Eigenspace arrangements

Eigenspace arrangements of reflection groups

Abstract.

The lattice of intersections of reflecting hyperplanes of a complex reflection group may be considered as the poset of -eigenspaces of the elements of . In this paper we replace with an arbitrary eigenvalue and study the topology and homology representation of the resulting poset. After posing the main question of whether this poset is shellable, we show that all its upper intervals are geometric lattices, and then answer the question in the affirmative for the infinite family of complex reflection groups, and the first 31 of the 34 exceptional groups, by constructing CL-shellings. In addition, we completely determine when these eigenspaces of form a (resp. free) arrangement.

For the symmetric group, we also extend the combinatorial model available for its intersection lattice to all other eigenvalues by introducing balanced partition posets, presented as particular upper order ideals of Dowling lattices, study the representation afforded by the top (co)homology group, and give a simple map to the posets of pointed -divisible partitions.

Key words and phrases:
Cohen-Macaulay, -divisible partitions, Dowling lattices, Eilenberg-MacLane spaces, homology, reflection groups, ribbon representations, Specht modules, subspace arrangements
Partially supported by NSF grant DMS-1001933

1. Main question and results

Let denote an -dimensional -vector space. A reflection in is any non-identity element in of finite order that fixes some hyperplane , and a finite subgroup of is called a reflection group if it is generated by reflections.

Example.

The action of the symmetric group on gives rise to a faithful action on via , where denote the standard basis vectors. Since the transpositions act as reflections and generate the group, this representation realizes as a reflection group in . We shall refer to it as the defining representation.

For an element and root of unity , let denote the -eigenspace of in . Define to be the -poset (partially ordered set) of all such -eigenspaces ordered by reverse inclusion, with -action given by . Choosing recovers the lattice of intersections of reflecting hyperplanes for ; see [29, Lemma 4.4]. The minimal elements of (i.e., inclusion-maximal -eigenspaces of ) are the focus of Springer’s theory of regular elements [35], and each has dimension equal to the number of degrees of that are divisible by , the order of ; see Proposition 3.7 below. When is crystallographic, the poset itself appears in Broué, Malle, and Michel’s -Sylow theory [8].

Example.

The degrees of are , and those of the dihedral group (whose cardinality is ) are . Hence the following table.

for 4 2 1 1 0

for
2 2 0 1 0

When , the poset has only one element, the -dimensional subspace. For all other listed cases, we have provided the Hasse diagram of in Figures 1-5, labeling each eigenspace by linear equations that define it, and adorning maximal eigenspaces with an additional integer label. For example, the maximal eigenspace labeled \⃝raisebox{-0.9pt}{$4\hskip0.426791pt$} in Figure 3 is the -eigenspace for the 3-cycle permutation . Note that the cases for coincide, since the scalar matrix is an element of ; see Corollary 3.8 below.

Figure 1. The poset of -eigenspaces for .
Figure 2. The poset of -eigenspaces for .
Figure 3. The poset of -eigenspaces for any primitive 3rd root of unity and .
Figure 4. The poset of -eigenspaces for any primitive 4rd root of unity and .
Figure 5. The poset of -eigenspaces for the dihedral group of order , and of order .

This paper concerns the following problem of Lehrer and Taylor [25, Problem 7].

Problem (Lehrer-Taylor).

Study connections between the structure and representations of and the topology of the posets .

In the case of , Stanley [38] used the work of Hanlon [17] to obtain an explicit expression for the top homology representation, which combined with Klyachko’s work to establish a connection with the Lie representation ; see [21, 19]. Lehrer and Solomon [22] extended Stanley’s result and conjectured an analogue for all other finite Coxeter groups. Hanlon’s work [18] on Dowling lattices gives an alternate extension and provides the top homology character of for , the complex reflection group of monomial matrices whose nonzero entries are roots of unity. N. Bergeron [1] gave a type- analogue of the abovementioned Lie correspondence that was subsequently generalized to Dowling lattices by Gottlieb and Wachs [16]. In addition to the above, analogous results have been obtained for various subposets of ; see [45]. However, the author is unaware of any analogous results for , even for .

Question A.

Is the following true for every and every reflection group ?
(Weak version) is homotopy Cohen-Macaulay. (Strong version) is CL-shellable.

The fact that CL-shellability implies homotopy CM-ness is well-known; see §2.

Our first main result answers affirmatively the strong version of Question A for all irreducible complex reflection groups except types (which, in the Shephard-Todd classification, are ). Since the question reduces (see §3 below) to the case where acts irreducibly, only these three Weyl groups remain.

Theorem 1.1.

