Michael Carr and Satyan L. Devadoss- Coxeter Complexes and Graph-Associahedra

8/3/2019 Michael Carr and Satyan L. Devadoss- Coxeter Complexes and Graph-Associahedra

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COXETER COMPLEXES AND GRAPH-ASSOCIAHEDRA

MICHAEL CARR AND SATYAN L. DEVADOSS

Abstract. Given a graph , we construct a simple, convex polytope, dubbed graph-associahedra , whose face poset is based on the connected subgraphs of . This providesa natural generalization of the Stasheff associahedron and the Bott-Taubes cyclohedron.Moreover, we show that for any simplicial Coxeter system, the minimal blow-ups of its associated Coxeter complex has a tiling by graph-associahedra. The geometric andcombinatorial properties of the complex as well as of the polyhedra are given. Thesespaces are natural generalizations of the Deligne-Knudsen-Mumford compactication of the real moduli space of curves.

1. Introduction

The Deligne-Knudsen-Mumford compactication of the real moduli space of curvesM 0,n (R ) appears in many areas, from operads [8, 16] , to combinatorics [ 10, 13], to grouptheory [5, 6]. One reason for this is an intrinsic tiling of M 0,n (R ) by the associahedron ,the Stasheff polytope [15]. The motivation for this work comes from a remarkable fact,rst noticed by Kapranov, involving Coxeter complexes: Blowing up certain faces of theCoxeter complex of type A yields a double cover of M 0,n (R ). Extending this to the Coxetercomplex of affine type A results in a moduli space tessellated by the cyclohedron [9], the

Bott-Taubes polytope associated to knot invariants [ 2]. Davis et al. have shown these spacesto be aspherical, where all the homotopy properties are completely encapsulated in theirfundamental groups [5]. This paper looks at analogues of M 0,n (R ) for all simplicial Coxetergroups W , which we denote as C(W )# .

Section 2 begins with the study of graph-associahedra. For any graph , we construct asimple, convex polytope whose face poset is based on the connected subgraphs of (Theo-rem 2.6). This provides a natural generalization of the associahedron and the cyclohedron.Some combinatorial properties of this polytope are also explored (Theorem 2.9).

Section 3 provides the background of Coxeter complexes and proves that graph-associa-hedra tile C(W )# (Theorem 3.7). A gluing map of these polytopes is also provided (Theo-rem 3.8). Section 4 nishes by looking at the geometry of C(W )

#. In particular, we show

that each blown-up cell of C(W )# resolves into a product of lower-dimensional blown-upCoxeter complexes (Theorem 4.2).

1991 Mathematics Subject Classication. Primary 14P25, Secondary 05B45, 52B11.Key words and phrases. Coxeter complexes, graph-associahedra, minimal blow-ups.Authors were partially supported by NSF grants DMS-9820570 and CARGO DMS-0310354.

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2 M. CARR AND S. DEVADOSS

2. Constructing Graph-Associahedra

2.1. The motivating example will be the associahedron.

Denition 2.1. Let A (n) be the poset of bracketings of a path with n nodes, ordered suchthat a a if a is obtained from a by adding new brackets. The associahedron K n is aconvex polytope of dimension n 2 whose face poset is isomorphic to A (n).

The associahedron K n was originally dened by Stasheff for use in homotopy theoryin connection with associativity properties of H -spaces [15, Section 2]. The constructionof the polytope K n is given by Lee [14] and Haiman (unpublished). The vertices of K nare enumerated by the Catalan numbers. Figure 1(a) shows the 2-dimensional K 4 as thepentagon. Each edge of K 4 has one set of brackets, whereas each vertex has two. Note thatFigure 7(a) shows C(A3)# tiled by 24 K 4 pentagons. We give an alternate denition of K nwith respect to tubings .

( a ) ( b )

Figure 1. Associahedron K 4 labeled with (a) bracketings and (b) tubings.

Denition 2.2. Let be a graph. A tube is a proper nonempty set of nodes of whoseinduced graph is a proper, connected subgraph of . There are three ways that two tubest1 and t2 may interact on the graph.

(1) Tubes are nested if t1 t2 .(2) Tubes intersect if t1 t2 = and t1 t2 and t2 t1 .(3) Tubes are adjacent if t1 t2 = and t1 t2 is a tube in .

Tubes are compatible if they do not intersect and they are not adjacent. A tubing T of isa set of tubes of such that every pair of tubes in T is compatible. A k-tubing is a tubingwith k tubes.

Lemma 2.3. Let be a path with n 1 nodes. The face poset of K n is isomorphic to theposet of all valid tubings of , ordered such that tubings T T if T is obtained from T

by adding tubes.

Figure 1(b) shows the faces of associahedron K 4 labeled with tubings. The proof of thelemma is based on a trivial bijection between bracketings and tubings on paths.


