aa r X i v : . [ m a t h . C O ] J un Cubical Convex Ear Decompositions
Russ Woodroofe
Department of MathematicsWashington University in St. LouisSt. Louis, MO 63130, USA [email protected]
Mathematics Subject Classification: 05E25Dedicated to Anders Bj¨orner in honor of his 60th birthday.
Abstract
We consider the problem of constructing a convex ear decomposition for a poset.The usual technique, introduced by Nyman and Swartz, starts with a CL -labelingand uses this to shell the ‘ears’ of the decomposition. We axiomatize the necessaryconditions for this technique as a “ CL -ced” or “ EL -ced”. We find an EL -ced of the d -divisible partition lattice, and a closely related convex ear decomposition of thecoset lattice of a relatively complemented finite group. Along the way, we constructnew EL -labelings of both lattices. The convex ear decompositions so constructedare formed by face lattices of hypercubes.We then proceed to show that if two posets P and P have convex ear decom-positions ( CL -ceds), then their products P × P , P ˇ × P , and P ˆ × P also haveconvex ear decompositions ( CL -ceds). An interesting special case is: if P and P have polytopal order complexes, then so do their products. Contents EL -ceds and CL -ceds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 The d -divisible partition lattice 9 EL -labeling for Π dn . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 An EL -ced for Π dn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 EL -labeling for C ( G ) . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 A convex ear decomposition for C ( G ) . . . . . . . . . . . . . . . . . . . . . 20 CL -labelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.4 CL -ceds of product posets . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Convex ear decompositions, introduced by Chari in [6], break a simplicial complex intosubcomplexes of convex polytopes in a manner with nice properties for enumeration. Acomplex with a convex ear decomposition inherits many properties of convex polytopes.For example, such a complex has a unimodal h -vector [6], with an analogue of the g -theorem holding [31], and is doubly Cohen-Macaulay [31].Nyman and Swartz constructed a convex ear decomposition for geometric latticesin [17]. Their proof method used the EL -labeling of such lattices to understand thedecomposition’s topology. Similar techniques were pushed further by Schweig [23]. InSection 2, we introduce the necessary background material, and then axiomatize theconditions necessary for these techniques. We call such a convex ear decomposition a“ CL -ced”, or “ EL -ced.”We then show by example in Sections 3 and 4 how to use these techniques on someposet families: d -divisible partition lattices, and coset lattices of a relatively comple-mented group. These posets have each interval [ a, ˆ1] supersolvable, where a = ˆ0. Findingthe convex ear decompositions will involve constructing a (dual) EL -labeling that re-spects the supersolvable structure up to sign, and showing that a set of (barycentriclysubdivided) hypercubes related to the EL -labeling is an EL -ced, or at least a convex eardecomposition. We will prove specifically: Theorem 1.1.
The d -divisible partition lattice Π dn has an EL -ced, hence a convex eardecomposition. Theorem 1.2.
The coset lattice C ( G ) has a convex ear decomposition if and only if G isa relatively complemented finite group. EL -shellable, the EL -labelings thatwe construct in these sections also seem to be new. The ideas used to find them may beapplicable in other settings, as briefly discussed in Section 6. Lemma 1.3. Π dn has a dual EL -labeling; C ( G ) has a dual EL -labeling if G is a comple-mented finite group. In Section 5 we change focus slightly to discuss products of bounded posets. Our firstgoal is:
Theorem 1.4.
If bounded posets P and P have convex ear decompositions, then so do P × P , P ˇ × P , and P ˆ × P . This is the first result of which I am aware that links poset constructions and convexear decompositions with such generality. A result of a similar flavor (but more restrictive)is proved by Schweig [23]: that rank selected subposets of some specific families of posetshave convex ear decompositions.A special case of Theorem 1.4 has a particularly pleasing form:
Lemma 1.5. If P and P are bounded posets such that | P | and | P | are isomorphicto the boundary complexes of simplicial polytopes, then so are | P × P | , | P ˇ × P | , and | P ˆ × P | . We then recall the work of Bj¨orner and Wachs [4, Section 10] on CL -labelings of posetproducts, which we use to prove a result closely related to Theorem 1.4: Theorem 1.6.
If bounded posets P and P have CL -ceds with respect to CL -labelings λ and λ , then P × P , P ˇ × P , and P ˆ × P have CL -ceds with respect to the labelings λ × λ , λ ˇ × λ , and λ ˆ × λ . We close by considering some additional questions and directions for further researchin Section 6.
All simplicial complexes, posets, and groups discussed in this paper are finite.A poset P is bounded if it has a lower bound ˆ0 and an upper bound ˆ1, so that ˆ0 ≤ x ≤ ˆ1for all x ∈ P .If P is a bounded poset, then the order complex | P | is the simplicial complex whosefaces are the chains of P \ { ˆ0 , ˆ1 } . (This is slightly different from the standard definition,in that we are taking only the proper part of the poset.) Where it will cause no confu-sion, we talk about P and | P | interchangeably: for example, we say P has a convex eardecomposition if | P | does.We denote by M ( P ) the set of maximal chains of P , which is in natural bijectivecorrespondence with the facets of | P | through adding or removing ˆ0 and ˆ1.3 .1 Convex ear decompositions A convex ear decomposition of a pure ( d − , . . . , ∆ m ⊆ ∆ with the following properties: ced-polytope ∆ s is isomorphic to a subcomplex of the boundary complex of a simplicial d -polytope for each s . ced-topology ∆ is a ( d − s is a ( d − s > ced-bdry ( S s − t =1 ∆ t ) ∩ ∆ s = ∂ ∆ s for each s > ced-union S ms =1 ∆ s = ∆.It follows immediately from the definition that any complex with a convex ear decompo-sition is pure. As far as I know, no one has tried generalizing the theory of convex eardecompositions to non-pure complexes. As many interesting posets are not graded (i.e.,have an order complex that is not pure), finding such a generalization could be useful.Convex ear decompositions were first introduced by Chari [6]. He used the unimodalityof the h -vector of a simplicial polytope to give a strong condition on the h -vector for acomplex with a convex ear decomposition. Swartz [31] showed that a ‘ g -theorem’ holds forany ( d − h -vectors, M -vectors,and the (original) g -theorem. Theorem 2.1. (Chari [6, Section 3])
The h -vector of a pure ( d − -dimensional complexwith a convex ear decomposition satisfies the conditions h ≤ h ≤ · · · ≤ h ⌊ d/ ⌋ h i ≤ h d − i , for ≤ i ≤ ⌊ d/ ⌋ . Theorem 2.2. (Swartz [31, Corollary 3.10]) If { h i } is the h -vector of a pure ( d − -dimensional complex with a convex ear decomposition, then ( h , h − h , . . . , h ⌊ d/ ⌋ − h ⌊ d/ ⌋− ) is an M -vector. An essential tool for us will be the theory of lexicographic shellability, developed byBj¨orner and Wachs in [1, 2, 3, 4]. We recall some of the main facts.We say that an ordering of the facets F , F , . . . , F t of a simplicial complex ∆ (with t facets) is a shelling if F i ∩ (cid:16)S i − j =1 F j (cid:17) is pure (dim F i − < i ≤ t .An equivalent condition that is often easier to use is:if 1 ≤ i < j ≤ t, then ∃ k < j such that (1) F i ∩ F j ⊆ F k ∩ F j = F j \ { x } for some x ∈ F j . shellable if it has a shelling.The existence of a shelling tells us a great deal about the topology of a pure d -dimensional complex: the complex is Cohen-Macaulay, with homotopy type a bouquet ofspheres of dimension d . A fact about shellable complexes that will be especially useful forus is that a shellable proper pure d -dimensional subcomplex of a simplicial d -sphere is a d -ball [7, Proposition 1.2].A cover relation in a poset P , denoted x ⋖ y , is a pair x (cid:12) y of elements in P suchthat there is no z ∈ P with x (cid:12) z (cid:12) y . Equivalently, a cover relation is an edge in theHasse diagram of P .An EL -labeling of P (where EL stands for edge lexicographic ) is a map from thecover relations of P to some fixed partially ordered set, such that in any interval [ x, y ]there is a unique increasing maximal chain (i.e., a unique chain with increasing labels,read from the bottom), and this chain is lexicographically first among maximal chainsin [ x, y ]. It is a well-known theorem of Bj¨orner in the pure case [1, Theorem 2.3], andmore generally of Bj¨orner and Wachs [3, Theorem 5.8], that any bounded poset P withan EL -labeling is shellable. As a result, the term EL -shelling is sometimes used as asynonym of EL -labeling.The families of posets that we study in this paper will have lower intervals [ˆ0 , x ] that‘look like’ the whole poset, but upper intervals [ x, ˆ1] of a different form. For induction,then, it will usually be easier for us to label the posets upside down, and construct dual EL -labelings , that is, EL -labelings of the dual poset. Dual EL -labelings have been usedin other settings, and are natural in many contexts [2, Corollary 4.4] [24, Corollary 4.10].A generalization of an EL -labeling which is sometimes easier to construct (thoughharder to think about) is that of a CL -labeling. Here, instead of labeling the coverrelations (edges), we label “rooted edges.” More precisely, a rooted edge , or rooted coverrelation, is a pair ( r , x ⋖ y ), where the root r is any maximal chain from ˆ0 to x . Also,if x ⋖ x ⋖ · · · ⋖ x n is a maximal chain on [ x , x n ], and r is a root for x ⋖ x , then r ∪ { x } is a root for x ⋖ x , and so on, so it makes sense to talk of a rooted chain c r on a rooted interval [ x , x n ] r . A CL -labeling is one where every rooted interval [ x, z ] r has a unique increasing maximal chain, and the