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Which nestohedra are removahedra?
Vincent Pilaud
To cite this version:
Vincent Pilaud. Which nestohedra are removahedra?. Revista Colombiana de Matematicas, 2017, 51
(1), pp.2142. �10.15446/recolma.v51n1.66833�. �hal02344112�
WHICH NESTOHEDRA ARE REMOVAHEDRA?
VINCENT PILAUD
Abstract. A removahedron is a polytope obtained by deleting inequalities from the facet de scription of the classical permutahedron. Relevant examples range from the associahedron to the permutahedron itself, which raises the natural question to characterize which nestohedra can be realized as removahedra. In this paper, we show that the nested complex of any con nected building set closed under intersection can be realized as a removahedron. We present two complementary constructions: one based on the building trees and the nested fan, and the other based on Minkowski sums of dilated faces of the standard simplex. In general, this closure condition is sufficient but not necessary to obtain removahedra. In contrast, we show that the nested fan of a graphical building set is the normal fan of a removahedron if and only if the graphical building set is closed under intersection, which is equivalent to the corresponding graph being chordful (i.e.any cycle induces a clique).
keywords. Building set, nested complex, nestohedron, graph associahedron, generalized per mutahedron, removahedron.
1. Introduction
Thepermutahedronis a classical polytope related to various properties of the symmetric group.
It is obtained as the convex hull of all permutations of [n] and its normal fan is the typeACoxeter fan. In [Pos09], A. Postnikov definedgeneralized permutahedraas the polytopes obtained from the classical permutahedron by gliding its facets orthogonally to its normal vectors without passing any vertex. Equivalently [PRW08], a generalized permutahedron is a polytope whose normal fan coarsens the typeACoxeter fan.
A particularly relevant example of generalized permutahedron is the associahedron constructed by S. Shnider and S. Sternberg [SS93] and J.L. Loday [Lod04]. An associahedron is a poly tope whose 1skeleton realizes the rotation graph on binary trees with nvertices. The polytope of [SS93,Lod04] is a remarkable realization of the associahedron which, besides being a generalized permutahedron, has the following three properties:
(i) it is obtained by deleting inequalities from the facet description of the permutahedron; we call such polytopesremovahedra;
(ii) its vertex coordinates enumerate paths between leaves in the corresponding binary trees;
(iii) it is the Minkowski sum of the faces4I of the standard simplex given by all intervalsIof [n].
This paper studies the connections between these three properties for a larger class of poly topes called nestohedra defined independently by A. Postnikov [Pos09] and by E.M. Feichtner and B. Sturmfels [FS05]. These polytopes extend the graph associahedra of M. Carr and S. De vadoss [CD06] as they realize the nested complex on an arbitrary building set, see Section 2.2for definitions. All nestohedra are generalized permutahedra, but not necessarily removahedra. In this paper, we investigate which nestohedra can be realized as removahedra. We show that the nested complex of a connected building set closed under intersection can always be realized as a removahedron. For graph associahedra, this closure condition is equivalent to the underlying graph being chordful (i.e.any cycle induces a clique). Conversely, we show that the nested fan of a graphical building set is the normal fan of a removahedron if and only if the graph is chordful.
We develop two complementary approaches to describe a nestohedron which is a removahedron.
First, we show that its vertex coordinates can be computed by enumerating paths between leaves in building trees, exactly as in Point (ii) above. Second, we show that it decomposes into a simple Minkowski sum of faces of the standard simplex given by certain paths in the building set, which corresponds to the intervals of [n] in Point (iii) above.
Partially supported by the Spanish MICINN grant MTM201122792 and by the French ANR grants EGOS (12 JS02 002 01) and SC3A (15 CE40 0004 01).
1
2. Preliminaries
2.1. Permutahedra, deformed permutahedra, and removahedra. All polytopes consid ered in this paper are closely related to the braid arrangement and to the classical permu tahedron. Therefore, we first recall the definition and basic properties of the permutahedron (see [Zie95, Lect. 0]) and certain relevant deformations of it. We fix a finite ground set S and denote by{ess∈S}the canonical basis ofR^{S}.
Definition 1. The permutahedronPerm(S)is the convex polytope obtained equivalently as (i) either the convex hull of the vectorsP
s∈Sσ(s)es∈R^{S} for all bijectionsσ:S→[S], (ii) or the intersection of the hyperplaneH^{:}=H^{=}(S)with the halfspacesH^{≥}(R)for∅6=R⊂S,
where H^{=}(R)^{:}=
x∈R^{S}
X
r∈R
x_{r}=
R+ 1 2
and H^{≥}(R)^{:}=
x∈R^{S}
X
r∈R
x_{r}≥
R+ 1 2
,
(iii) or the Minkowski sum of all segments[er, es]for(r, s)∈ ^{S}_{2} .
The normal fan of the permutahedron is the fan defined by thebraid arrangementinH,i.e.the arrangement of the hyperplanes {x∈Hxr=xs} forr 6=s ∈S. Itskdimensional cones corre spond to the surjections fromSto [k+ 1], or equivalently to the ordered partitions ofSintok+ 1 parts. In this paper, we are interested in the following deformations of the permutahedronPerm(S).