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COXETER COMPLEXES AND GRAPH-ASSOCIAHEDRA 3

2.2. For a graph with n nodes, let be the n 1 simplex in which each facet (codimen-sion 1 face) corresponds to a particular node. Each proper subset of nodes of correspondsto a unique face of , dened by the intersection of the faces associated to those nodes.The empty set corresponds to the face which is the entire polytope .

Denition 2.4. For a given graph , truncate faces of which correspond to 1-tubingsin increasing order of dimension. The resulting polytope P is the graph-associahedron .

This denition is well-dened: Theorem 2.6 below guarantees that truncating any orderingof faces of the same dimension produces the same poset/polytope. Note also that P is asimple, convex polytope.

Example 2.5. Figure 2 shows a 3-simplex tetrahedron truncated according to a graph.The facets of P ( ) are labeled with 1-tubings. One can verify that the edges correspond

to all possible 2-tubings and the vertices to 3-tubings.

Figure 2. Iterated truncations of the 3-simplex based on an underlying graph.

Theorem 2.6. P is a simple, convex polytope whose face poset is isomorphic to set of valid tubings of , ordered such that T T if T is obtained from T by adding tubes.

The proof of this theorem is given at the end of the section. Note that simplicity andconvexity of P follows from its construction. Stasheff and Schnider [16, Appendix B]proved the following motivating examples. They follow immediately from Theorem 2.6.

Corollary 2.7. When is a path with n 1 nodes, P is the associahedron K n . When is a cycle with n 1 nodes, P is the cyclohedron W n .

2.3. For a given tube t and a graph , let t denote the induced subgraph on the graph .By abuse of notation, we sometimes refer to t as a tube.

Denition 2.8. Given a graph and a tube t, construct a new graph t called the recon-nected complement : If V is the set of nodes of , then V t is the set of nodes of t . Thereis an edge between nodes a and b in t if either {a, b} or {a, b} t is connected in .


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Figure 3. Examples of 1-tubings and their reconnected complements.

Figure 3 illustrates some examples of 1-tubings on graphs along with their reconnectedcomplements.

Theorem 2.9. The facets of P correspond to the set of 1-tubings on . In particular,the facet associated to a 1-tubing {t} is combinatorially equivalent to P t P t .

Proof. We know from Theorem 2.6 that a facet of P is given by a 1-tubing {t}. The facescontained in this facet are the tubings T of that contain t. Now if t i t is a tube of tthen it is also a tube of . Consider the map

: {tubes of t } {tubes of containing t}

where

(t ) =t t if t t is connected in t otherwise .

Note that is a bijection and it preserves the validity of tubings. That is, two tubes t1 andt2 are compatible in t if and only if (t1) and (t2 ) are compatible. Dene the naturalmap

: {tubings on

t } { tubings on t } {tubings on }

where

(T i T j ) = {t} t i T i {(t i )} t j T j {t j }.It is straightforward to show that this is an isomorphism of posets.Example 2.10. Figure 4 shows the Schlegel diagram of the 4-dimensional polytope P ( ).It is obtained from the 4-simplex by rst truncating four vertices, each of which become a3-dimensional facet, as depicted in Figure 4(d) along with its 1-tubing. Then six edges aretruncated, becoming facets of type Figure 4(c); note that Theorem 2.9 shows the structure

of the facet to be the product of the associahedron K 4 of Figure 1(b) and an interval. Finallyfour 2-faces of the original 4-simplex are truncated to result in the polytope of Figure 4(b);this is the product of the cyclohedron W 3 (hexagon) and an interval. Four of the originalve facets of the 4-simplex have become the polyhedron of Figure 4(d), whereas the fth(external) facet is the 3-dimensional permutohedron , as shown in Figure 4(a).


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( a )

( b )

external face ( c )

( d )

Figure 4. The Schlegel diagrams of a 4-polytope along with its four typesof facets.

2.4. The remaining section is devoted to the proof of Theorem 2.6, which follows directlyfrom Lemmas 2.14 and 2.15 below. First we must dene a poset operation analogous totruncation.

We dene an initial partial ordering 0 on tubes by saying that t i 0 t j if and only if t i t j . We also dene a partial ordering on a set of tubings T induced by any partialordering of tubes of : Given tubings T I , T J T , then T I T J if and only if for all t j T J ,there exists t i such that t j t i T I . We write this partially ordered set of tubings as(T ,). Note that is isomorphic to ( T 0 ,0): the set of nonnested tubings of with

order induced by 0.

Denition 2.11. Given a poset of tubings ( T ,), we can produce a set ( T , ) by pro-moting the tube t. Let

T = {T {t} | T T and T {t} is a valid tubing of }

and let T = T T . Let be dened so that t is incomparable to any other tube, andfor any tubes ta , t b not equal to t, let ta tb if and only if t a tb.

