L (n, d)$ for $d \leq 6$ . This section is divided into four subsections.

In Section 4.1 we derive an explicit formula for the value of the Möbius function of $L (n, d)$ and show that it only depends on the type of $T \in L (n, d)$ . Next in Section 4.2 we derive a formula for the number of elements of the same type as $T \in L (n, d)$ . 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 $d \leq 6$ .

4.1. Möbius function of the intersection lattice

We first obtain the value of Möbius function of $L (n, d)$ .

Proposition 4.1.

For $T \in L (n, d)$ , $T \neq \emptyset$ ,

μn,d(T)=∏Ti∈Tμn−#Ti,d−#Ti(\Set∅).

Note that for $T = \emptyset$ we have $μ_{n, d} (\emptyset) = 1$ . Also, as discussed at the beginning of Section 2.1, $μn−#Ti,d−#Ti(\Set∅)=−1$ if $d = # T_{i}$ .

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 $P$ and $P^{'}$ be posets, and $P \times P^{'}$ the direct product of posets $P$ and $P^{'}$ . Then $μ_{P} (S, T) \cdot μ_{P^{'}} (S^{'}, T^{'}) = μ_{P \times P^{'}} ((S, S^{'}), (T, T^{'}))$ for $S, T \in P$ and $S^{'}, T^{'} \in P^{'}$ , where $μ$ denotes the Möbius function for each poset.

Proposition 4.1 shows that the Möbius function of $L (n, d)$ is completely determined by the values of $μn+k−d,k(\Set∅)$ , $0 \leq k \leq d$ . In particular for $T≠\Set∅$ , $μ_{n, d} (T)$ is a product of $μn+k−d,k(\Set∅)$ for $k$ smaller than $d$ . As seen in the examples of Section 2.1, $μn′,d′(\set∅)$ is a polynomial in $n^{'}$ . Hence $μ_{n, d} (T)$ , $T≠\set∅$ , can be immediately obtained from $μn′,d′(\set∅)$ for $d^{'} < d$ . Therefore for the recursion on $d$ , the essential step is to compute $μn,d(\set∅)$ 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 $T=\SetT1,…,Tl∈L(n,d)$ and $T′=\SetT′1,…,T′l\allowbreak∈L(n,d′)$ satisfy ${codim}_{d} (T_{i}) = {codim}_{d^{'}} (T_{i}^{'})$ for each $i$ . Define ${¯ μ}_{u, d} (T) = μ_{d + u, d} (T)$ , $u \geq 1$ . Then

{¯ μ}_{u, d} (T) = {¯ μ}_{u, d^{'}} (T^{'}) .

In this sense the value of the Möbius function depends only on the multiset of codimensions, i.e., the type $γ_{d} (T)$ of $T$ . Therefore from now on we denote $μ_{n, d} (T) = μ_{n, d} (γ)$ if $γ_{d} (T) = γ$ .

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 $L (n, d)$ , denoted by $λ_{n, d} (γ)$ below. In this subsection we give an explicit expression for $λ_{n, d} (γ)$ in Proposition 4.7.

Let

λn,d(γ)=#\setT∈L(n,d)γd(T)=γ

denote the number of $T \in L (n, d)$ of type $γ$ . Then (4) and (5) are written as follows.

(8)		$χ_{n, d} (t)$	$=μn,d(\set∅)+d∑i=0∑γ⊢iλn,d(γ)μn,d(γ)td+1−i,$
(9)		$μn,d(\set∅)$	$= - d \sum i = 0 \sum γ ⊢ i λ_{n, d} (γ) μ_{n, d} (γ) .$

For stating Proposition 4.7 we need some more definitions. For a partition $γ = (γ_{1}, \dots, γ_{l})$ of a nonnegative integer let

StabSl(γ)=\Setσ∈Slγi=γσ(i),i=1,…,l

denote the stabilizer of the symmetric group $S_{l}$ fixing $γ$ . Then we have

# {Stab}_{S_{l}} (γ) = \prod k m_{k} (γ)!,

where $m_{k} (γ)$ denotes the multiplicity $#\Setiγi=k$ of $k∈\set1,…,d$ in $γ$ .

Denote the elements of $2\Set1,…,l$ as