Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are boolean forms a simplicial poset under the Bruhat order. This simplicial poset defines a cell complex, called the boolean complex. In this paper it is shown that, for any Coxeter system of rank , the boolean complex is homotopy equivalent to a wedge of -dimensional spheres. The number of such spheres can be computed recursively from the unlabeled Coxeter graph, and defines a new graph invariant called the boolean number. Specific calculations of the boolean number are given for all finite and affine irreducible Coxeter systems, as well as for systems with graphs that are disconnected, complete, or stars. One implication of these results is that the boolean complex is contractible if and only if a generator of the Coxeter system is in the center of the group.
Keywords. Coxeter system; Bruhat order; boolean; boolean number; homotopy; cell complex; discrete Morse theory
2000 Mathematics Subject Classification:Primary 20F55; Secondary 05E15, 06A07, 55P15
The boolean complex of a finitely generated Coxeter system arises from the Bruhat order on . Regarding as a poset in the Bruhat order, we define the boolean ideal to be the subposet consisting of those elements whose principal (lower) order ideals are boolean. The boolean ideal is a simplicial poset, and, as the name suggests, it is an order ideal of . In fact, it is maximal among order ideals that are simplicial posets. The boolean complex is defined as the regular cell complex whose face poset is the (simplicial) poset .
The elements in are easily described: an element in is an element of that can be written as a product of distinct elements from the generating set . Consequently the boolean complex is pure, with each maximal cell having dimension . These elements play an important role in the study of Coxeter groups because their boolean nature has a variety of consequences related to -polynomials, Kazhdan-Lusztig polynomials, and -polynomials (see [brenti]).
As described above, elements of the boolean ideal are products of distinct elements of the generating set , and thus are governed by the commutativity of elements of . Consequently, this ideal is determined by the Coxeter graph of . Recall that the Coxeter graph has vertex set , with an edge between vertices and if and only if and do not commute in . An edge is labeled by the order of when . Since we are only concerned with the commutativity of generators in this paper, we suppress the labels and consider the underlying unlabeled graph. Because elements of are involutions, the elements and commute if and only if , and hence if and only if the vertices and are non-adjacent in . For more information about Coxeter systems, see [bjornerbrenti].
From the graph , one constructs a simplicial poset whose elements are equivalence classes of strings of distinct elements of , where two strings are equivalent if one can be transformed into the other by commuting elements that are non-adjacent in . The partial order on is induced by substring inclusion. Of course, when , the poset is isomorphic to the poset , so this construction recovers the boolean ideal from the (unlabeled) Coxeter graph. We refer to as the boolean ideal of , and to the associated regular cell complex as the boolean complex of .
If is the complete graph, then is the complex of injective words, which has previously been studied by Farmer [farmer], Björner and Wachs [bjornerwachs], and Reiner and Webb [reiner-webb]. The complete graph is treated in Corollary LABEL:cor:complete.
If the graph consists of two vertices and a single edge between them, then the poset and the boolean complex are depicted in Figure 1.
Because their unlabeled Coxeter graphs are the same, Example 1.1 applies to the Coxeter groups and , and shows that in each case the geometric realization of the boolean complex is homotopy equivalent to the unit circle . In this paper, we prove more generally that the boolean complex of any finite simple graph, and hence of any Coxeter system, has geometric realization homotopy equivalent to a wedge of top dimensional spheres, and give a recursive formula for calculating the number of these spheres. In specific cases, including the finite and affine irreducible Coxeter systems, we calculate this number explicitly.
The subsequent organization of this paper begins with a section precisely defining the primary objects and putting this project in the greater context of the study of the Bruhat order of Coxeter group. Section 3 states the main result of the article, that the boolean complex for any finite simple graph has geometric realization homotopy equivalent to a wedge of a particular number of top dimensional spheres. The homotopy types for the boolean complexes of the finite and affine irreducible Coxeter systems are given as a corollary in this section. Section 4 discusses discrete Morse theory, which is the main tool in the proof presented in Section 5. A selection of corollaries to the main theorem are given in Section LABEL:section:corollaries. Section LABEL:sec:Homology contains a discussion on how generating cycles for the homology of Boolean complex enumerate the spheres occurring in the wedge sum representing the homotopy type of its geometric realization. The paper concludes with suggestions for follow-up questions in Section LABEL:section:follow-up.
2. Motivation and definitions
As noted above, the motivation for this work is the study of Coxeter systems and Coxeter graphs, and the importance of their boolean elements. Henceforth, all Coxeter systems are assumed to be finitely generated.