Let {t i } be the set of tubes in for i {1, . . . , k } ordered in decreasing size. Notice thesecorrespond to the faces of in increasing order of dimension. Let ( T i ,i ) be the resulting

set after consecutively promoting the tubes t1 , . . . , t i in (T 0 ,0). The following two lemmasexplicitly dene the tubings and the ordering of ( T i ,i ). Both are trivial inductions fromthe denition of promotion.

Lemma 2.12. T i is the set of all valid tubings of the form T 0mj =1 {tq j } where T 0 T 0 and

{qj } {1,...,i }.


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Lemma 2.13. If a or b is less than or equal to i , then t a i tb if and only if a = b. If both a and b are greater than i, then ta i tb if and only if t a 0 tb.

As a special case we can state the following:

Lemma 2.14. (T k ,k ) is isomorphic as a poset to the set of tubings of , ordered such that T T if and only if T can be obtained by adding tubes to T .

Proof. Applying Lemma 2.12 to the case i = k shows T k is the set of all tubings of .Lemma 2.13 shows that T k T if and only if T T .

T {t } is not valid

T { t } is valid but t T

t T

{t }

{t}

T {t } is not valid

T { t } is valid but t T

t T

T

T {t }

relationship for a general T

Figure 5. A sketch of the poset lattice before and after promotion of tube

{t}. Regions shaded with like colors are isomorphic as posets.

The only step that remains is to show the equivalence of promotion to truncation whenperformed in this order. The following lemma accomplishes this.

Lemma 2.15. Let f i be a face of corresponding to the tube t i . Let P i be the polytopecreated by consecutively truncating faces f 1 ,...,f i of . Then (T k ,k ) = P k .

Proof. For consistency, we refer to by P 0 . Since P 0 is convex, so is P i . Thus we maydene these polytopes as intersections of halfspaces. Denote the hyperplane that denesthe halfspace H +a by H a . If X is the halfspace set for a polytope P then there is a naturalposet map

: P (X )op : f X f where (X )op is the set of subsets of X ordered under reverse inclusion and X f is the subsetsuch that f = P a X f H a . Note that is an injection with its image as all the sets X

such that P a X H a in nonempty. By truncating P at f , a new halfspace H + is addedwith the following properties:


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(1) A vertex of P is in H + if and only if it is not in f .(2) No vertices of P are in H .

This produces the truncated polytope P = H +

a X

H +a

. Let P 0

be dened by

a X 0 H +a where X 0 is the set of indices for the dening halfspaces. Let H

+i be the halfspace

with which we intersect P i 1 to truncate f i . The halfspace set for P i is

X i = X i 1 {i} = X 0 {1, . . . , i }.

We dene the map i : P i (X i )op which takes a face of P i to the set of hyperplanesthat contain it.

We now produce an order preserving injection i from T i to (X i )op . Let 0 be the mapfrom tubes of to ( X 0) that takes a tube t i to 0(f i ). Dene

i (t j ) ={ j } if j i,

0(t j ) if j > i.

This allows us to dene a new map

i : T i (X i )op : T J t j T J

i (t j ).

It follows from the denition that this is an order preserving injection. An induction argu-ment shows that i (T i ) = i (P i ). Since i and i are order preserving and injective, wehave that 1i i : T i P i is an isomorphism of posets.

3. Tiling Coxeter Complexes

3.1. We begin with some standard facts and denitions about Coxeter systems. Most of the background used here can be found in Bourbaki [3] and Brown [4].

Denition 3.1. Given a nite set S , a Coxeter group W is given by the presentation

W = s i S | s2i = 1 , (s i s j )m ij = 1 ,

where m ij = m ji and 2 m ij .

Associated to any Coxeter system ( W,S ) is its Coxeter graph W : Each node represents anelement of S , where two nodes s i , s j determine an edge if and only if m ij 3. A Coxetergroup is irreducible if its Coxeter graph is connected and it is locally nite if either W isnite or each proper subset of S generates a nite group. A Coxeter group is simplicial if it is irreducible and locally nite. The classication of simplicial Coxeter groups and their

Coxeter graphs are well-known [ 3, Chapter 6]. Unless stated otherwise, the Coxeter groupsdiscussed below are assumed to be simplicial.Every simplicial Coxeter group has a realization as a group generated by reections

acting faithfully on a variety [ 4, Chapter 3]. The geometry of the variety is either spherical,Euclidean, or hyperbolic, depending on the group. Every conjugate of a generator s i acts onthe variety as a reection in some hyperplane, dividing the variety into simplicial chambers.


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This variety, along with its cellulation is the Coxeter complex corresponding to W , denotedCW . The hyperplanes associated to the generators s i of W all border a single chamber,called the fundamental chamber of CW . The W -action on the chambers of CW is transitive,and thus we may associate an element of W to each chamber; generally, the identity isassociated to the fundamental chamber.