Let or one of the first 31 exceptional groups , and let be a primitive root of unity. Then is CL-shellable. In particular, the order complex is a pure bouquet of spheres.

Our second main result is central to the first.

Theorem 1.2.

Let be a reflection group and let be a primitive root of unity. Then

(1)

In particular, each upper interval in is a geometric lattice.

We will see in Section 3 below that this theorem also has interesting consequences of its own, namely, that depends only on (see Theorem 7.1 for a much sharper result) and is built from copies (conjugates) of the intersection lattice of Lehrer and Springer’s reflection subquotient (see Proposition 3.9).

We also answer the natural and freeness questions for eigenspaces, that is, we determine exactly when the complement of the proper -eigenspaces of a reflection group is , and exactly when the -eigenspaces of codimension 1 form a free (hyperplane) arrangement. For brevity, define to be the set of all proper -eigenspaces of the reflection group that are maximal under inclusion, for a fixed but arbitrary choice of primitive root of unity , so that the complement of the proper -eigenspaces of is

(2)

The two questions were resolved affirmatively for the highly nontrivial case of the reflection arrangement by Bessis and Terao, respectively, and in contrast to Question A, the non-classical case is surprisingly simple.

Theorem 1.3.

Let be a complex reflection group and let .
Then the following are equivalent:

  1. One has or . That is, contains a hyperplane.

  2. The arrangement is a free hyperplane arrangement.

  3. The complement is a space.

Furthermore, if and only if .

The reader should be warned that no new examples of free or arrangements occur in Theorem 1.3. Nevertheless, the result is essentially all one could hope for, and provides a pleasant extension of the classical picture to arbitrary eigenvalues.

Acknowledgements. This work was partly supported by NSF grant DMS-1001933, and forms part of the author’s doctoral work at the University of Minnesota under the supervision of Victor Reiner, whom the author thanks for many helpful conversations. He is also grateful to Gustav Lehrer and Donald Taylor for writing [25], in which they posed the above problem, and to Anders Björner and Volkmar Welker for their helpful comments.

2. Preliminaries

Recall that a -poset is a poset with a -action that preserves order, i.e., implies for all and . It is bounded if it has a unique minimal element (called the bottom element and denoted ) and a unique maximal element (called the top element and denoted ). Write for the poset obtained from by adjoining a new element , regardless of whether has a bottom element. Similarly, is obtained by adjoining a new element . Appending both yields .

The order complex of a poset is the (abstract) simplicial complex consisting of all totally ordered sets in . Because it has a cone point (and is therefore contractible) if has a top or bottom element, one often considers the proper part of the poset, which is simply if neither a top nor bottom element is present.

Recall that a (finite) simplicial complex is -connected if its homotopy groups are trivial for . Define the link of a face to be the subcomplex

and say is homotopy Cohen-Macaulay (abbreviated HCM) if for each face the link is -connected. For a field or , the simplicial complex is said to be Cohen-Macaulay over (abbreviated CM/ or simply CM when ) if for each face the homology groups vanish for . The property of being Cohen-Macaulay is topologically invariant and implies that the complex is homologically a bouquet of ()-spheres, whereas the stronger property of being homotopy Cohen-Macaulay is not topologically invariant but implies that the complex is homotopically a bouquet of -spheres; see [27] and [32, p. 117], respectively, or surveys [4, 45].

Though there are many techniques for establishing Cohen-Macaulayness, we shall be concerned with (pure) CL-shellability. A simplicial complex which is pure -dimensional (i.e., each maximal face under inclusion has dimension ) is said to be shellable if its maximal faces (called facets) can be ordered so that for each the subcomplex generated by the first facets intersects the st facet in a pure -dimensional subcomplex. A poset is called shellable (resp. HCM, CM) if its order complex is shellable (resp. HCM, CM). Finally, a CL-shellable1 poset is a bounded poset that admits a recursive atom ordering, as defined in Section 4. The following implications for poset shellability are strict:

Note that a poset is shellable (resp. HCM, CM) if and only if , or just , is shellable (resp. HCM, CM).

3. General reductions

Shephard and Todd classified all irreducible reflection groups in [34]. There are 34 exceptional groups in their classification, labeled , and 3 infinite families, explained below:

  • .

  • with , a divisor of , and .

  • .

denotes the representation of one obtains from the defining representation of §1 after modding out by the fixed space . For the set of roots of unity (and a divisor or ), the group consists of all monomial matrices with nonzero entries in whose product lies in . General reflection groups decompose into irreducible ones as follows.

Proposition 3.1 (Theorem 1.27 in [25]).