We use standard poset terminology throughout this paper, and refer the reader to [ec1] for more background.
Let be a group defined on generators . The pair is a Coxeter system if the relations in are of the form for all , and for and .
Because consists of involutions, two elements commute if and only if . The involution condition also implies that . Since generates , any can be written as a word on letters in . That is, admits an expression , where . The product is a reduced expression for if it is of minimal length , in which case is the length of .
For a Coxeter system , the (strong) Bruhat order is the partial order on where if and only if admits a reduced expression that is a subword of a reduced expression of .
The Bruhat order makes a ranked poset, with rank determined by length. Because the minimal element in a simplicial poset corresponds to the empty face in the geometric realization of that poset, we make the the convention that this minimal element has rank , thus emphasizing that the face data in the poset is contained in the non-negative ranks.
The structure of the Bruhat order for finite Coxeter groups was studied by the second author in [tenner]. One aspect of this study was a description of elements with boolean principal order ideals.
Let be a Coxeter system and regard as a poset under the Bruhat order. An element is boolean if its principal (lower) order ideal in is isomorphic to a boolean algebra. The boolean ideal is the subposet of boolean elements.
It is clear from the definition that is an order ideal in the Bruhat order, thus justifying the terminology. It is also clear, by construction, that is a simplicial poset.
Because the boolean poset is simplicial, the interval between any two comparable elements is boolean and, in particular, a lattice. By [brenti], this has several implications for the Kazhdan-Lusztig polynomials, -polynomials, and -polynomials. For example, for any with , we have
where is the Kazhdan-Lusztig polynomial, and
where is the length function. Further structural and computational consequences can be found in [brenti].
The results in [tenner] state that for the finite Coxeter groups of types , , and , boolean elements can be characterized by pattern avoidance. Moreover, the boolean elements of these groups are enumerated by length. For example, the number of boolean elements of length in the finite Coxeter group is
As the unlabeled Coxeter graphs for and are identical, the boolean elements of length in are also enumerated by (1). For the group , the enumeration is more complicated, and a recursive formula is given in [tenner].
The following lemma is immediate from the description of the Bruhat order above, and gives a useful characterization of boolean elements.
Let be a Coxeter system. An element of is boolean if and only if it has no repeated letters in its reduced expressions.
It follows from the lemma that every maximal element in has the same rank, equal to .
Let be a Coxeter system. The boolean complex of is the regular cell complex whose face poset is the simplicial poset .
The existence of such a complex follows from a well-known result about simplicial posets, and in fact about CW-posets (see [bjorner]).
The minimal element of represents the empty cell, and an element of rank represents a -dimensional cell (see Remark 2.3). One can think of the cells in as simplices, because the minimal subcomplex containing each cell is isomorphic to a simplex of the same dimension. Nevertheless, the boolean complex itself is not a simplicial complex because the cells are not determined by the vertices they contain; for instance, there are two 1-cells, and , with the same vertices in Example 1.1.
One obtains a geometric realization of the boolean complex in the standard way, by taking one geometric simplex of dimension for each cell of dimension , and gluing them together according to the face poset. The homotopy type of a complex is understood to mean the homotopy type of its geometric realization.
The main result of this paper, Theorem 3.4, shows that has the homotopy type of a wedge of spheres of dimension . Moreover, we give a recursive formula for computing the number of spheres in the wedge. To describe this recursion, we present an alternative construction of the boolean complex, in terms of the unlabeled Coxeter graph of .
The Coxeter graph of a Coxeter system has vertex set and an edge between and if and only if . An edge corresponding to is labeled by . The unlabeled Coxeter graph is the underlying simple graph obtained by omitting all edge labels.
A Coxeter system can be recovered from its Coxeter graph . Taking the vertex set of to be , one forms the group generated by subject to the relations mandated by the edges in , and the condition that should consist of involutions. An unlabeled Coxeter graph, however, contains less information, and only allows one to determine when two elements in commute. Thus, if one is only concerned with commutativity of generators, this graph suffices.
Let be the group generated by , with relations for all , and , . The (unlabeled) Coxeter graph is shown below.
For a finite simple graph with vertex set , define the poset as follows. First, let be the set of words on with no repeated letters, ordered by the subword order relation. A typical element in is thus of the form , where are distinct elements of . Next, consider the equivalence relation generated by the requirement that
if is not an edge in . Let be the set of equivalence classes of with respect to this equivalence relation. A preimage of an element is called a word representative. Note that the set of letters occurring in each word representative of is the same. We say that contains a letter if occurs in the string representatives of . A partial ordering is induced on the set from the subword order. That is, in if some word representative of is a subword of a word representative of .