Notation. For a spherical Coxeter complex CW , we dene the projective Coxeter complex P C(W ) to be CW with antipodal points on the sphere identied. These complexes arisenaturally in blow-ups, as shown in Theorem 4.2.

Example 3.2. The Coxeter group of type An has n generators, and m ij = 3 if i = j 1and 2 otherwise. Thus An is isomorphic to the symmetric group S n +1 and acts on theintersection of the unit sphere in R n +1 with the hyperplane x1 + x2 + + xn +1 = 0. Eachs i is the reection in the plane x i = x i +1 . Figure 6(a) shows the Coxeter complex CA3 , a2-sphere cut into 24 triangles.

The Bn Coxeter group has n generators with the same m ij as An except that m12 = 4.The group Bn is the symmetry group of the n-cube, and acts on the unit sphere in R n . Eachgenerator s i is a reection in the hyperplane x i 1 = xi , except s1 which is the reection inx1 = 0. Figure 6(b) shows the Coxeter complex CB3 , the 2-sphere tiled by simplices.

The An Coxeter group has n + 1 generators, with m ij = 3 if i = j 1, and m (1)( n +1) = 3.Every other m ij equals two. The group An acts on the hyperplane dened by x1 + x2 + + xn +1 = 0 in R n +1 . Each s i is the reection in xi = x i +1 , except sn +1 which is thereection in xn +1 = x1 +1. Figure 6(c) shows the Coxeter complex CA2 , the plane with thecorresponding hyperplanes.

( a ) ( b ) ( c )

Figure 6. Coxeter complexes CA3 , CB3 , and CA2 .

3.2. The collection of hyperplanes {xi = 0 | i = 1 , . . . , n } of R n generates the coordinatearrangement. A crossing of hyperplanes is normal if it is locally isomorphic to a coordinatearrangement. A construction which transforms any crossing into a normal crossing involvesthe algebro-geometric concept of a blow-up; see Section 4.1 for a denition.


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A general collection of blow-ups is usually noncommutative in nature; in other words, theorder in which spaces are blown up is important. For a given arrangement, De Concini andProcesi [7, Section 3] establish the existence (and uniqueness) of a minimal building set , acollection of subspaces for which blow-ups commute for a given dimension, and for whichevery crossing in the resulting space is normal. We denote the minimal building set of anarrangement A by Min( A). Let be an intersection of hyperplanes in an arrangement A.Denote H to be the set of all hyperplanes that contain . We say H is reducible if it isa disjoint union H H , where = for intersections of hyperplanes and .

Lemma 3.3. [7, Section 2] Min (A) if and only if H is irreducible.

If reections in H generate a Coxeter group (nite reection group), it is called thestabilizer of and denoted W . For a Coxeter complex CW , we denote its minimal buildingset by Min( CW ). The relationship between the set Min( CW ) and the group W is given bythe following.

Lemma 3.4. [5, Section 3] Min (CW ) if and only if W is irreducible.

Denition 3.5. The minimal blow-up of CW , denoted as C(W )# , is obtained by blowingup along elements of Min( CW ) in increasing order of dimension.

The construct C(W )# is well-dened: Lemma 4.9 below guarantees that blowing-up anyordering of subspaces in Min( CW ) of the same dimension produces the same cellulation.

Example 3.6. Figure 7(a) shows the blow-ups of the sphere CA3 of Figure 6(a) at nonnor-

mal crossings. Each blown up point has become a hexagon with antipodal identication andthe resulting manifold is C(A3 )# . Figure 8 shows the local structure at a blow-up, whereeach crossing is now normal. The minimal blow-up of the projective Coxeter complex of type A3 is shown in Figure 7(b), with the four points blown up in RP 2 . Figure 7(c) showsthe minimal blow-up of CA2 of Figure 6(c).

( a ) ( b ) ( c )

Figure 7. Minimal blow-ups of (a) CA3 , (b) P C(A3) and (c) CA2 .


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3.3. Given the construction of graph-associahedra above, we turn to applying them to thechambers tiling C(W )# .

Theorem 3.7. Let W be a simplicial Coxeter group and W be its associated Coxeter graph.Then P W is the fundamental domain for C(W )# .

Proof. It is a classic result of geometric group theory that each chamber of a simplicialCoxeter complex CW is a simplex. The representation of W can be chosen such thatthe generators correspond to the reections through the supporting hyperplanes of a xedchamber. In other words, a fundamental chamber of CW is the simplex W such that eachfacet of W is associated to a node of W .

Let f be a face of W and let be the support of f , the smallest intersection of hyperplanes of CW containing f . As in the previous section, the face f corresponds to asubset S of the nodes of W . The nodes in S represent the generators of W that stabilize .