increasing chain is lexicographically firstamong all chains in [ x, z ] r . An in-depth discussion of CL -labelings can be found in [2, 3]:the main fact is that EL -shellable = ⇒ CL -shellable = ⇒ shellable. We will make real useof the greater generality of CL -labelings only in Section 5, and the unfamiliar reader isencouraged to read “ EL ” for “ CL ” everywhere else.The homotopy type of bounded posets with a CL -labeling (including an EL -labeling)is especially easy to understand, as discussed in [3]. Such a poset is homotopy equivalent toa bouquet of spheres, with the spheres in one-to-one correspondence with the descendingmaximal chains. These descending chains moreover form a cohomology basis for | P | . The upper intervals [ x, ˆ1] in the posets we look at will be supersolvable, so we mentionsome facts about supersolvable lattices. For additional background, the reader is referred5o [26] or [15].An element x of a lattice L is left modular if for every y ≤ z in L it holds that( y ∨ x ) ∧ z = y ∨ ( x ∧ z ). This looks a great deal like the well-known Dedekind identityfrom group theory, and in particular any normal subgroup is left modular in the subgrouplattice.A graded lattice is supersolvable if there is a maximal chain ˆ1 = x ⋗ x ⋗ · · · ⋗ x d = ˆ0,where each x i is left modular. Thus the subgroup lattice of a supersolvable group isa supersolvable lattice. In fact, supersolvable lattices were introduced to generalize thelattice properties of supersolvable groups.A supersolvable lattice has a dual EL -labeling λ ss ( y ⋗ z ) = min { j : x j ∧ y ≤ z } = max { j − x j ∨ z ≥ y } , which we call the supersolvable labeling of L (relative to the given chain of left modularelements). This labeling has the property:Given an interval [ x, y ], every chain on [ x, y ] has the same setof labels (in different orders). (2)McNamara [14] has shown that having an EL -labeling that satisfies (2) characterizes thesupersolvable lattices. If F is a face in a simplicial complex ∆, then the link of F in ∆ islink ∆ F = { G ∈ ∆ : G ∩ F = ∅ and G ∪ F ∈ ∆ } . A simplicial complex ∆ is
Cohen-Macaulay if the link of every face has the homology of abouquet of top dimensional spheres, that is, if ˜ H i (link ∆ F ) = 0 for all i < dim(link ∆ F ).The Cohen-Macaulay property has a particularly nice formulation on the order com-plex of a poset. A poset is Cohen-Macaulay if every interval [ x, y ] has ˜ H i ([ x, y ]) = 0 forall i < dim( | [ x, y ] | ). In particular, every interval in a Cohen-Macaulay poset is Cohen-Macaulay. It is well-known that every shellable complex is Cohen-Macaulay. For a proofof this fact and additional background on Cohen-Macaulay complexes and posets, see [28].The Cohen-Macaulay property is essentially a connectivity property. Just as we say agraph G is doubly connected (or 2-connected) if G is connected and G \ { v } is connectedfor each v ∈ G , we say that a simplicial complex ∆ is doubly Cohen-Macaulay (2-CM) if1. ∆ is Cohen-Macaulay, and2. for each vertex x ∈ ∆, the induced complex ∆ \ { x } is Cohen-Macaulay of the samedimension as ∆.Doubly Cohen-Macaulay complexes are closely related to complexes with convex ear de-compositions: 6 heorem 2.3. (Swartz [31]) If ∆ has a convex ear decomposition, then ∆ is doublyCohen-Macaulay. Thus, convex ear decompositions can be thought of as occupying an analogous roleto shellings in the geometry of simplicial complexes: a shelling is a combinatorial reasonfor a complex to be (homotopy) Cohen-Macaulay, and a convex ear decomposition is acombinatorial reason for a complex to be doubly Cohen-Macaulay. Of course, convex eardecompositions also give the strong enumerative constraints of Theorems 2.1 and 2.2.Intervals in a poset with a convex ear decomposition are not known to have convex eardecompositions. However, intervals do inherit the 2-CM property, as intervals are links inthe order complex, and intervals inherit the Cohen-Macalay property. Thus, Theorem 2.3is particularly useful in proving that a poset does not have a convex ear decomposition. EL -ceds and C L -ceds
Nyman and Swartz used an EL -labeling in [17] to find a convex ear decomposition forany geometric lattice. The condition on an EL -labeling says that ascending chains areunique in every interval, and that the lexicographic order of maximal chains is a shelling.Starting with the usual EL -labeling of a geometric lattice, Nyman and Swartz showedthat descending chains are unique in intervals of an ear of their decomposition, and thatthe reverse of the lexicographic order is a shelling. Schweig used similar techniques in [23]to find convex ear decompositions for several families of posets, including supersolvablelattices with complemented intervals.In this subsection, we axiomatize the conditions necessary for these techniques. Al-though we state everything in terms of CL -labelings, one could just as easily read ‘ EL ’for the purposes of this section, and ignore the word ‘rooted’ whenever it occurs.Suppose that P is a bounded poset of rank k . Let { Σ s } be an ordered collection ofrank k subposets of P . For each s , let ∆ s be the simplicial subcomplex generated byall maximal chains that occur in Σ s , but not in any Σ t for t < s . (Informally, ∆ s is all“new” maximal chains in Σ s .) Recall that M (Σ s ) refers to the maximal chains of Σ s ,and let M (∆ s ) be the maximal chains of ∆ s . As usual, maximal chains are in bijectivecorrespondence with facets of the order complex via removing or adding ˆ1 and ˆ0.The ordered collection { Σ s } is a chain lexicographic convex ear decomposition (or CL -ced for short) of P with respect to the CL -labeling λ , if it obeys the following properties: CLced-polytope
For each s , Σ s is the face lattice of a convex polytope. CLced-desc
For any ∆ s and rooted interval [ x, y ] r in P , there is at most one descendingmaximal chain c on [ x, y ] r which is a face of ∆ s . CLced-bdry If c is a chain of length < k , such that c can be extended to a maximalchain in both of ∆ s and ∆ t , where t < s ; then c can be extended to a chain in M (Σ s ) \ M (∆ s ). CLced-union
Every chain in P is in some Σ s .7 ote . We note the resemblance of (CLced-desc) with the increasing chain conditionfor a CL -labeling (under the reverse ordering of labels); but though ∆ s is a simplicialcomplex corresponding with chains in P , it is not itself a poset. Note . By analogy with CL -labelings, it would seem that we should require the de-scending chain in (CLced-desc) to be lexicographically last. But this would be redundant:suppose c is the lexicographically last maximal chain in [ x, y ] r that is also in ∆ s , but that c has an ascent at c i . Then Lemma 2.7 below gives that we can replace the ascent witha descent, obtaining a lexicographically later chain, a contradiction. Note . As previously mentioned, we will usually refer to EL -ceds in this paper, i.e.,the special case where λ is an EL -labeling. Similarly, we may refer to dual EL -ceds, thatis, EL -ceds of the dual poset. Lemma 2.7. (Technical Lemma)
Let { Σ s } be a CL -ced of a poset, with { ∆ s } as above,and let c = { x ⋖ c ⋖ · · · ⋖ c j − ⋖ y } be a maximal chain on a rooted interval [ x, y ] r , with c a face in ∆ s . Suppose that c has an ascent at c i . Then ∆ s contains a c ′′ = ( c \ { c i } ) ∪ c ′′ i which descends at c ′′ i , and is lexicographically later than c .Proof. Let c − = c \ { c i } , and let Σ t be the first subposet in the CL -ced that contains c − .Since Σ t is the face lattice of a polytope, it is Eulerian, so c − has two extensions in Σ t .By the uniqueness of ascending chains in CL -labelings, at most one is ascending at rank i ; by (CLced-desc), at most one is descending. Thus, there is exactly one of each. Theextension with the ascent is c , call the other extension c ′′ .We have shown that c is in Σ t and (since Σ t is the first subposet containing c − ) that s = t , so that c ′′ is in ∆ s . Finally, c ′′ is lexicographically later than c by the definition of CL -labeling.We also recall a useful lemma from undergraduate point-set topology [16, Exercise17.19]: Lemma 2.8. If B is a closed subset of X , then ∂B = B ∩ X \ B . Although they did not use the terms “ CL -ced” or “ EL -ced” in their paper, the essenceof the following theorem was proved by Nyman and Swartz in [17, Section 4], where theyused it to construct convex ear decompositions of geometric lattices. Theorem 2.9. If { Σ s } is an CL -ced for P , then the associated subcomplexes { ∆ s } forma convex ear decomposition for | P | .Proof. (Nyman and Swartz [17, Section 4] ) The property (ced-union) follows directly from(CLced-union), and (ced-polytope) follows from (CLced-polytope) because the barycentricsubdivision of a polytope is again a polytope.For (ced-bdry), we first note that ∂ ∆ s = ∂ (cid:16) | Σ s | \ ∆ s (cid:17) (the topological closure), hence ∂ ∆ s ⊆ ∆ s ∩ ( S t
1, and so c ′′ is lexicographically later than c , as Condition (1) requires for a shelling.We now check that ∆ s is a proper subcomplex of | Σ s | for s ≥
2. Suppose that∆ s = | Σ s | . Then by Notes 2.4 and 2.5, λ is a CL -labeling on Σ s with respect to the reverseordering of its label set. Since | Σ s | is a sphere, there is an ascending chain (descendingchain with respect to the reverse ordering) in Σ s . Since the ascending chain in P is unique,we have s = 1.By definition ∆ = | Σ | is a ( k − s is shellable and a propersubcomplex of the ( k − | Σ s | for s ≥
2, we get that ∆ s is a ( k − Note . Each non-empty ear of { ∆ s } contains exactly one descending chain. This isno accident: see the discussion at the end of Section 2.2. Corollary 2.11.
The following families of posets have EL -ceds, thus convex ear decom-positions.1. (Nyman and Swartz [17, Section 4]) Geometric lattices.2. (Schweig [23, Theorem 3.2])
Supersolvable lattices with M¨obius function non-zero onevery interval.3. (Schweig [23, Theorems 5.1 and 7.1])
Rank-selected subposets of supersolvable andgeometric lattices.