The boolean ideal of a finite simple graph is the poset .
The motivation for the construction of is of course the following obvious fact, which we record as a lemma.
If is the unlabeled Coxeter graph of a Coxeter system , then .
It can be shown directly, or via the relationship between graphs and Coxeter systems, that is a simplicial poset for all finite simple graphs .
The boolean complex of a finite simple graph is the regular cell complex associated to .
The main result of this paper, as mentioned above, can be equivalently stated as saying that for any finite simple graph with vertex set , the geometric realization is homotopy equivalent to a wedge of -dimensional spheres. It is this version of the result that we shall prove. The promised recursive formula for the number of spheres is given in terms of basic graph operations. These results are stated precisely in Theorem 3.4.
The Euler characteristic of a regular cell complex , and likewise the Euler characteristic of its geometric realization , is the alternating sum of the number of faces of each rank in :
Given Remark 2.3, this can also be computed by enumerating each non-negative rank in the corresponding simplicial poset. In particular, the enumeration from [tenner] cited in (1) enables the calculation of the Euler characteristic of the boolean complex for the Coxeter group .
For all ,
where are the Fibonacci numbers.
Note the relationship between and sequences A008346 and A119282 in [oeis]: sequence A008346 is equal to , while sequence A119282 is equal to . Also, Corollary 2.12 foreshadows the fact that is homotopy equivalent to the wedge of -spheres.
Before stating the main results precisely, it is informative to mention similar work which has been done for the independence complex. Ehrenborg and Hetyei [ehrenborg] and Kozlov [kozlov] prove, each in the context of different results and frameworks, that the complex of sparse sets of is contractible in some cases and homotopy equivalent to a sphere in the remaining cases. In the context of the Bruhat order, the sparse subsets of correspond to the fully commutative elements in the Coxeter group . That is, all letters in such a reduced expression commute with each other. Thus, these results show that the complex formed from the subposet of consisting of the fully commutable elements is either contractible or homotopy equivalent to a sphere.
3. Main results
In this section, we state the main result of the article and draw consequences for the classical Coxeter groups. The proof of the main theorem is rather technical, and is postponed until Section 5.
It is convenient to use the notation
for a wedge sum of spheres of dimension . Since the wedge sum is the coproduct in the category of pointed spaces, then denotes a single point.
Graph-theoretic notation will also be used in the statement of the theorem and in its proof.
For a finite graph , let denote the number of vertices in .
Let be a finite simple graph and an edge in .
Deletion: is the graph obtained by deleting the edge .
Simple contraction: is the graph obtained by contracting the edge and then removing all loops and redundant edges.
Extraction: is the graph obtained by removing the edge and its incident vertices.
For , let be the graph consisting of disconnected vertices.
We can now state the main theorem of this paper. We use the symbol to denote homotopy equivalence.
For every nonempty, finite simple graph , there is an integer so that
Moreover, the values can be computed using the recursive formula
if is an edge in such that is nonempty, with initial conditions
where is the graph with two vertices and one edge.
The integer will be called the boolean number of the graph . Notice that by formally setting , the boolean number can be extended so that the recursive formula in equation (2) holds for .
As discussed in Section 2, the above theorem implies that the geometric realization of the boolean complex of a Coxeter system is homotopy equivalent to a wedge of spheres of dimension . The number of spheres occurring in the wedge can be calculated recursively using equation (2). This process can be greatly expedited by the following proposition, which shows that the boolean number is multiplicative with respect to connected components.
If for graphs and , then
where denotes simplicial join, and consequently
In particular, .
Since and are disjoint, every element of commutes with every element of . Thus, the complex is formed by taking the simplicial join of the complexes and , and hence the geometric realization is the topological join of and (see [hatcher]). The last claim now follows from Theorem 3.4. ∎
The homotopy types of the boolean complexes associated to the finite and affine irreducible Coxeter systems can be calculated as a corollary to Theorem 3.4.
The homotopy types of the boolean complexes for the finite and affine irreducible Coxeter systems are listed below, where is the sequence of Fibonacci numbers (with ) and is the sequence defined by and .