These elements generate W , and the subgraph induced by S is the Coxeter graph of W .By Lemma 3.4, is an element of Min( CW ) if and only if W is irreducible. But W

is irreducible if and only if W is connected, that is, when the set of nodes of W is atube of W . Note that blowing up in CW truncates the face f of W . Thus performingminimal blow ups of CW is equivalent to truncating the faces of W that correspond totubes of W . By denition, the resulting polytope is P W .

Remark. The maximal building set is the collection of all crossings, not just the nonnormalones. The fundamental chambers of the maximal blow-up of CW will be tiled by permuto-hedra, obtained by iterated truncations of all faces of the simplex.

Remark. The generalized associahedra of Fomin and Zelevinsky [11] are fundamentally dif-ferent than graph-associahedra. Although both are motivated from type An (the classicalassociahedra of Stasheff), they are distinct in all other cases. For example, the cyclohedronis the generalized associahedron of type Bn , whereas it is the type An graph-associahedron.

3.4. The construction of the Coxeter complex CW implies a natural W -action. This action,restricted to the chambers is faithful and transitive, so we can identify each chamber withthe group element that takes the fundamental chamber to it. The faces of the chambers of CW have different types (according to their associated tubings in W ). A transformation istype preserving if it takes each face to a face of the same type. We call the W -action typepreserving because each w induces a type preserving transformation of CW .

We may use this action to dene a W -action on C(W )# . There is a hyperplane-preservingisomorphism between CW Min(CW ) and C(W )# Min(CW ). We dene the W -actionon C(W )# to agree with the W -action on CW in C(W )# Min(CW ). We dene the actionon the remainder of C(W )# by requiring that for all subvarieties V of C(W )# Min(CW ),the action of w takes the closure of V to the closure of wV . The W -action dened this wayis type preserving, and the stabilizer of each hyperplane is the group W .


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Given C(W )# , we may associate an element s f W to each facet f of the fundamentalchamber. We call this element the reection in that facet. If is the hyperplane of C(W )#that contains f , then s f is a reection in . This corresponds to the reection across in CW , which is the longest word in W [4, Section 3]. For a face f of the fundamentalchamber, dene W f = s f i and s f = s f i , where the f i s are the facets of the fundamentalchamber that contain f . We denote the face corresponding to f of the chamber labeled wby w(f ).

Theorem 3.8. C(W )# can be constructed from |W | copies of P W , labeled by the elementsof W and with the face w(f ) identied to w (f ) whenever w 1w W f . The facet directly opposite w through w(f ) is ws f .

Remark. One may be tempted to think that whenever w(f ) is identied with w (f ), themap between them is the restriction of the identity map between the chambers w and w .However, Davis et al. [ 6, Section 8] show that this is not the case, and compute the actualgluing maps between faces. For this reason they call the elements s f mock reections.The gluing map may also be computed by applying the theorem above to subfaces of f .

Proof. Since the W -action is type preserving, a chamber w contains a face f if and only if w preserves f . Recall that W f is generated by reections in facets that contain f . Thusf is contained only in chambers whose elements correspond to W f . The chamber that liesdirectly across f from the fundamental chamber corresponds to the longest word in W f .Minimal blow ups of CW resolve nonnormal crossings, so W f is isomorphic to ( Z / 2Z )d ,where F has codimension d. Thus the longest word in W f is the product of generators s f i .

For every subspace Min( CW ) and every w W , the subspace w() is also inMin(CW ). Thus we may extend the adjacency relation to chambers other than the fun-damental chamber analogously. Since the W -action preserves containment, a face w (f ) isidentied with w(f ) if and only if w 1w W f . Similarly, w respects reection across F sothe chamber directly across w(f ) from w is ws f .

Example 3.9. Consider the Coxeter group A3 . Denote two facets of the fundamentalchamber of CA3 by x and y, whose reections have the property that ( sx sy )3 = 1. Notethat C(A3)# is tiled by 24 copies of the associahedron P (A3). Let f be the facet adjacentto x and y in the fundamental chamber of C(A3)# and let be the intersection of y andf , as in Figure 8. Then s f = sx sy sx , and the fundamental chamber meets sy sx , sx sy sx ,

and sy at . If we travel directly across from the fundamental chamber, we arrive ins = sx sy sx sy = sy sx .


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y y

f

f

x

s

s

y s

f s

s

x

y

f f

y s

f s

s

Figure 8. Reections locally around C(A3)# .

4. Geometry of Minimal Blow-ups

4.1. One of our objectives is to describe the geometric structures of Min( CW ) before andafter blow-ups. This nal section proves Theorem 4.2 which describes C(W )# seen from theviewpoint of CW .