Let be a reflection group. Let denote the nontrivial irreducible submodules of , so that the restriction of to is irreducible. Then and one has an orthogonal sum decomposition with respect to a chosen -invariant positive definite Hermitian form.

Corollary 3.2.

Maintain the notation of Proposition 3.1 and let be a root of unity. Then

(3)

In particular, the family of posets obtained by letting vary over the three infinite families above is the same as that obtained by letting vary over the single infinite family . Another consequence is that Question A and Theorem 1.2 reduce to irreducible reflection groups.

Corollary 3.3.

Maintain the notation of Corollary 3.2. Then the following hold.

  1. is HCM (resp. CM) if and only if each is HCM (resp. CM).

  2. is CL-shellable if and only if each is CL-shellable.

Proof.

Note that each has a top element . Then (i) follows from (3) and a homeomorphism of Quillen (see [32, Ex. 8.1] and [46, Thm. 5.1(b)]), while (ii) follows from (3) and [6, Thm. 10.16]. ∎

Because of the important role that maximal eigenspaces play in what follows, we make the following convention before proceeding.

Convention 3.4.

A maximal -eigenspace for is one that is not properly contained in any other. Because such a space is minimal with respect to the poset order of given by reverse inclusion, in order to avoid confusion we shall always take minimal and maximal to be with respect to inclusion when dealing with subspaces. For example, “ maximal” thus means that is not properly contained in any .

A reflection group acts on the algebra of polynomial functions via , and Shephard and Todd showed that the subalgebra of -fixed polynomials is again polynomial, generated by algebraically independent homogeneous polynomials , called basic invariants, the degrees of which are uniquely determined by the group and denoted . We shall always assume an indexing such that . For , write for the number of divisible by as in §1; see Springer [35].

Proposition 3.5 (Springer).

Let be a reflection group, let be a primitive root of unity, and let be a set of basic invariants for . Set . Then one has .

Consequentially, the collection of maximal -eigenspaces of depends only on (the group and) the multiset of degrees that are divisible by , which we shall denote by , so that ; see Theorem 7.1 below for a much sharper result.

Corollary 3.6.

Let be a reflection group. Let and be roots of unity of orders and such that . Then the set of maximal -eigenspaces of coincides with the set of maximal -eigenspaces of .

Proposition 3.7 (Springer).

Let be a reflection group, let be a primitive root of unity, and let be maximal eigenspaces. Then

  1. , and

  2. for some .

Corollary 3.8.

Let be a reflection group and let be a primitive root of unity. Then the following are equivalent.

  1. .

  2. .

  3. .

Proof.

Clearly (i)(ii)(iii). Assume so that , and hence . Writing for each , it follows that . ∎

Theorem 3.9 (Lehrer-Springer [23, 24]).

Let be a reflection group, let be a primitive root of unity, and let be maximal with normalizer

and centralizer

Then acts as a reflection group on , and the following hold:

  1. If form a set of basic invariants for , then the restrictions of those whose degree is divisible by form a set of basic invariants for .

  2. The reflecting hyperplanes of on are the intersections of with the reflecting hyperplanes of that do not contain .

  3. If is irreducible, then acts irreducibly on .

  4. is uniquely determined by and , up to conjugation by .

Proposition 3.10.

Let be a reflection group and let be a root of unity. Then one has an inclusion

(4)

and the following are equivalent.

  1. Equality in (4).

  2. There exists maximal such that for all one has
    .

  3. For every maximal and one has .

  4. There exists maximal such that .

  5. For every maximal, one has

Proof.

For the inclusion (4), let and choose a maximal -eigenspace such that . Write and for some . Then if and only if , i.e., if and only if . Hence for some . As for the equivalences, clearly (i) is equivalent to (iii), which is equivalent to (v) by Theorem 3.9(ii), and the remaining two equivalences (ii)(iii) and (iv)(v) follow from Proposition 3.7(ii). ∎

4. case of Theorem 1.2

Recall that denotes the collection of all roots of unity, and that denotes the group of all monomial matrices with nonzero entries in whose product lies in . Note that is the defining representation of given in §1, and that . When the set of reflecting hyperplanes for coincides with that for and is given by the union of the following two sets:

(5)
(6)

When the set of reflecting hyperplanes for is simply given by (5).

For roots of unity and an -set , identify the -line array

with the linear map that fixes each with and that sends to for , while . Because multiple arrays may represent the same map, in the next section we will require that , but we postpone the restriction until then. Call such an element a (colored) cycle, and define

  • (the length of ),

  • (the support of ), and

  • (the multiset of colors of ).