The sequence is entry A014739 in [oeis]. It can be written in closed form as
4. Discrete Morse theory
The primary tool in the proof of Theorem 3.4 is discrete Morse theory, which gives an expedient way to analyze the homotopy type of the geometric realization of a regular cell complex through combinatorial properties of its face poset. Discrete Morse theory is a rich subject, and in this section we present only those ideas and results necessary for our argument. The reader is encouraged to read [forman-complexes, forman-user] for a detailed background.
Let be a ranked poset. A matching on is a collection of pairs where is a covering relation in , and each element of occurs in at most one pair in . If , then is a matched edge in . If occurs in a matched edge in , then is matched. Otherwise, is unmatched. In the case of a simplicial poset , we require that a matching leaves the minimal element unmatched.
Let be a ranked poset, and let be a matching on . Consider the Hasse diagram of as a directed graph, with an edge if . Reverse the direction of each edge in the Hasse diagram which corresponds to a matched edge in . Let be the resulting directed graph. The matching is acyclic if there are no directed cycles in . In the Hasse diagram for , those edges whose directions have been changed will be said to point up, while unchanged edges point down.
Discrete Morse theory allows one to reduce regular cell complexes without changing the homotopy type of their geometric realizations. Roughly speaking, when is a regular cell complex with face poset , and is an acyclic matching on , one can collapse cells along matched edges in without changing the homotopy type of . We will apply this method to regular cell complexes associated to simplicial posets.
The convention that the minimal element of a simplicial poset be left unmatched circumvents a minor technical issue, as the minimal element in a simplicial poset is represented by the empty cell in the associated cell complex and thus plays no role in the geometric realization. Indeed the minimal element will henceforth be ignored. This convention is also taken in [forman-complexes, forman-user], and it should bring to mind Remark 2.3.
The particular result which we will use to analyze the boolean complex is stated below.
Theorem 4.3 (see [forman-complexes, forman-user]).
Let be a simplicial poset, and let be an acyclic matching on . For each , let denote the number of elements of rank that are unmatched. Then the geometric realization of the regular cell complex associated to is homotopy equivalent to a CW-complex with exactly cells of dimension for each .
Theorem 3.4 will be proved inductively by constructing an acyclic matching of the simplicial poset with all elements of non-negative rank matched except for one element of rank 0 and some number, which we will denote by , of maximal elements. There are two important points to make about such a matching, summarized in the following remark.
If is an acyclic matching on a simplicial poset , and the only elements on non-negative rank unmatched by are one element of rank and elements of rank , then the geometric realization of the regular cell complex associated to is homotopy equivalent to . Also, note that the number is determined by the homotopy type of the cell complex with face poset , and is therefore independent of the matching . In fact, the number is determined by the Euler characteristic of and the formula
The following lemma is useful for proving the acyclicity of matchings.
Consider a matching on a ranked poset . If has a cycle, then the elements in the cycle lie in two adjacent ranks of .
Because is a matching, there cannot be two incident upward pointing edges in . Thus, after moving upward, one must move downward at least once before moving upward again. So if is an element in a directed cycle, then no element in the cycle can be more than one rank higher than in . ∎
5. Proof of Theorem 3.4
In this section we prove Theorem 3.4 by constructing, for each nonempty finite simple graph , an acyclic matching of where the only unmatched elements of non-negative rank are one element of rank and some number of maximal elements. As noted in Remark 4.4, this implies that is homotopy equivalent to a wedge of spheres of maximal dimension. Furthermore, the number of such spheres, the boolean number , is determined by the Euler characteristic of , prompting the next definition.
For a nonempty finite simple graph on vertices, set
Although this definition gives a way to calculate the value in Theorem 3.4, computing the Euler characteristic requires knowing a significant amount about the structure of the poset , as opposed to the recursive formula in equation (2) which requires only basic graph operations.
We construct the matchings by induction on the number of edges in . The inductive step is somewhat complicated, and we will in fact produce matchings with more specific properties than are actually needed for the desired conclusion. The inductive hypothesis is stated below, after introducing the following notation.
For a nonempty finite simple graph and a vertex in , let be the subposet of elements containing the letter .
The goal of this section is to show inductively that the following statement holds for every integer .
Inductive Hypothesis ().
For every nonempty graph with at most edges, and for every vertex in , there exists an acyclic matching on the poset with the following properties:
The only unmatched elements in of non-negative rank are one element of rank 0 and maximal elements;
If is nonempty, then, in the restriction of to the subposet , the only unmatched elements are maximal elements; and
If is a matched edge in , and contains , then there exist word representatives for and such that one of the following conditions holds:
is obtained from by deleting a letter appearing to the left of .