We recall elementary notions of local structures, along with xing notation: The tangentspace of a variety V at p is denoted T p(V ). For a Coxeter complex CW , the tangent spacehas a natural Euclidean geometry which it inherits from the embedding of CW in R n (withthe hyperbolic simplicial Coxeter groups being viewed as acting on the hyperboloid model

inside R n ). Two nonzero subspaces of T p(V ) are perpendicular if each vector in the rst isperpendicular to each vector in the second, under the Euclidean geometry of T p (V ). Thetangent bundle of a variety V on a subvariety U is

T U (V ) = {( p,v) | p U, vT p (V )}.

If U = V , we write T (V ). The normal space of U at p is

N p (U ) = {v | v T p(V ), v T p (U )}

and the normal bundle of U at a subvariety W U is

N W (U ) = {( p,v) | p U, v N p (V )}.

If W = U , we write N (U ).

Denition 4.1. [12] The blow-up of a variety V along a codimension k intersection of hyperplanes is the closure of {(x, f (x)) | x V } in V P k 1 . The function f : V P k 1 isdened by f : p [f 1( p) : f 2( p) : : f k ( p)], where the f i dene hyperplanes of H whoseintersection is .


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We denote the blow-up of V along by V # . There is a natural projection map

: V # V : (x, y ) x

which is an isomorphism on V . The hyperplanes of V # are the closures 1 (h ) foreach hyperplane h of V and one additional hyperplane 1(). Thus V and V # 1()are isomorphic not only as varieties but as cellulations. 1 The hyperplane of V # has anatural identication with the projectied normal bundle of in V . The intersection of ahyperplane h with is the part of that corresponds to T (h) N ().

4.2. A arrangement of hyperplanes of a variety V cut V into regions. We say that thehyperplanes give a cellulation of V . Two cellulations are equivalent if there is a hyperplane-preserving isomorphism between the two varieties. Let be an intersection of hyperplanes.We say that hyperplanes h i cellulate to mean the intersections h i give a cellulation of , denoted by C. The notation C will always refer to the cellulation of in the originalcomplex, rather than its image in subsequent blow-ups. Let Min( C) denote the minimalbuilding set of C, and let C()# denote the blow-up of the minimal building set of .

Theorem 4.2. Let CW be the Coxeter complex of a simplicial Coxeter group W and let Min (CW ). The blow-up of in C(W )# is equivalent to the product

C()# P C(W )# .

Example 4.3. There are 2 n +1n k dimension k elements of Min( CAn ). Each of these el-ements become C(Ak +1 )# P C(An k 1 )# in C(An )# . Figure 9(d) shows the projectiveCoxeter complex P C(A4)# after minimal blow-ups. This is the Deligne-Knudsen-Mumfordcompactication M 0,6(R ) of the real moduli space of curves with six marked points. Itis the real projective sphere RP 3 with ve points and ten lines blown-up. Each of the veblown-up points are P C(A3)# , shown in Figure 9(b) as C(A4)# before projecting through theantipodal map. Each of the ten lines, each line dened by two distinct points in Min( CA4 ),becomes P C(A2)# P C(A2)# , a 2-torus depicted in Figure 9(c). Note that there are also tencodimension 1 subspaces P C(A3)# pictured in Figure 9(a), dened by three distinct pointsin Min(CA4).

Remark.Lemmas 4.5 and 4.6 are enough to provide the results of Theorem 4.2 for themaximal blow-up of CW .

Remark. Extensions of these results to conguration spaces are given in [1, Section 3].

1We give each hyperplane h of V # the same name as its projection (h ).


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( a )

( b )

( c )

( d )

Figure 9. The pro jective Coxeter complex (d) P C(A4)# along with com-ponents (a) P C(A3)# , (b) C(A3)# and (c) P C(A2)# P C(A2)# .

4.3. The proof of Theorem 4.2 requires two denitions and four preliminary lemmas.

Denition 4.4. Let and be intersections of hyperplanes in a cellulation of V . We saythat is strongly perpendicular to and write if for all p in , all three of thefollowing subspaces span T p(V ) and any two of them are perpendicular:

(1) T p( ), (2) N p ( ), and (3) N p ( ).

Note that this directly implies that T p( ) is the span of T p ( ) N p ( ). For anintersection of hyperplanes , the normal space N p ( ) is the span of the normal spaces of the elements of H at p; if contains , then N p ( ) contains N p ( ). This shows immediatelythat if H reduces to H 1 H 2 , then 1 2 .

Lemma 4.5. For every intersection of hyperplanes in a Coxeter complex CW , the set H has a unique maximal decomposition H = H 1 H k where

(1) each H i is irreducible,

(2) i j for all i = j , and (3) i S i properly contains for any proper subset S {1, 2, . . . , k }.

Proof. If the normal spaces of two hyperplanes h1 , h2 of H are not perpendicular, writeh1 h2 . Then is a symmetric, reexive relation on H . Let be the unique smallestequivalence relation containing as a subset of H H .