In the following two sections, we will exhibit an EL -ced for the d -divisible partitionlattice, and (using only slightly different techniques) a convex ear decomposition for thecoset lattice of a relatively complemented group. d -divisible partition lattice The d -divisible partition poset , denoted Π dn , is the set of all proper partitions of [ n ] = { , . . . , n } where each block has cardinality divisible by d . The d -divisible partition lattice ,denoted Π dn is Π dn with a ‘top’ ˆ1 and ‘bottom’ ˆ0 adjoined. Π dn is ordered by refinement9which we denote by ≺ ), as in the usual partition lattice Π n (= Π n ). In general, Π dn isa subposet of Π n , with equality in the case d = 1; on the other hand, intervals [ a, ˆ1] areisomorphic to Π n/d for any atom a ∈ Π dn . We refer frequently to [33] for information aboutthe d -divisible partition lattice.As Π n is a supersolvable geometric lattice, and hence quite well understood, we restrictourself to the case d >
1. It will sometimes be convenient to partition a different set S = [ n ]. In this case we write Π S to be the set of all partitions of S , and Π dS the set of all d -divisible partitions of S , so that Π dn = Π d [ n ] is a special case.Wachs found a homology basis for Π dn in [33, Section 2]. We recall her construction.By S n we denote the symmetric group on n letters. We will write a permutation α ∈ S n as a word α (1) α (2) . . . α ( n ), and define the descent set of α to be the indices where α descends, i.e., des α = { i ∈ [ n −
1] : α ( i ) > α ( i + 1) } .Then a split of α ∈ S n at di divides α into α (1) α (2) . . . α ( di ) and α ( di + 1) . . . α ( n ).A switch-and-split at position di does the same, but first transposes (‘switches’) α ( di )and α ( di + 1).These operations can be repeated, and the result of repeated applications of splitsand switch-and-splits at d -divisible positions is a d -divisible partition. For example, if α = 561234, then the 2-divisible partition 56 | |
24 results from splitting at position 2and switch-and-splitting at position 4.Let Σ α be the subposet of Π dn that consists of all partitions that are obtained bysplitting and/or switch-and-splitting the permutation α at positions divisible by d . Let A dn = { α ∈ S n : α ( n ) = n, des α = { d, d, . . . , n − d }} . Wachs proved
Theorem 3.1. (Wachs [33, Theorems 2.1-2.2]) Σ α is isomorphic to the face lattice of the ( nd − -cube for any α ∈ S n .2. { Σ α : α ∈ A dn } is a basis for H ∗ (Π dn ) . After some work, this basis will prove to be a dual EL -ced. EL -labeling for Π dn In addition to the homology basis already mentioned, Wachs constructs an EL -labelingin [33, Section 5], by taking something close to the standard EL -labeling of the geometriclattice on intervals [ a, ˆ1] ∼ = Π n/d (for a an atom), and “twisting” by making selected labelsnegative. While her labeling is not convenient for our purposes, we use her sign idea toconstruct our own dual EL -labeling starting with a supersolvable EL -labeling of [ a, ˆ1].Partition lattices were one of the first examples of supersolvable lattices to be studied[26]. It is not difficult to see that the maximal chain with j th ranked element1 | | . . . | j | ( j + 1) . . . n
10s a left modular chain in Π n .Let y ·≻ z be a cover relation in Π n . Then y is obtained by merging two blocks B and B of the partition z , where without loss of generality max B < max B . Thesupersolvable dual EL -labeling (relative to the above chain of left modular elements) isespecially natural: λ ss ( y ·≻ z ) = min { j : (1 | . . . | j | ( j + 1) . . . n ) ∧ y ≺ z } = max B . We now construct the labeling that we will use for Π dn . Let y ·≻ z be a cover relationin Π dn , where z = ˆ0. As above, y is obtained by merging blocks B and B of z , wheremax B < max B . Label λ ( y ·≻ z ) = ( − max B if max B < min B , max B otherwise, and λ ( y ·≻ ˆ0) = 0 . When discussing dual EL -labelings, any reference to ascending or descending chains is inthe dual poset, so that the inequalities go in the opposite direction from normal. Note . Let a ∈ Π dn be an atom. Then a has n/d blocks, and every block has d elements. Order the blocks { B i } so that max B < max B < · · · < max B n/d , and let B = { max B , . . . , max B n/d } . Then [ a, ∼ = Π B , and we recognize | λ | as the supersolvabledual EL -labeling λ ss on Π B . Note . We also can view Π dn as a subposet of Π n . A cover relation y ·≻ z in Π dn is acover relation in Π n unless z = ˆ0. Thus, | λ | is the restriction of λ ss on Π n , except at thebottom edges y ·≻ ˆ0. Note . The cover relation x ·≻ x gets a negative label if and only if B | B is a non-crossing partition of B = B ∪ B . We will call this a non-crossing refinement of x . Theposet of all non-crossing partitions has been studied extensively [25, 13], although thisseems to have a different flavor from what we are doing. Also related is the connectivity set of a permutation [29], the set of positions at which a split yields a non-crossing partition.Recall that if P and P are posets, then their direct product P × P is the Cartesianproduct with the ordering ( x , x ) ≤ ( y , y ) if x ≤ x and y ≤ y . The lower reducedproduct P ˇ × P of two bounded posets is (cid:0) ( P \ { ˆ0 } ) × ( P \ { ˆ0 } ) (cid:1) ∪ { ˆ0 } . Although thedefinition of the lower reduced product may appear strange at first glance, it occursnaturally in many settings, including the following easily-proved lemma: Lemma 3.5.
Let y ≻ x be elements of Π dn \ { ˆ0 } , with y = B | . . . | B k . Then1. [ˆ0 , y ] ∼ = Π dB ˇ × Π dB ˇ × . . . ˇ × Π dB k .2. [ y, ˆ1] ∼ = Π k .3. [ x, y ] is the direct product of intervals in Π dB i . ote . We discuss (lower/upper reduced) products of posets at much more length inSection 5. Although the situation with Π dn is simple enough that we do not need to referdirectly to product labelings (introduced in Section 5.3), they are the underlying reasonwe can look at partitions block by block in the proofs that follow. Theorem 3.7. λ is a dual EL -labeling of Π dn .Proof. We need to show that each interval has a unique (dual) increasing maximal chainwhich is lexicographically first. There are two forms of intervals we must check:
Case
1. Intervals of the form [ˆ0 , x ].Since the bottommost label on every chain in [ˆ0 , x ] is a 0, every other label in an in-creasing chain must be negative. Hence, every edge x i ·≻ x i +1 in an increasing chain mustcorrespond to a non-crossing refinement of x i .In such a chain, any block B of x is partitioned repeatedly into non-crossing sub-blocks.At the atom level, this block B is sub-partitioned as B | . . . | B k , where max B i < min B i +1 . Thus, any increasing chain on [ˆ0 , x ] passes through this single atom, andwe have reduced the problem to Case 2. Case
2. Intervals of the form [ x m , x ], where x m = ˆ0.By Lemma 3.5 and the discussion following, it suffices to examine a single block B of x .(The labels on disjoint blocks are independent of each other.)In x m , let B be subpartitioned as B | . . . | B k , with max B s = b s and b < b < · · · < b k .The edges we consider correspond with subpartitioning B between itself and B | . . . | B k .First, we show that the lexicographically first chain c = x ·≻ x ·≻ . . . ·≻ x m is unique.If there are any negative labels down from x i , the edge x i ·≻ x i +1 will have the label − b s with greatest absolute value among negative labels. Thus, x i +1 = x i ∧ ( B . . . B s | B s +1 . . . B k ) , and hence x i ·≻ x i +1 is the unique edge down from x i with this label. Otherwise, x i ·≻ x i +1 will have the least possible (positive) label, which is unique since | λ | is a dual supersolvable EL -labeling on [ x m , x ].Next, we show that the lexicographically first chain is increasing. Suppose that c has adescent at x i − ·≻ x i ·≻ x i +1 , with λ ( x i − ·≻ x i ) = α and λ ( x i ·≻ x i +1 ) = β , correspondingto dividing a block C as C ·≻ C | C ∪ C ·≻ C | C | C . Since | λ | is a dual EL -labeling, both labels cannot be positive. Thus, β <
0. If then | α | < | β | , we have max C < max C < min C , and then C ·≻ C ∪ C | C is noncrossing,with a β label, and so lexicographically before x i − ·≻ x i . Otherwise, | α | > | β | . Sincewe have a descent at i , we see α >
0, and so the ± β < α label on the edge obtained bypartitioning C ·≻ C ∪ C | C is again lexicographically before x i − ·≻ x i . In either case,we have shown that any c with a descent is not lexicographically first.12inally, we show that any increasing chain is lexicographically first. Suppose that thereis an edge x ·≻ y ( ≻ x m ) that receives a − b s label. Then y = B . . . B s | B s +1 . . . B k isa non-crossing partition of B , and in particular B s < B s +1 , . . . , B k . We see that anysubpartion of x separating B s from B t for t > s is non-crossing, thus every chain on[ x m , x ] has a − b s label. This fact, combined with (2) shows that any increasing chain onan interval must be constructed inductively by repeatedly taking the least-labeled edgedown, hence be lexicographically first.The descending chains of Wachs’s EL -labeling are { r σ : σ ∈ A dn } , where r σ corre-sponds to successively splitting σ at the greatest possible σ ( id ) [33, Theorem 5.2]. It iseasy to see that each r σ is also descending with respect to our dual EL -labeling, and adimension argument shows us that { r σ : σ ∈ A dn } is exactly the set of descending chains. EL -ced for Π dn Order { Σ α } lexicographically by the reverse of the words α according to the reverseordering on [ n ]. That is, order lexicographically by the words α ( n ) α ( n − · · · α (1),where n ⊳ n − ⊳ · · · ⊳
1. We refer to this ordering as rr -lex , for “reverse reverselexicographic.” For example, 132546 is the first permutation in A with respect to rr -lex,while 231546 < rr − lex > α be as in the text precedingTheorem 3.1, and λ as in Section 3.1. Theorem 3.8. { Σ α : α ∈ A dn } ordered by rr -lex is a dual EL -ced of Π dn with respect to λ . We introduce some terms. If B | . . . | B k is a partition of [ n ], then we say that α ∈ S n has the form B B . . . B k if the first | B | elements in the word α are in B , the next | B | are in B , and so forth. When k = 2, we say that α has switched form B B if α ′ hasthe form B B for α ′ = α ◦ ( | B | | B | + 1), that is, for α ′ equal to α composed with thetransposition of adjacent elements at | B | .We can also talk of α having form B B . . . B k up to switching , by which we meansome α ′ has the form B . . . B k , where α ′ is α up to transpositions at the borders ofsome (but not necessarily all) of the blocks. Finally, if B ⊆ [ n ], then α | B is the word α = α (1) α (2) . . . α ( n ) with all α ( i )’s that are not in B removed. Example 3.9. If B = { , , } and B = { , , } , then 123456, 321654, and 213465 allhave the form B B . 124356 and 135246 have switched form B B , while 152346 does nothave the form B B , even up to switching.Clearly, the d -divisible partition B | . . . | B k is in Σ α if and only if α has the form B B . . . B k up to switching. Lemma 3.10.