With two cycles said to be disjoint if , note that any element of may be decomposed as a product of disjoint cycles, and that such a product is an element of if and only if is an element of . The following lemma is a straightforward calculation.

Lemma 4.1.

Let be a cycle and write .
Let be a root of unity. Then

Moreover, in the former case is the solution set of the following equations:

(7)
(8)

The crux of Theorem 1.2 is Proposition 4.3 below, for which we will need the following.

Lemma 4.2.

Let , let , and let be a primitive root of unity. Suppose that and that . Then there exists an such that for all one has .

Proof.

It suffices to assume that is maximal. Since , we have that contains

which must have by Proposition 3.7(i), and therefore for each . Since acts transitively on its maximal -eigenspaces by Proposition 3.7(ii), the result follows. ∎

In the next proposition we define another group within the family, that contains . In particular, . We do so to obtain a stronger version of Theorem 1.2 (Theorem 4.4 below) which will be used in Sections 8 and 9.

Proposition 4.3.

Let , let be a primitive root of unity, and let denote the least common multiple of and . Set

Then for every and .

Proof.

If , then one has , and the result follows.

Assume that . It suffices to show that for every and every reflecting hyperplane of , since a general element of is an intersection of reflecting hyperplanes. Let and let be a reflection with fixed space . We show that is a -eigenspace of by exhibiting an such that .

Set if . Assume otherwise so that . Write as a maximal product of nonempty disjoint cycles so that and , and define . Since is of the form or , we see from Lemma 4.1 that it contains all but exactly one or exactly two of the eigenspaces .

Case 1. There exists exactly one cycle such that .

Subcase 1a. .

Write and let be the element of that one obtains from by replacing with the product of the cycles

Then if and only if . Clearly , since . Suppose that . Since also, Lemma 4.1 tells us that . However, .

For example, if and , then for

one has .

Subcase 1b. .

Then for some , implying that , and so and . Applying Lemmas 4.1 and 4.2, it follows that for some . Note that necessarily , as their -eigenspaces disagree. Write

and obtain from by replacing the two cycles with the single cycle

Applying Lemma 4.1 shows that ; indeed, one has , and so . It follows that . That is in is clear.

For example, if and , then for and , one has .

Case 2. There exist exactly two cycles such that .

Then for some and a suitable indexing, we have

Since is given by , Lemma 4.1 implies that consists of the points that satisfy for and

(9)

Let be such that the coefficient of in (9) is an element of . (For existence, note that the cosets cover the group , since generates and , then observe that permutes these cosets, since from basic algebra.) Then

(10)

for the reflecting hyperplane of .

We claim that satisfies , or in other words, that . To see this, employ (10) to rewrite the equality as

(11)

Now observe that, on one hand, is clearly contained in and has dimension by hypothesis. On the other hand, is necessarily a cycle, and therefore has -eigenspace of dimension at most by Lemma 4.1.

For example, if and , then for and

one has ,  . ∎

Proof of Theorem 1.2 for .

Note that in Proposition 4.3, since is a subgroup of , then invoke Proposition 3.10. ∎

Another consequence of Proposition 4.3 is the following stronger result, which will play an important role in Sections 8 and 9 below.

Theorem 4.4.

Let , let be a primitive root of unity, and let be as in Proposition 4.3. Then

  1. , and

  2. for every and .

In other words, is an upper order ideal of .

Proof.

Observe that contains the scalar matrix , and that

whenever . The first claim follows, and the second is Proposition 4.3. ∎

5. Maximal eigenspaces of

In this section we associate a certain word, denoted , to each maximal eigenspace for and . In the next section we show that lexicographically ordering these words gives a recursive atom ordering for . In addition to , the following number plays an important role in our discussion.

(12)
(13)

The crux of our construction is that each maximal eigenspace determines a unique set of many -cycles in whose product has -eigenspace . It is from this set that we construct in Corollary 5.4 below. We establish the correspondence by first showing that any product of many nontrivial -cycles is uniquely determined by its eigenspace , and then showing that each maximal may be realized as the -eigenspace of such a product.

Recall from Section 4 the identification of a 2-line array of the form

(14)

and a particular element of , and note that the element determines the 2-line array up to cyclically permuting columns. Thus, by requiring that the smallest come first, the array is uniquely determined. We adopt this convention for the remainder of the section, i.e., that .

Lemma 5.1.

Let be a primitive root of unity for , and set . Suppose that are two -cycles such that . Then .

Proof.

We show that the map is a bijection when restricted to the -cycles such that by constructing its inverse. Fix such a cycle and label its entries as in (14) so that its image is defined by equations (7) and (8) of Lemma 4.1. Working backwards, first note that