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No two hyperplanes h1 h2 can be separated by any reduction of H . To prove this,suppose they could, and let H reduce to H 1 H 2 with h1 in 1 and h2 in 2 . Thensince W is a Coxeter group, the reection of h1 across h2 must be in H . By hypothesis,the resulting hyperplane must contain 1 or 2 . The former implies that 1 h2 and thelatter implies 2 h1 , yielding a contradiction. Since is the smallest transitive relationcontaining , the hyperplanes h1 , h 2 cannot be separated whenever h1 h2 .

However, if partitions H into at least two classes, then we may separate H intoH 1 H 2 with each partition contained in either H 1 or H 2 . Clearly H 1 H 2 = . Toverify that H( H i ) = H i , note that no element h1 of H 1 may contain H 2 . If it does,then for all p in , we have N p (h1 ) contained in N p ( H 2), and thus in the span of {N p (h2)}for h2 in H 2 . This violates the pairwise perpendicularity in our choice of H 1 , H 2 . Thus theequivalence relation partitions H into a unique maximal decomposition and thereforethe H i s are irreducible. Also, no proper subset S of the i can intersect in exactly , sincethen H i for i in S would be H . But H must reduce to H i for i S and H jfor j /S by the argument above.

When H reduces to H 1 H 2 , we have 1 2 . Furthermore, since N p( 1) is thespan of the normal spaces of H 1 , and H 1 H , then for any 3 (with nonemptyintersection), it follows that 3 1 . Thus by induction, i j for i = j .

4.4. The following two lemmas describe the effect of a blow-up on a cellulation. The rstlemma combines several facts that follow directly from the denitions of hyperplanes andblow-ups. Note that as we perform blow-ups of CW , the set of hyperplanes that contain agiven may change. However, H is always assumed to refer to the set of hyperplanes thatcontain in CW .

Lemma 4.6. Let be a subvariety of V with cellulation C 1 . Suppose the tangent spaces of the hyperplanes H cellulate the normal bundle at each point p with cellulation C 2 .

(1) The subvariety of V # is a product C 1 P C 2 .(2) The tangent space T p (V # ) for p retains a local Euclidean structure. Roughly

speaking, n 1 of the coordinate vectors are in T p ( ), and the other is parallel tothe 1-dimensional subspace of N ( p) ( ) that corresponds to p.

(3) For each hyperplane h of V that meets at a subvariety = , the hyperplane h of V # meets at P C 2 .

(4) For each hyperplane h of V that properly contains , the hyperplane h of V # meets

at C 1 h

, where h

is the image of T p(h) in P

C 2 . Also h .Lemma 4.7. Let be a subvariety of V with cellulation C 1 . If in V , then C in V # is equivalent to (C 1 )#( ) .

Proof. The normal bundle N ( ) is contained in T ( ) since . Thus N ( )and N ( ) have the same intersection with T ( ). Since blow-ups replace a variety with its


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projectied normal bundle, the blow-ups along and produce equivalent cellulationsof .

Finally we establish the tools that will allow us to change the order in which we blowup elements of Min( CW ). The following denition and lemma give a class of orderings thatproduce the same cellulation as minimal blow-ups.

Denition 4.8. Given a variety V with intersections of hyperplanes , , the blow-upsalong and commute if the cellulations ( V # )# and ( V # )# are equivalent, and theinduced map on the hyperplanes preserves their labels.

Lemma 4.9. Let x1 , x2 , . . . , x k be an ordering of the elements of Min (CW ) such that i jwhenever x i is contained in xj . Then blowing up CW along the xi in order gives a cellulation equivalent to C(W )# . The induced map on the hyperplanes also preserves labels.

Proof. First we verify that if , then the blow-ups along and commute. Since ,the bundle N ( ) is contained in T ( ) and N ( ) is contained in T ( ). Denethe maps : V # V and : (V # )# V # . Then 1 (V ) =

1 (V )

1 ( )

since N ( ) T ( ). Thus ( V # )# is the closure of ( 1 (V )), which is the

closure of 1 ( 1 (V )). Similar reasoning shows that ( V # )# is the closure of

1 ( 1 (V )). Since the s are isomorphisms on V , we have a natural

isomorphism between ( V # )# and ( V # )# .Now take , to be elements of Min( CW ) such that neither contains the other. By

Lemma 3.3, the arrangements H and H are irreducible. Applying Lemma 4.5 shows thatif H( ) is reducible, then and H H is the unique reduction.

If H ( ) is irreducible, then is in Min(CW ). After the blow-up along , theresulting spaces and do not intersect by Lemma 4.6, and thus (vacuously) . If is not in Min( CW ), then . In either case, the blow-ups along and commute.Thus we may transpose any two elements that do not contain each other in the ordering of Min(CW ) and get an equivalent cellulation (with matching hyperplane labels) after blowingup all of Min(CW ). Repeating this procedure proves the statement of the lemma.