Every maximal chain c in Π dn is in Σ α for some α ∈ A dn . roof. We will in fact construct the earliest such α according to the rr-lex ordering, whichwill in turn help us with Corollary 3.11. The proof has a similar feel to the well-knownquicksort algorithm. Let c = { ˆ1 = c ·≻ . . . ·≻ c n/d = ˆ0 } .Consider first the edge ˆ1 ·≻ c in c . The edge splits [ n ] into B | B , and clearly such α ,if it exists, must have the form B B or B B up to switching. If max B < max B , thenall permutations in A dn of the (possibly switched) form B B come before permutations ofthe (possibly switched) form B B , so the rr-lex first α with c in Σ α has the form B B up to switching.Apply this argument inductively down the chain. At c i , we will have shown thatthe rr-lex first α with c in Σ α must have the form B B · · · B i +1 up to switching. Thenif c i ·≻ c i +1 splits block B j into B j, and B j, , with max B j, < max B j, , an argumentsimilar to that with ˆ1 ·≻ c gives that α must in fact have the form B B . . . B j − B j, B j, B j +1 . . . B i +1 up to switching.At the end, we have shown the earliest α having c in Σ α must have the form B . . . B n/d up to switching. Conversely, it is clear from the above that for any α of this form, c is inΣ α . Sort the elements of each B i in ascending order to get a permutation α . This α isin S n but not necessarily in A dn , so we perform a switch at each d -divisible position wherethere is an ascent (i.e., where B i < B i +1 ). This gives us an element α ∈ A dn of the givenform up to switching, and finishes the proof of the statement.We continue nonetheless to finish showing that α is the first element in A dn with c in Σ α . We need to show that if β is another element of A dn with the same form up toswitching of B B . . . B n/d (but different switches), then β > rr − lex α . If B i < B i +1 , thenboth α and β are switched at id (as otherwise we are not in A dn ). Otherwise, if β is a switchat id , then the switch exchanges β ( id ) and β ( id + 1) (up to resorting the blocks). Since β ( id ) > β ( id + 1), “unswitching” moves a larger element of [ n ] later in the permutation,yielding an rr-lex earlier element of the given form up to switching.Let ∆ α be the simplicial complex generated by maximal chains that are in Σ α ( α ∈ A dn ),but in no Σ β for β ∈ A dn with β < rr − lex α . In the following corollary, we summarize theinformation from the proof of Lemma 3.10 about the form of α with c in ∆ α . Corollary 3.11.
Let c be a maximal chain in ∆ α , with y ·≻ x an edge in c which mergesblocks B and B into block B ( max B < max B ). Then1. α has the form . . . B B . . . , up to switching.2. Let τ id be the transposition exchanging id and id + 1 . If y ·≻ x corresponds to aswitch-and-split at id , then the permutation α ◦ τ id is ascending between positions ( i − d + 1 and ( i + 1) d .3. α | B = . . . max B , i.e., max B is rightmost in α | B . roof. (1) and (2) are clear from the proof of Lemma 3.10.For (3), suppose that max B is not rightmost in α | B . Then since α is ascending on d -segments, we have that max B is rightmost in some d -segment of α | B . If a switch-and-split occurs at max B then we have a contradiction of (2), while a split contradicts(1).Every chain passing through an atom a has the same labels up to sign, and Corollary3.11 tells us what the labels are. It is now not difficult to prove (CLced-desc) and (CLced-bdry). Proposition 3.12.
Let [ x m , x ] be an interval with x m , x ∈ ∆ α . Then there is at mostone (dual) descending maximal chain c on [ x m , x ] which is in ∆ α .Proof. There are two cases:
Case x m = ˆ0.It suffices to consider a block B of x . Partitions of B corresponding to edges in Σ α musteither split or switch-and-split α | B at d -divisible positions, and as every chain on [ˆ0 , x ]has bottommost label 0, all other edges of a descending chain must have positive labels(and so correspond to crossing partitions). Claim . All edges of such a descending chain correspond to splittings of α . Proof. (of Claim) Suppose otherwise. Without loss of generality we can assume that x ·≻ x in c corresponds to a switch-and-split of B into B | B , with B the block ofsmallest size which is switch-and-split by an edge in c . We will show that c has an ascent.Corollary 3.11 part 2 tells us that the first d letters in α | B are strictly greater than thelast d in α | B , and since max B is rightmost in α | B , that the first d letters of α | B arestrictly greater than all of B . Since λ ( x m − ·≻ ˆ0) = 0, any negative label gives an ascent.If | B | = d , then we have shown that B | B is non-crossing (giving a negative label). Ifon the other hand | B | > d , then any subdivision of B gives a label with absolute value > max B , hence an ascent. In either case, we contradict c being a descending chain.It follows immediately that a descending chain on [ˆ0 , x ] is unique. Case x m = ˆ0.As usual, we consider what happens to a block B of x . In x m , let B partition as B | . . . | B k , where α | B has the form B B . . . B k up to switching. Then every edge in ∆ σ comes from subdividing at some B i , i.e., as shown at the dotted line here · · · ∪ B j | B j +1 ∪ · · · ∪ B i ... B i +1 ∪ · · · ∪ B l | . . . Let b i = max B i , so every chain on [ x m , x ] has labels ± b , ± b , . . . , ± b k − .15uppose that · · · ∪ B i | B i +1 ∪ . . . is crossing, but B i | B i +1 is non-crossing. Corollary 3.11part 3 tells us that max( · · · ∪ B i ) = max B i , so thatmin( B i +2 ∪ · · · ∪ B k ) < max B i < min B i +1 < max B i +1 . It follows that B i +1 | B i +2 ∪ . . . is also crossing. Thus, if B i | B i +1 is non-crossing, then(positive) b i is not the label of the first edge of a descending chain c , since b i +1 > b i would then be the label of a later edge. That is, if B i | B i +1 is non-crossing, then adescending chain has a − b i label. The “only if” direction is immediate, thus there is aunique permutation and set of signs for the ± b , . . . , ± b k − that could label a descendingchain. Proposition 3.14.
Let c be a (non-maximal) chain with extensions in both Σ α and Σ β , β < rr − lex α . Then c has maximal extensions in M (Σ α ) \ M (∆ α ) .Proof. Let c = { ˆ1 = c > c · · · > c m > c m +1 = ˆ0 } . The first β with c in Σ β obeys thefollowing two conditions:1. For each c i = ˆ0, each block B in c i − splits into sub-blocks B , . . . , B k in c i , wheremax B < · · · < max B k . The restriction β | B is of the form B B . . . B k . (Byrepeated application of Corollary 3.11 .)2. For each block B of c m , β | B is the permutation { b b . . . b d +1 b d . . . b id +1 b id . . . b k } , where B = { b , . . . , b k } for b < · · · < b k . That is, β | B is the ascending permutationof the elements of B , with transpositions applied at d -divisible positions. (Theproof is by starting at the end and working to the front, greedily taking the greatestpossible element for each position.)Since α is not the first permutation in A dn such that c ∈ Σ α , α must violate at least oneof these. If it violates (1) for some B , then α | B has the form B . . . B k with max B j > max B j +1 . Merge B j and B j +1 to add an edge down from c i − that is in Σ α , otherwiseextend arbitrarily in Σ α . By Corollary 3.11 part 1, the resulting chain is not in ∆ α .If α | B violates (2) for some B , then extend c by switch-and-splitting at every d -divisible position of B , otherwise arbitrarily in Σ α . At the bottom, B is partitionedinto some B | . . . | B k . Since (2) is violated, applying transpositions to α at d -divisiblepartitions gives a descent. But this contradicts the conclusion of Corollary 3.11 part 2,and the resulting chain is not in ∆ α .We check the CL -ced properties: Wachs had already proved (CLced-polytope) aspresented in Theorem 3.1, Lemma 3.10 gives us (CLced-union), Proposition 3.12 gives(CLced-desc), and Proposition 3.14 gives (CLced-bdry). We have completed the proof ofTheorem 3.8. 16 The coset lattice
The coset poset of G , denoted C ( G ), is the set of all right cosets of all proper subgroups of G , ordered under inclusion. The coset lattice of G , denoted C ( G ), is C ( G ) ∪ {∅ , G } , thatis, C ( G ) with a top ˆ1 = G and bottom ˆ0 = ∅ added. With our definitions, it makes senseto look at the order complex of C ( G ) (which is the set of all chains of C ( G )), and so wetalk about the coset lattice, even though “coset poset” has a better sound to it. We noticethat C ( G ) has meet operation Hx ∧ Ky = Hx ∩ Ky and join Hx ∨ Ky = h H, K, xy − i y (so it really is a lattice.) General background on the coset lattice can be found in [22,Chapter 8.4], and its topological combinatorics have been studied in [5, 19, 36].The subgroup lattice of G , denoted L ( G ) is the set of all subgroups of G , ordered byinclusion. General background can be found in [22], and its topological combinatoricshave been studied extensively, for example in [24, 32].Notice that for any x ∈ G , the interval [ x, G ] in C ( G ) is isomorphic to L ( G ). It is atheorem of Iwasawa [11] that L ( G ) is graded if and only if G is supersolvable, hence C ( G )is graded under the same conditions. As we have only defined convex ear decompositionsfor pure complexes, we are primarily interested in supersolvable groups in this paper.Schweig proved the following: Proposition 4.1. (Schweig [23])
For a supersolvable lattice L , the following are equiva-lent:1. L has a convex ear decomposition.2. L is doubly Cohen-Macaulay.3. Every interval of L is complemented.Note . A construction very much like Schweig’s convex ear decomposition was earlierused by Th´evenaz in [32] on a subposet of L ( G ) to understand the homotopy type andthe conjugation action on homology of L ( G ) for a solvable group G .It is easy to check that any normal subgroup N ⊳ G is left modular in L ( G ), so asupersolvable group has a supersolvable subgroup lattice with any chief series as its leftmodular chain. Let G ′ denote the commutator subgroup of G . The following collectedclassification of groups with every interval in their subgroup lattice complemented is pre-sented in Schmidt’s book [22, Chapter 3.3], and was worked out over several years byZacher, Menegazzo, and Emaldi. Proposition 4.3.