We refer to a matching with properties – as an -matching of at , or just as an -matching. Notice that the condition in implies that is not maximal in and must therefore be matched. When is a graph with a single vertex , an -matching of is obtained trivially. Property can be extended to this trivial -matching if we formally define .
For a nonempty finite simple graph , the maximal elements of are products of letters. Therefore, if has an -matching, then Theorem 4.3 and imply that . Property is needed to preserve for the inductive step, and also to prove the recursive formula in Theorem 3.4. Property is needed purely for the purposes of the induction, specifically, to prove acyclicity of the constructed matching.
The next lemma establishes the base case of the induction.
Lemma 5.3 (Base case).
holds, and for all .
We show that for each vertex , there exists an -matching at . Let the vertices of be labeled . Without loss of generality, suppose that the vertex is . Let be the matching consisting of all covering relations of the form for . First notice that the only unmatched element of non-negative rank in is the vertex , so has property , with . Secondly, there are no unmatched elements in , unless , proving . Property follows from the fact that the Coxeter group with graph is commutative, so the letters in and can be permuted at will.
It remains to show that is acyclic. This is straightforward, due to the fact that all the matched edges represent adding or removing the letter from an element in the poset. Recall from Lemma 4.5 that a cycle is contained in two adjacent ranks of , so an “up” edge would have to be followed by a “down” edge . The next step must be an up edge , so cannot contain . However, this implies that , contradicting the directions of the edges. Therefore is acyclic. ∎
The remainder of the section is devoted to proving the inductive step. Henceforth we assume that holds for some and consider a graph with edges. We show that for an arbitrary vertex , there exists an -matching of at . If is the endpoint of an edge , then we construct in Lemma LABEL:lem:Edge the -matching of from -matchings of and , which exist by the induction hypothesis. The case when is an isolated vertex is treated in Lemma LABEL:lem:vIsolated by constructing the required matching from an -matching of .
5.1. Considering a non-isolated vertex
Suppose that is the endpoint of an edge , and set . By the induction hypothesis, there exists an -matching of at . We want to use this matching to produce an -matching of at . To this end, we first compare the complexes and . The complex is obtained from by identifying all elements that can be written as with the elements represented by , respectively. Consequently, there is a canonical projection of posets
Now, let be the subset of elements that can be written in the form ; that is, consists of those elements in which both and appear, and where it is possible to write immediately to the left of . Let be the set complement of in . This gives a decomposition of sets
where denotes the disjoint union. However, this is not a decomposition of posets, as there are covering relations between and , as recorded in the following lemma. The proof of the lemma is not difficult, and is left to the reader.
If is a covering relation in such that and , then is obtained from by deleting either or .
If is a covering relation in such that and , then can be written as , where is obtained from by removing a letter between and , and the vertices form a path in the graph .
It is easy to see that restricts to a bijective order-preserving map
However this is not a bijection of posets, as can have more covering relations than , corresponding to elements of covering elements of . The following lemma nevertheless allows us to pull back the -matching of at to a matching on along the map .
If and are elements in such that is a matched edge in , then is a covering relation in .
By Lemma 5.4, if is a covering relation in but is not a covering relation in , then can be written in the form , where the vertices form a path in the graph , and is obtained by deleting a letter between and . Property prohibits the covering relation from being a matched edge in this case. ∎
Lemma 5.5 allows us to define a matching on by declaring to be a matched edge in if and only if is a matched edge in . Property for implies that the elements of non-negative rank in that are unmatched in are exactly one element of rank and maximal elements. This matching constitutes a part of our -matching of at , and what remains is to produce matched edges among the elements in .
Let denote the graph with denoting the new vertex representing this edge. Then has at most edges, so there is an -matching of at by the induction hypothesis. The following lemma allows us to pull the restriction of to back to a matching on .
There is an isomorphism of posets
The proof mainly consists of checking that is a well-defined, order-preserving map. The same reasoning then gives an inverse to . This is left to the reader.
We now define a matching on by declaring to be a matched edge in if and only if is a matched edge in . The following example illustrates this procedure for the graph considered in Example 1.1.
Let be the graph with two vertices, and , and one edge between them. Then is the graph with two disconnected vertices and , and is the graph on a single vertex . The picture below shows a Hasse diagram for the poset , excluding the empty word, illustrating how decomposes into parts and , and how these parts can be related to and , respectively.