4.5. We have now assembled all the lemmas needed for the proof of the theorem.

Proof of Theorem 4.2. We begin by applying Lemma 4.9. Divide the elements of Min( CW )into three sets:

(1) {},(2) X = { : }, and(3) Y = { : }.

We reorder the elements of Min( CW ) as follows: First we blow up the elements of X ,ordered by the dimension of , followed by blowing up along . Finally blow up the


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elements of Y in order of dimension, as usual. Note that this is a valid application of Lemma 4.9, since if contains , then contains .

We next produce a bijection between the set X of elements xi in X that intersect in( ((CW )# x 1 )# x 2 )# x i 1 and the elements of Min( C) in CW . We show that the map : X Min(C) : is a bijection, and that blowing up the elements of X hasthe same effect on the cellulation of as blowing up the elements of Min( C).

(1) Suppose X and , and thus H H . Since is in Min(CW ), the groupW is an irreducible spherical Coxeter group by Lemma 3.4, and the arrangementH is irreducible in H by Lemma 3.3. For all spherical Coxeter groups W , theelements of H intersect in an irreducible arrangement. 2 Therefore = Min(C).

(2) Suppose X and , then is not in Min( CW ), thus H ( ) reduces.Lemma 4.5 guarantees that . Thus by Lemma 4.7, blowing up is equivalentto blowing up in the cellulation of .

(3) We now produce an function : Min(C) X that will be the inverse to . For Min(C), either Min(CW ) or H is reducible. If Min( CW ), then let( ) = . If not, then H must reduce to H 0 H1 H m . Without lossof generality, assume contains 0 .

Since N ( i ) is contained in T ( 0) for i = 0, the normal spaces of the elementsof H H 0 are the same in as they are in CW . Thus in , we know that i j for i, j = 0 , i = j . Furthermore, the normal space N p( 0) in is a subsetof N p ( 0) in V . Thus if N p(0) in is nonzero, it is perpendicular to each N p( i )in . Thus the set of hyperplanes of induced by H H reduces to the disjoint

union induced by ( H 0 H ) H1 Hm . To satisfy the hypothesis that Min(C), it is necessary that 0 = and m = 1. Thus we dene ( ) = 1 . Itis straightforward to check that is the inverse of , so is a bijection.

(4) By our choice of ordering, the elements of X are blown up in the same order aselements of Min( C) under minimal blow-ups. Furthermore, the subvariety hasequivalent cellulations in the blow-up along ( ) and in the blow-up along . Thisfollows trivially if , and from Lemma 4.7 if not.

Thus, after blowing up all the elements of X in CW , the cellulation of is equivalentto C()# . By Lemma 4.6, the result after blowing up is equivalent to C()# P C(W ).Furthermore, for each element y Y , we have y and y = C()# y , where y is

the image of y in P C(W ). Since the elements of Y are ordered by dimension, they are alsoordered by their dimension in CW . Lemma 4.7 guarantees that blowing up the elementsof Y produces a cellulation of equivalent to C()# P C(W )# .

2This can be checked by hand for the simpler cases, and a detailed decomposition of types A n , B n andD n is given in [1, Section 5]. For the larger complexes ( E 6 , E 7 , E 8 ), we can exploit the appearance of A nas a subgroup.


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Acknowledgments. We thank Mike Davis, Rick Scott and Jim Stasheff for advice and clar-ications. We also thank Zan Armstrong, Eric Engler, Ananda Leininger and MichaelManapat for numerous discussions.

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2. R. Bott and C. Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994), 5247-5287.3. N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4-6, Springer-Verlag, Berlin, 2002.4. K. S. Brown, Buildings, Springer-Verlag, New York, 1989.5. M. Davis, T. Januszkiewicz, R. Scott, Nonpositive curvature of blowups, Selecta Math. 4 (1998),

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620-628.11. S. Fomin, A. Zelevinsky, Y -systems and generalized associahedra, Ann. Math. 158 (2003), 977-1018.12. R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.13. M. M. Kapranov, The permutoassociahedron, MacLanes coherence theorem, and asymptotic zones

for the KZ equation, J. Pure Appl. Alg. 85 (1993), 119-142.14. C. Lee, The associahedron and triangulations of the n -gon, Euro. J. Combin. 10 (1989), 551-560.15. J. D. Stasheff, Homotopy associativity of H -spaces, Trans. Amer. Math. Soc. 108 (1963), 275-292.16. J. D. Stasheff (Appendix B coauthored with S. Shnider), From operads to physically inspired

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M. Carr: University of Michigan, Ann Arbor, MI 48109E-mail address : [email protected]

S. Devadoss: Williams College, Williamstown, MA 01267E-mail address : [email protected]

Michael Carr and Satyan L. Devadoss- Coxeter Complexes and Graph-Associahedra

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Transcript of Michael Carr and Satyan L. Devadoss- Coxeter Complexes and Graph-Associahedra