The following are equivalent for a (finite) group G :1. Every interval of L ( G ) is complemented.2. If H is any subgroup on the interval [ H , H ] , then there is a K such that HK = H and H ∩ K = H . . L ( G ) is coatomic, i.e., every subgroup H of G is an intersection of maximal sub-groups of G .4. G has elementary abelian Sylow subgroups, and if H ⊳ H ⊳ H ⊆ G , then H ⊳ H .5. G ′ and G/G ′ are both elementary abelian, G ′ is a Hall π -subgroup of G , and everysubgroup of G ′ is normal in G .Note . The classification of finite simple groups is used in the proof that (3) is equivalentto the others.We will follow Schmidt and call such a group a relatively complemented group.We notice that relatively complemented groups are complemented , that is, satisfy thecondition of Proposition 4.3 Part (2) on the interval [1 , G ]. On the other hand, S × Z isan example of a complemented group which is not relatively complemented. The comple-mented groups are exactly the groups with (equivalently in this case) shellable, Cohen-Macaulay, and sequentially Cohen-Macaulay coset lattice [36]. Computation with GAP[10] shows that there are 92804 groups of order up to 511, but only 1366 complementedgroups, and 1186 relatively complemented groups.We summarize the situation for the subgroup lattice regarding convex ear decomposi-tions: Corollary 4.5.
The following are equivalent for a group G :1. L ( G ) has a convex ear decomposition.2. L ( G ) is doubly Cohen-Macaulay.3. G is a relatively complemented group. As a consequence, we get one direction of Theorem 1.2.
Corollary 4.6. If C ( G ) is doubly Cohen-Macaulay (hence if it has a convex ear decom-position), then G is a relatively complemented group.Proof. Every interval of a 2-Cohen-Macaulay poset is 2-Cohen-Macaulay, and the interval[1 , G ] in C ( G ) is isomorphic to L ( G ).The remainder of Section 4 will be devoted to proving the other direction. EL -labeling for C ( G ) As with the d -divisible partition lattice, the first thing we need is a dual EL -labeling of C ( G ). We will construct one for the more general case where G is complemented. Themain idea is to start with the EL -labeling of an upper interval and “twist” by addingsigns, similarly to our EL -labeling for Π dn . The resulting labeling is significantly simplerthan the one I described in [36]. 18et G be a complemented group, and fix a chief series G = N ⊲ N ⊲ · · · ⊲ N k +1 = 1for G throughout the remainder of Section 4. Our labeling (and later our convex eardecomposition) will depend on this choice of chief series, but the consequences for thetopology and h -vector of C ( G ) will obviously depend only on G .For each factor N i /N i +1 , choose a complement B i , i.e., a subgroup such that N i B i = G but N i ∩ B i = N i +1 . (Such a B i exists, as every quotient group of a complemented groupis itself complemented [22, Lemma 3.2.1].) From Section 2.3, the usual dual EL -labelingof the subgroup lattice of a supersolvable group is λ ss ( K ⊃· K ) = max { i : N i K ⊇ K } = min { i : N i +1 ∩ K ⊆ K } . Remember that λ ss labels every chain on a given interval with the same set of labels (upto permutation).We now define a labeling λ of C ( G ) as follows. For K ⊃· K labeled by λ ss with i , let λ ( K x ⊃· K x ) = (cid:26) − i if K x = K x ∩ B i ,i otherwise, and λ ( x ⊃· ∅ ) = 0 . It is immediate from this construction that | λ | [ x,G ] = λ ss (up to the “dropping x ” iso-morphism), much like the situation discussed in Section 3.1 for the d -divisible partitionlattice. Lemma 4.7.
Let G be any supersolvable group. Then:1. If KB = G where B ⊂· G , then K ∩ B ⊂· K .2. If λ ss ( K ⊃· K ) = i , then for any complement B i of N i /N i +1 and K ⊇ K we have K B i = KB i = G .Proof. For part 1, count: | K ∩ B | = | K || B || G | = | K | [ G : B ] and by supersolvability, [ K : K ∩ B ] =[ G : B ] is a prime.For part 2, by the definition of the labeling, N i ∩ K K but N i +1 ∩ K ⊆ K . We seethat N i ∩ K N i +1 , and since N i +1 ⊂· N i , that ( N i ∩ K ) N i +1 = N i and so K N i +1 ⊇ N i .Then K B i = K N i +1 B i ⊇ N i B i = G . Theorem 4.8. If G is a complemented group, then λ is a dual EL -labeling of C ( G ) .Proof. We need to show that every interval has a unique increasing maximal chain whichis lexicographically first. There are two kinds of intervals we need to check:
Case
1. [ ∅ , H x ]As the last label of any chain on this interval is 0, in an increasing chain the others must benegative (in increasing order). Since every chain has the same labels up to permutation,uniqueness of the increasing chain is clear from the definition of λ . Existence follows fromapplying Lemma 4.7 to the maximal subgroups B i . Finally, the chain takes the edge withthe least possible label down from each Hx , so it is lexicographically first.19 ase
2. [ H n x, H x ]Let S be the label set of λ ss restricted to the interval [ H n , H ]. We notice that a − i label ispossible on [ H n x, H x ] only if H n x ⊆ B i and i ∈ S . Thus, the lexicographically first chainis labeled by all possible negative labels (in increasing order), followed by the remaining(positive) labels, also in increasing order. Such a chain clearly exists and is increasing.The negative-labeled part is unique since a − i label corresponds with intersection by B i ,while the positive-labeled part is unique since λ ss = | λ | is an EL -labeling.It remains to check that there are no other increasing chains. We have already shown thatthere is only one increasing chain which has a − i label for each B i containing H n x , soany other increasing chain would need to have a + i label for some i ∈ S where H n x ⊆ B i .Without loss of generality, let this edge H x ⊃· H x be directly down from H x . Then i = min S , and since λ ss is a dual EL -labeling, we have that there is a unique edge downfrom H x with label ± i . But then H x = H x ∩ B i , so the edge gets a − i label, givingus a contradiction and completing the proof.Though we do not need it for our convex ear decomposition, let us briefly sketch thedecreasing chains of λ . Following Th´evenaz [32], a chain of complements to a chief series G = N ⊃· N ⊃· . . . ⊃· N k +1 = 1 is a chain of subgroups G = H k +1 ⊃· H k ⊃· . . . ⊃· H = 1 where for each i, H i is a complement to N i . Th´evenaz showed that the chains ofcomplements in G correspond to homotopy spheres in | L ( G ) | . The following propositionis the EL -shelling version of Th´evenaz’s result for a supersolvable group, and is a specialcase of [37, Proposition 4.3]. Proposition 4.9.
The decreasing chains in L ( G ) with respect to λ ss are the chains ofcomplements to G = N ⊲ . . . ⊲ N k +1 = 1 .Proof. If G = H k +1 ⊃· H k ⊃· . . . ⊃· H = 1 is a chain of complements, then N i H i = G ⊇ H i +1 , while N i +1 H i ∩ H i +1 = ( N i +1 ∩ H i +1 ) H i = 1 · H i = H i by left modularity (the Dedekind identity). Thus λ ss ( H i +1 ⊃· H i ) = i , and the chain isdescending.Conversely, any descending chain corresponds to a sphere in | L ( G ) | , and by Th´evenaz’scorrespondence, there can be no others. Corollary 4.10.
The decreasing chains in C ( G ) with respect to λ are all cosets of chainsof complements { G = H k +1 x ⊃· . . . ⊃· H x = x ⊃· ∅} to the chief series G = N ⊲ . . . ⊲N k +1 = 1 such that no H i x = H i +1 x ∩ B i . C ( G ) Recall that subgroups H and K commute if HK = KH is a subgroup of G . Lemma 4.11. (Warm-up Lemma)
Let G be a solvable group with chief series G = N ⊲N ⊲ · · · ⊲ N k +1 = 1 , and B i and B j be complements of normal factors N i /N i +1 and N j /N j +1 where i = j . Then B i and B j commute. roof. Suppose j < i . Then N i +1 ( N i ⊆ N j +1 ( N j , and N j +1 ⊆ B j . Thus, B j B i ⊇ N i B i = G .Recalling G = N ⊲ N ⊲ · · · ⊲ N k +1 = 1 as the chief series we fixed in Section 4.2, let B = { B i : B i is a complement to N i /N i +1 , ≤ i ≤ k } be a set of complements to N i , one complement for each chief factor (so that |B| = k ).For any x ∈ G , let B x = { B i x : B i ∈ B} . We will call B a base-set for C ( G ).The first step is to show that intersections of certain cosets of B give us a cube, usinga stronger version of Lemma 4.11. Lemma 4.12. If B is a base-set, then ( B i ∩ · · · ∩ B i l ) B i ℓ +1 = G .Proof. We count | ( B i ∩ · · · ∩ B i ℓ ) B i ℓ +1 | = | ( B i ∩ · · · ∩ B i ℓ ) || B i ℓ +1 || B i ∩ · · · ∩ B i ℓ ∩ B i ℓ +1 | = | B i ∩ · · · ∩ B i ℓ − || B i ℓ || B i ℓ +1 || ( B i ∩ · · · ∩ B i ℓ − ) B i ℓ || B i ∩ · · · ∩ B i ℓ ∩ B i ℓ +1 | . By induction on ℓ , this is= | B i ∩ · · · ∩ B i ℓ − || B i ℓ || B i ℓ +1 || G || B i ∩ · · · ∩ B i ℓ ∩ B i ℓ +1 | , and by symmetry, | ( B i ∩ · · · ∩ B i ℓ ) B i ℓ +1 | = | ( B i ∩ · · · ∩ B i ℓ − ∩ B i ℓ +1 ) B i ℓ | . Repeating this argument shows that | ( B i ∩ · · · ∩ B i ℓ ) B i ℓ +1 | is independent of the orderingof the B i j ’s, or of the choice of i ℓ +1 .Then take i ℓ +1 to be the largest index of any such B i j , so that N i ℓ +1 ⊆ B i ∩ · · · ∩ B i ℓ .In particular, ( B i ∩ · · · ∩ B i ℓ ) B i ℓ +1 ⊇ N i ℓ +1 B i ℓ +1 = G . Since the ordering of the i j ’sdoesn’t affect the cardinality, | ( B i ∩ · · · ∩ B i ℓ ) B i ℓ +1 | = | G | for any choice of i ℓ +1 , provingthe lemma. Corollary 4.13. If B is a base-set and x is such that the elements of B and B x aredistinct from one another (i.e., B i = B i x for all i ), then the meet sublattice generated by B ∪ B x is isomorphic to the face lattice of the boundary of a k -cube.Proof. Any B j commutes with any intersection of B i ’s, j = i , and the result follows fromLemma 4.7 and since B i ∩ B i x = ∅ for all i .21e henceforth assume that G is relatively complemented.Let B be a base-set for C ( G ) as above, and x ∈ G be such that B i x = B i (for each i ).Then we define Σ B x to be the meet sublattice of C ( G ) generated by B x ∪ { B i : B i x = B i x } , and the larger meet sublattice Σ + B x to be generated by B x ∪ { B i : B i x = B i x } ∪ { B i y i : B i = B i } , where the y i ’s are some elements such that B i y i = B i x . By Lemma 4.7 and the proof ofCorollary 4.13, \ { i : B i x = B i x } B i ∩ \ { i : B i = B i } B i y i = y (for some y ),so Σ + B x is given by all intersections of B x ∪ B y . Thus (also by Corollary 4.13) | Σ + B x | is aconvex polytope with subcomplex | Σ B x | . Lemma 4.14.
Let H x ⊃· H x be an edge in C ( G ) with λ ( H x ⊃· H x ) = i . Then H x = H x ∩ B i x for some complement B i to N i /N i +1 .Proof. Since every maximal chain in C ( G ) has exactly one edge with λ ( H x ⊃· H x ) = ± i for each i ∈ [ k ], it suffices to show that H is contained in some complement B i to N i /N i +1 .Then H cannot be contained in B i , as that would give two ± i edges, and so H = H ∩ B i .Since G is relatively complemented, every interval in L ( G ) is complemented. In par-ticular, any interval of height 2 has both increasing and decreasing chains, so for any H − ⊃· H there is an H +1 ⊂· H − with λ ss ( H − ⊃· H +1 ) = i .Repeat this argument inductively on H − ⊃· H +1 until H − = G . The final H +1 is thedesired B i , and the definition of λ ss shows that B i is a complement to N i /N i +1 . Corollary 4.15.
Every maximal chain in C ( G ) is in some Σ B x . Corollary 4.15 would not hold if we replaced ‘relatively complemented’ with any weakercondition, since the result implies coatomicity, and Proposition 4.3 tells us that relativelycomplemented groups are exactly those with coatomic subgroup lattice.Now that we have a set of cubes that cover C ( G ), the next step is to assign an orderto them. For any base-set B , let ρ i ( B ) be 0 if B i = B i , and 1 otherwise. We put the ρ i ’s together in a binary vector ρ ( B ), which we will call the pattern of B . Order the B x ’s(and hence the Σ B x ’s) in any linear extension of the lexicographic order on ρ ( B ). Let ∆ B x be the simplicial complex with facets the maximal chains that are in Σ B x , but not in anypreceding Σ B ′ x ′ .The Σ B x ’s are generally proper subsets of face lattices of convex polytopes, so (CLced-polytope) does not hold and we do not have an EL -ced. We can use the same sort ofargument, however, to prove the following refinement of Theorem 1.2: Theorem 4.16. { ∆ B x } is a convex ear decomposition for C ( G ) under the pattern ordering. Corollary 4.15 shows that the ears cover C ( G ), that is, that (CLced-union) holds. Ournext step is to show that an analogue of (CLced-desc) holds.It will be convenient to let S ([ a, b ]) be the label set of | λ | on the interval [ a, b ], that is,the set of nonnegative i ’s such that λ gives ± i labels on cover relations in [ a, b ].22 emma 4.17. For any interval [ a, b ] in C ( G ) , there is at most one (dual) descendingmaximal chain c on [ a, b ] which is in ∆ B x .Proof. If a = ∅ , then the unique descending chain on [ a, b ] in Σ B x is given by intersectingwith each B i x ( i ∈ S ([ ∅ , b ]) \ { } ) in order.If a = ∅ , then the interval [ a, b ] in Σ B x is Boolean, with a maximal chain for eachpermutation of S ([ a, b ]). If there is a − i label on a chain in ∆ B x , then the edge can beobtained by intersecting with B i . But since Σ B x is the first such complex containing thechain, we must have ρ i ( B ) = 0 (otherwise, replace B i x with B i ). But this tells us thatevery label with absolute value i on [ a, b ] in ∆ B x is negative. Every such chain thus hasthe same set of labels, and at most one permutation of these labels is descending. Corollary 4.18. ∆ B x is shellable.Proof. Suppose a maximal chain c = { G = c ⊃· . . . ⊃· c k +1 ⊃· c k +2 = ∅} in Σ B x has anascent at j . If j = k + 1, then it is immediate that c \ { c j } has two extensions in Σ B x ,and we argue exactly as in Lemma 2.7 and Theorem 2.9.If j = k + 1, then the ascent at j has labels − i,
0, and hence ρ i ( B ) = 0 and Σ B x is thefirst cube containing c \ { c k +1 } . Intersecting with B i x instead of B i at c k gives anotherchain c ′ in ∆ B x with a descent at k + 1, and we again argue as in Theorem 2.9.Finally, we show directly that (ced-bdry) holds. We start with a lemma. Lemma 4.19.
Given any chain c = { G = c ⊃ · · · ⊃ c m ⊃ c m +1 = ∅} , there is anextension to a maximal chain c ++ such that if c is in Σ B x , then c ++ is in some Σ + B x . If B x is the first such with c in Σ B x , then c ++ is in Σ B x .Proof. We make the extension in two steps. First, let c + be the extension of c by aug-menting each c j ⊃ c j +1 for j = m with the chain on [ c j +1 , c j ] that is increasing accordingto | λ | . Intersecting c j iteratively with B i x or B i (as appropriate, for each i in S ([ c j +1 , c j ]))in increasing order gives this chain, thus, c + is also in Σ B x .In a similar manner, let c ++ be the extension of c + at c m ⊃ ∅ by intersecting witheach B i for i ∈ S ( m ) in increasing order. Suppose c m = Hx . Then uniqueness of thelexicographically first chain in [1 , H ] gives that H ∩ B i = H ∩ B i , so there is some B i y i with Hx ∩ B i = Hx ∩ B i y i . Repeated use of this gives us a Σ + B x containing c ++ : thegenerating elements for this cube include those for Σ B x and the B i y i ’s found here. Noticethat if ρ ( i ) = 0 for each i ∈ S ([ ∅ , c m ]), then B i is already in the generating set for Σ B x ,thus c ++ is also in Σ B x . Proposition 4.20. ∆ B x ∩ (cid:0)S B ′ x ′ ≺B x ∆ B ′ x ′ (cid:1) = ∂ ∆ B x .Proof. Suppose that c is in ∆ B x ∩ (cid:0)S B ′ x ′ ≺B x ∆ B ′ x ′ (cid:1) , and let c ++ be as in Lemma 4.19.Then c ++ is an extension in Σ + B x , but since c ++ is contained in Σ B ′ x for the first suchcomplex containing c , we get that c ++ is in M (Σ + B x ) \ M (∆ B x ). Lemma 2.8 then givesthat c is in ∂ ∆ B x .Conversely, let c be in ∆ B x , but not in a previous Σ B ′ x ′ . Since c is not in any previousΣ B ′ x ′ , no extensions of it are either, so any extension of c that is in Σ B x is in ∆ B x . As23e have ordered the base-sets by pattern, we get that ρ i ( B ) = 0 for i ∈ S ([ ∅ , c m ]), thus,by the special treatment of B i in the definition of Σ B x , every extension of c in any Σ + B x is in Σ B x . Combining these two statements, we see that there is no extension of c in M (Σ + B x ) \ M (∆ B x ), and so by Lemma 2.8 that c is not in ∂ ∆ B x .We have now finished the proof of Theorem 4.16. Let us review: Corollary 4.13 gave us(ced-polytope), Proposition 4.20 was (ced-bdry), and Corollary 4.15 gave us (ced-union).We notice that the base-set with the earliest pattern is B = { B i } , and that each Σ B x is the face lattice of a cube. Thus the first ∆ B x is a polytope, while all subsequent onesare proper subcomplexes of polytopes. Since we proved in Corollary 4.18 that each ∆ B x is shellable, we have (ced-topology). Note . As previously mentioned, the convex ear decomposition we have constructedis not a (dual) EL -ced. Although we would rather find an EL -ced than a general convexear decomposition, this is not in general possible with the cubes we are looking at here.For example C ( Z ) has exactly three possible Σ + B x ’s, but the homotopy type of the wedgeof 6 1-spheres, so some | Σ + B x | \ | Σ + B ′ x ′ | must be disconnected. The example of C ( Z ) is ageometric lattice, so does have an EL -ced (for a different EL -labeling), but I have notbeen able to extend this to an EL -ced for other relatively complemented groups.The reader may have noticed that the constructed convex ear decomposition is notfar from being an EL -ced – the difference is that each Σ + B x gives several “new” ears – andthat another possibility would be to extend the definition of EL -ced to cover this case.However, as this would make the definition more complicated, and as the gain seemsrelatively small, I have chosen to leave the definition as presented. Throughout this section, let P and P be bounded posets.In Section 3.1, we defined the product P × P and lower reduced product P ˇ × P of P and P . It should come as no surprise that the upper reduced product P ˆ × P of P and P is defined as (cid:0) ( P \ { ˆ1 } ) × ( P \ { ˆ1 } ) (cid:1) ∪ { ˆ1 } . There is a natural inclusion of P ˇ × P (and of P ˆ × P ) into P × P .Our goal in Section 5 is to explain the background and give proofs for Theorems 1.4and 1.6. The flavor and techniques of this section are different from the previous two,so we pause to justify its connection with “Cubical Convex Ear Decompositions”. Lowerreduced products come up fundamentally both in the d -divisible partition lattice, as wediscussed in Section 3.1, as well as in the coset lattice, where C ( G × G ) ∼ = C ( G ) ˇ × C ( G )for groups G and G of co-prime orders. And some of the decompositions in productposets are cubical after all: a cube is the direct product of intervals, so if C d is theboundary of the d -cube, with face lattice L ( C d ), then L ( C d ) = ˇ Q d L ( C ). I am told that the following proposition is folklore. It is also discussed briefly in [12].24 roposition 5.1. If Σ and Σ are the face lattices of convex polytopes X and X , then1. Σ × Σ is the face lattice of the “free join” X ⊛ X , a convex polytope.2. Σ ˇ × Σ is the face lattice of the Cartesian product X × X , a convex polytope.3. Σ ˆ × Σ is the face lattice of the “free sum” of X and X , a convex polytope. Proposition 5.1 guides us to a proof of Lemma 1.5. Our main tool will be stellarsubdivision.If ∆ is a convex polytope with a proper face σ , then a stellar subdivision of ∆ at σ ,denoted stellar σ ∆, is conv (∆ ∪ { v σ } ), where v σ = w σ − ε ( w ∆ − w σ ) for some point w σ in the relative interior of σ , some point w ∆ in the interior of ∆, and a small number ε .In plain language, we “cone off” a new vertex lying just over σ . Note that the relativeinterior of a vertex is the vertex itself. Stellar subdivisions are discussed in depth in [9,III.2] and [8].The main fact [9, III.2.1, III.2.2] that we will need is that the faces of the boundarycomplex of stellar σ ∆ are { τ : σ τ } ∪ { v σ ∗ τ : τ ∈ ∆ with τ, σ ⊆ τ ′ for some τ ′ ∈ ∆ , but σ τ } . Thus the stellar subdivision replaces the faces containing σ with finer subdivisions. Example 5.2. [8, Section 2] The barycentric subdivision of a polytopal d -complex ∆ isthe repeated stellar subdivision of ∆ along a reverse linear extension of its face lattice L (∆). That is, subdivide each d -dimensional face, then each ( d − X is the boundary complex of a polytope, then let X denote conv X , that is, thepolytope of which X is the boundary complex. Lemma 5.3.
Suppose P and P are bounded posets and that | P | and | P | are the bound-ary complexes of polytopes. Then | P ˇ × P | can be obtained from the boundary complex of | P | × | P | by a sequence of stellar subdivisions.Proof. Let ∆ be the boundary complex of | P | × | P | . The faces of ∆ are exactly theproducts F (1) × F (2) , where each F ( i ) is a non-empty face in P i , and at least one is proper.In particular the vertices are products of vertices v (1) × v (2) , where v ( i ) is in P i \ { ˆ0 , ˆ1 } .We write this product of vertices as ( v (1) , v (2) ), and think of it as sitting in | P ˇ × P | .We start by ordering the elements { v (2) } of P by a reverse linear extension, andstellarly subdividing at each σ = | P | × v (2) in this order. Inductively assume that thefaces containing σ are those of the form ( | P | × F (2) ) ∗ C , where F (2) is a face of | P | withtop-ranked vertex v (2) , and C is a simplex corresponding to (the simplicial join of) a chainof elements of the form (ˆ1 , w (2) ) (with each w (2) > v (2) ). Subdivision replaces these faceswith those of the form ( | P | × F (2)0 ) ∗ C ∗ { v σ } , where F (2)0 is a face having top-rankedvertex < v (2) . We abuse notation to call the newly introduced vertex v σ as (ˆ1 , v (2) ), whichputs us in the situation required to continue our induction.25e next do the same procedure for the faces v (1) × | P | . That is, we order { v (1) } bya reverse linear extension of P , and repeatedly perform stellar subdivision at each suchface according to this order. Since a face cannot contain both | P | and | P | , these stellarsubdivisions are independent of the ones at | P | × v (2) .After subdividing at all | P | × v (2) and v (1) × | P | , we obtain a complex ∆ . The vertexset of ∆ is exactly P ˇ × P \ { ˆ0 , ˆ1 } . The faces of ∆ are { ( F (1) × F (2) ) ∗ C } , where F ( i ) isa face of | P i | , and C is a simplex corresponding to either a chain of elements (ˆ1 , w (2) ) ora chain of elements ( w (1) , ˆ1).Finally, we perform stellar subdivision at the vertices v = ( v (1) , v (2) ), where v ( i ) ∈ P i \{ ˆ0 , ˆ1 } , in the order of a reverse linear extension of P ˇ × P . We make an induction argumentparallel to the one above: at the step associated with vertex v , the faces containing v are { ( F (1) × F (2) ) ∗ C } . As before, F ( i ) is a face of | P i | with top-ranked vertex v ( i ) , and C corresponds to (the simplicial join of) elements in a chain greater than v in P ˇ × P . Stellarsubdivision at v replaces these faces with { ( F (1)0 × F (2)0 ) ∗ C ∗ { v }} , where F ( i )0 has greatestvertex < v ( i ) , and we continue the induction.When we have subdivided at every vertex, we obtain a complex ∆ . The faces of ∆ are simply { C } , where C is the simplicial join of vertices in a chain of P ˇ × P , which isthe definition of the order complex | P ˇ × P | . Corollary 5.4.
If If P and P are bounded posets such that | P | and | P | are the boundarycomplexes of polytopes, then | P ˇ × P | and (by duality) | P ˆ × P | are also boundary complexesof polytopes. For P × P , a similar result holds. Recall that the free join ∆ ⊛ ∆ of two polytopes∆ and ∆ is obtained by taking the convex hull of embeddings of ∆ and ∆ into skewaffine subspaces of Euclidean space (of high enough dimension). The faces of ∆ ⊛ ∆ , ashinted in Proposition 5.1, are F (1) ⊛ F (2) , and dim F (1) ⊛ F (2) = dim F (1) + dim F (2) + 1. Lemma 5.5.
Suppose P and P are bounded posets and that | P | and | P | are the bound-ary complexes of polytopes. Then | P × P | can be obtained from the boundary complex of | P | ⊛ | P | by a sequence of stellar subdivisions.Proof. Since the details of the proof are very similar to the preceding Lemma 5.3, weprovide a sketch only. Let ∆ = | P | ⊛ | P | . Notice that the vertices of ∆ are {∅ ⊛ v (2) } ∪{ v (1) ⊛ ∅} , while the edges are { v (1) ⊛ v (2) } .As in Lemma 5.3, we begin by ordering the facets | P | ⊛ v (2) and v (1) ⊛ | P | accordingto reverse linear extensions of P and P , and inductively performing stellar subdivision.Each such subdivision creates a vertex, which we name (ˆ1 , v (2) ) or ( v (1) , ˆ1). We obtaina complex ∆ with faces { ( F (1) ⊛ F (2) ) ∗ C } where F ( i ) is a proper face of P i (possiblyempty), and C corresponds to a chain in the elements { (ˆ1 , v (2) ) } or { ( v (1) , ˆ1) } .We then order the edges v (1) ⊛ v (2) by a linear extension of P × P , and inductively per-form stellar subdivision to create vertices ( v (1) , v (2) ). The resulting complex is isomorphicto | P × P | . Corollary 5.6. If P and P are bounded posets such that | P | and | P | are the boundarycomplexes of polytopes, then | P × P | is also the boundary complex of a polytope. Let P and P be bounded posets with respective convex ear decompositions { ∆ (1) s } and { ∆ (2) t } . Let P be either P × P , P ˇ × P , or P ˆ × P ; with coordinate projection maps p and p . Take d = dim | P | , d = dim | P | , and d = dim | P | .We define ∆ s,t to be the simplicial complex generated by the maximal chains of P that project to ∆ (1) s in the first coordinate, and ∆ (2) t in the second. Order these complexeslexicographically by ( s, t ). Theorem 5.7. { ∆ s,t } is a convex ear decomposition for | P | .Proof. Lemma 1.5 gives that ∆ s,t is a subcomplex of the boundary complex of a polytope,so (ced-polytope) is satisfied.The topology of various poset products is nicely discussed in Sundaram’s [30, Section2]. There are homeomorphisms | P ˇ × P | ≈ | P ˆ × P | ≈ | P | ∗ | P | , where ∗ is the join of topological spaces. This result goes back to Quillen [18, Proposition1.9], although his notation was much different – Sundaram makes the connection in [30,proof of Proposition 2.5]. Walker [34, Theorem 5.1 (d)] extends this to show that | P × P | ≈ susp( | P | ∗ | P | ) , where susp denotes the topological suspension. Identical proofs to Quillen’s and Walker’sshow that ∆ s,t ≈ ∆ s ∗ ∆ t in the upper/lower reduced case, and that ∆ s,t ≈ susp(∆ s ∗ ∆ t )in the direct product case. In particular, ∆ s,t is a d -ball for ( s, t ) > (1 ,
1) and a d -spherefor ( s, t ) = (1 ,
1) by results in PL-topology [20, Proposition 2.23]. We have shown that(ced-topology) is satisfied.It is clear that (ced-union) holds. It remains to check (ced-bdry).
Claim . ∂ ∆ s,t is exactly the set of all faces in ∆ s,t that project to either ∂ ∆ (1) s or ∂ ∆ (2) t (or both). Proof.
The boundary of a simplicial d -ball ∆ is generated by the d − c is a d − s,t (i.e., a chain of length d − p ( c ) and p ( c ) also has codimension 1.Since ∆ s,t is defined to be the chains which project to ∆ (1) s and ∆ (2) t , we see that if p ( c )is d − c is d − p ( c ) has exactly one extensionin ∆ (1) s if and only if c has exactly one extension in ∆ s,t . The argument if p ( c ) hascodimension 1 is entirely similar. 27e now show both inclusions for (ced-bdry). If d is any chain in ∂ ∆ s,t with p ( d ) in ∂ ∆ (1) s , then p ( d ) is in ∆ (1) u for some u < s by (ced-bdry), so d is in ∆ u,t ; similarly if p ( d )is maximal and p ( d ) is in ∂ ∆ (2) t . Thus ∂ ∆ s,t ⊆ ∆ s,t ∩ (cid:16)S ( u,v ) < ( s,t ) ∆ u,v (cid:17) .In the other direction: if c is in ∆ s,t and ∆ u,v (for ( u, v ) < ( s, t )), then p ( c ) is inboth ∆ (1) s and ∆ (1) u . If s = u , then p ( c ) is in ∂ ∆ (1) s , so c is in ∂ ∆ s,t . A similar argumentapplies for p when s = u . Thus, ∂ ∆ s,t ⊇ ∆ s,t ∩ (cid:16)S ( u,v ) < ( s,t ) ∆ u,v (cid:17) , and we have shown(ced-bdry), completing the proof. C L -labelings
In this subsection, we explicitly recall the product CL -labelings introduced by Bj¨ornerand Wachs in [4, Section 10], and hinted at in Section 3.1. Since there is no particularreason to work with dual labelings in Section 5, I’ve chosen to work with standard (notdual) CL -labelings, so that everything is “upside down” relative to Sections 3 and 4. Sincethe root of an edge of the form ˆ0 ⋖ x is always ∅ , we suppress the root from our notationin this case.Let P be a bounded poset with a CL -labeling λ that has label set S λ . A label s ∈ S λ is atomic if it is used to label a cover relation ˆ0 ⋖ x (for any atom x ), and non-atomic if itis used to label any other rooted cover relation. (In an arbitrary CL -labeling, a label canbe both atomic and non-atomic.) A CL -labeling is orderly if S λ is totally ordered andpartitions into S − λ < S Aλ < S + λ , where every atomic label is in S Aλ , and every non-atomiclabel is either in S − λ or S + λ . There are similar definitions of co-atomic , non-co-atomic , and co-orderly , and of course we can generalize to talk of orderly and co-orderly chain edgelabelings, even if the CL -property is not met. Lemma 5.9. (Bj¨orner and Wachs [4, Lemma 10.18])
Let P be a bounded poset with a CL -labeling λ . Then P has an orderly CL -labeling λ ′ , and a co-orderly CL -labeling λ ′′ ,such that any maximal chain c in P has the same set of ascents and descents under eachof the three labelings λ , λ ′ , and λ ′′ . The proof involves constructing a recursive atom ordering from λ , and then construct-ing a CL -labeling with the desired properties from the recursive atom ordering. Note . The result of Lemma 5.9 is not known to be true if ‘ CL ’ is replaced by ‘ EL ’.To find a CL -labeling of P × P , we label each edge in P × P with the edge in P or P to which it projects. More formally, notice that any rooted cover relation ( r , x ⋖ y )projects to a cover relationship in one coordinate, and to a point in the other. Then the product labeling , denoted λ × λ , labels ( r , x ⋖ y ) with λ i ( p i ( r ) , p i ( x ⋖ y )), where i is thecoordinate where projection is nontrivial. It is straightforward to show that λ × λ is a CL -labeling if λ and λ are CL -labelings of P and P , and where we order S λ ∪ S λ byany shuffle of S λ and S λ [4, Proposition 10.15].The idea behind finding a CL -labeling of P ˇ × P (or similarly P ˆ × P ) is to restrict λ × λ to P ˇ × P . For a cover relation x ⋖ y where x = ˆ0, this works very well, as x ⋖ y P ˇ × P is also a cover relation in P × P , and the roots project straightforwardly. Theproblem comes at cover relations ˆ0 ⋖ y , which project to a cover relation in both P and P . Here, we need to combine the labels λ (cid:0) ˆ0 ⋖ p ( y ) (cid:1) and λ (cid:0) ˆ0 ⋖ p ( y ) (cid:1) .The orderly labelings constructed in Lemma 5.9 are a tool to perform this combinationin a manner that preserves the CL -property. Let P and P have orderly CL -labelings λ and λ , with disjoint label sets S and S . Suppose the label sets are shuffled together as S − < S − < S A < S A < S +1 < S +2 . Then the lower reduced product labeling λ ˇ × λ labels an edge ˆ0 ⋖ y with the word λ (cid:0) p (ˆ0 ⋖ y ) (cid:1) λ (cid:0) p (ˆ0 ⋖ y ) (cid:1) in S A S A (lexicographically ordered), while all other rootededges ( r , x ⋖ y ) (for x = ˆ0) are labeled with the nontrivial projection λ i ( p i ( r ) , p i ( x ⋖ y ))as in λ × λ . Bj¨orner and Wachs proved [4, Theorems 10.2 and 10.17] that λ ˇ × λ is a CL -labeling of P ˇ × P .Similarly, if λ and λ are co-orderly CL -labelings of P and P , with disjoint label setsshuffled together as for the orderly labelings above, we define the upper reduced productlabeling λ ˆ × λ as follows. Label an edge of the form ( r , x ⋖ ˆ1) with the word λ (cid:0) p ( r ) , p ( x ⋖ ˆ1) (cid:1) λ (cid:0) p ( r ) , p ( x ⋖ ˆ1) (cid:1) in S A S A , and all other edges ( r , x ⋖ y ) (for y = ˆ1) as in λ × λ . Then [4, Theorems 10.2and 10.17] gives us that λ ˆ × λ is a CL -labeling of P ˆ × P . Example 5.11.
The labeling λ div we constructed for the d -divisible partition lattice wasan co-orderly EL -labeling of the dual lattice: actually, S A was just { } . As discussed inLemma 3.5, intervals split as products, and the restriction of λ div to an interval splits asthe appropriate product labeling.We summarize in the following theorem: Theorem 5.12. (Bj¨orner and Wachs [4, Proposition 10.15 and Theorem 10.17])
Let P and P be posets, with respective labelings λ and λ .1. If λ and λ are CL -labelings ( EL -labelings), then λ × λ is a CL -labeling ( EL -labeling) of P × P .2. If λ and λ are orderly CL -labelings, then λ ˇ × λ is a CL -labeling of P ˇ × P .3. If λ and λ are co-orderly CL -labelings, then λ ˆ × λ is a CL -labeling of P ˆ × P . C L -ceds of product posets
Fix our notation as in Section 5.2, but suppose in addition that P and P have CL -ceds { Σ (1) s } and { Σ (2) t } with respect to the CL -labelings λ and λ . Denote the resulting ears ofnew chains as { ∆ (1) s } and { ∆ (2) t } , as in Section 2.5. Then take Σ s,t to be the appropriateproduct of Σ (1) s and Σ (2) t , and ∆ s,t to be the associated ear of new chains.We first notice that there is no inconsistency with the notation used in Section 5.2:29 emma 5.13. A maximal chain c is in ∆ s,t if and only if p ( c ) is in ∆ (1) s and p ( c ) isin ∆ (2) t .Proof. The statement follows straightforwardly from the fact that the maximal chains ofΣ s,t are those that project to Σ (1) s and Σ (2) t .As we did in Section 5.2, order the { Σ s,t } according to the lexicographic order of ( s, t ).Let λ be the appropriate product CL -labeling, where we assume without loss of generalityvia Lemma 5.9 that λ and λ are orderly or co-orderly. Then we will prove: Theorem 5.14. { Σ s,t } is a CL -ced for | P | with respect to λ .Proof. Proposition 5.1 tells us that (CLced-polytope) is satisfied, and (CLced-union) isimmediate from the definitions.For (CLced-bdry), we work backwards, and notice that we have already shown inTheorem 5.7 that ∂ ∆ s,t = ∆ s,t ∩ (cid:16)S u,v
Question . Are there other families of posets with similar structure to Π dn and C ( G )?Can the techniques used in Sections 3 and 4 be used to construct dual EL -labelings and EL -ceds? 30hat we mean by ‘similar’ here is not clear. At the least, we need a poset P whereevery interval of the form [ a, ˆ1] is supersolvable, and where the supersolvable structureis canonically determined, i.e., such that we can label all edges of P \ { ˆ0 } in a way thatrestricts to a supersolvable labeling on each such [ a, ˆ1] interval. We then need a way tosign the edges giving an EL -labeling, and the poset has to somehow be ‘wide’ or ‘rich’enough to have an EL -ced.One possible source of such examples is the theory of exponential structures. An exponential structure is a family of posets with each upper interval isomorphic to thepartition lattice, and each lower interval isomorphic to a product of smaller elements inthe same family. Exponential structures were introduced in [27], where the family of d -divisible partition lattices was shown to be one example. Shellings are constructed forsome other examples in [21, 35]. Question . Can techniques like those used in Section 3 (and Section 4) be used to con-struct dual EL -labelings and/or EL -ceds of exponential structures besides the d -divisiblepartition lattice?However, it is not a priori clear how to construct a labeling that restrict to a super-solvable labeling on any [ a, ˆ1] for exponential structures. In examples even finding an EL -labeling often seems to be a difficult problem.A question suggested by the results of Section 5 is: Question . Are there other operations on posets that preserve convex ear decompositionsand/or CL -ceds?For example, Schweig shows [23, Theorem 5.1] that rank-selected supersolvable andgeometric lattices have convex ear decompositions. Do all rank-selected subposets ofposets with convex ear decompositions have a convex ear decomposition? Are there anyother useful constructions that preserve having a convex ear decomposition and/or EL -ced? A place to start looking would be in Bj¨orner and Wach’s papers [2, 3, 4], where theyanswer many such questions for EL / CL -labelings. Acknowledgements
Thanks to Ed Swartz for introducing me to convex ear decompositions; and to him, JaySchweig, and my graduate school advisor Ken Brown for many helpful discussions aboutthem. Tom Rishel listened to and commented on many intermediate versions of the resultsand definitions of this paper. Sam Hsiao helped me in understanding the material ofProposition 5.1, and in looking for its extension to Lemma 1.5. Vic Reiner pointed out thatthe labeling based on pivots of C ( G ) that I used in [36] was really a supersolvable labeling,which suggested the improved EL -labeling used in Section 4. Volkmar Welker suggestedexponential structures as a possible area for further exploration. The anonymous refereegave many helpful comments. 31 eferences [1] Anders Bj¨orner, Shellable and Cohen-Macaulay partially ordered sets , Trans. Amer.Math. Soc. (1980), no. 1, 159–183.[2] Anders Bj¨orner and Michelle L. Wachs,
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