These polytopes were called generalized permutahedra by A. Postnikov [Pos09, PRW08], but we prefer the termdeformedto distinguish from other natural generalizations of the permutahedron (to finite Coxeter groups, to removahedra, ...).
Definition 2 ([Pos09, PRW08]). A deformed permutahedron is a polytope whose normal fan coarsens that of the permutahedron. Equivalently [PRW08], it is a polytope defined as
Defo(z)^{:}=
x∈R^{S}
X
s∈S
xs=zS and X
r∈R
xr≥zR for all∅6=R⊂S
,
for somez^{:}= (zR)_{R⊆S}∈(R>0)^{2}^{S}, such that zR+zR^{0} ≤z_{R∪R}^{0} +z_{R∩R}^{0} for anyR, R^{0}⊆S.
As the permutahedron itself, all deformed permutahedra can be decomposed as Minkowski sums and differences of dilates of faces of the standard simplex [ABD10]. For our purposes, we only need here the following simpler fact, already observed in [Pos09].
Remark 3([Pos09]). For anyS⊆S, we consider the face4S:= conv{ess∈S}of the standard simplex4S. For anyy^{:}= (yS)S⊆Swhere allyS are nonnegative real numbers, the Minkowski sum
Mink(y)^{:}= X
S⊆S
yS4S
of dilated faces of the standard simplex is a deformed permutahedron Defo(z), and the values z= (zR)_{R⊆S}of the right hand sides of the inequality description ofMink(y) =Defo(z) are given by
zR= X
S⊆R
yS.
Remark 4. As the normal fan of a Minkowski sum is just the common refinement of the normal fans of its summands, and the normal fan is invariant by dilation, the combinatorics of the face lattice of the Minkowski sumMink(y) only depends on the set{S⊆Sy_{S}>0}of nonvanishing dilation factors. When we want to deal with combinatorics only, we denote generically byMink[C]
any Minkowski sumMink(y) with dilation factorsy= (yS)_{S⊆S} such thatC={S ⊆SyS >0}.
Among these deformed permutahedra, some are simpler than the others as all their facet defining inequalities are also facet defining inequalities of the classical permutahedron. In other words, they are obtained from the permutahedron by removing facets, which motivates the following name.
Definition 5. A removahedron is a polytope obtained by removing inequalities from the facet description of the permutahedron, i.e. a polytope defined for some B⊆2^{S} by
Remo(B)^{:}=H∩ \
B∈B
H^{≥}(B) =
x∈H
X
s∈B
x_{s}≥
B+ 1 2
for allB∈B
.
Our motivation to study removahedra is that they are completely determined by their nor mal fan (in fact even just by their normal vectors). In general, many relevant simplicial spheres (associahedra [Lod04, HL07], graph associahedra [CD06], nestohedra [Pos09, FS05], signed tree associahedra [Pil13], Coxeter associahedra [HLT11], accordion complexes [MP17], ...) are easy to realize by fans that coarsen the normal fan of a zonotope (often the classical permutahedron, but also Coxeter permutahedra [HLT11], or even other zonotopes [HPS17,MP17]). Polytopal realiza tion problems would thus be simpler if all these fans would be realized by removing the undesired facets of these zonotopes. This paper studies when this approach is possible for nestohedra.
2.2. Building set, nested complex, and nested fan. We now switch to building sets and their nested complexes. We only select from [CD06,Pos09,FS05,Zel06] the definitions needed in this paper. More details and motivation can be found therein.
Definition 6. Abuilding setBon a ground setSis a collection of nonempty subsets ofSsuch that (B1) ifB, B^{0} ∈BandB∩B^{0}6=∅, then B∪B^{0} ∈B, and
(B2) Bcontains all singletons {s} fors∈S.
A building set is connectedifS is the unique maximal element. Moreover, we say that a building setB is closed under intersectionifB, B^{0} ∈BimpliesB∩B^{0}∈B∪ {∅}.
All building sets in this manuscript are assumed to be connected and we will study the relation between removahedra and building sets closed under intersection. We first recall a general example of building sets, arising from connected subgraphs of a graph.
Example 7. Given a graph G with vertex set S, we denote by BG the graphical building set on G, i.e. the collection of all nonempty subsets ofS which induce connected subgraphs of G.
The maximal elements ofBG are the vertex sets of the connected components of G, and we will therefore always assume that the graph G is connected. We call a graph Gchordful if any cycle of G induces a clique. Observe in particular that every tree is chordful. The following statement describes the graphical building sets of chordful graphs.
Lemma 8. A (finite connected) graph G is chordful if and only if its graphical building set is closed under intersection.
Proof. Assume that G is chordful, and consider B, B^{0} ∈ BG and s, t ∈ B∩B^{0}. As B and B^{0} induce connected subgraphs of G, there exists pathsP andP^{0} betweensandtin G whose vertex sets are contained inB andB^{0}, respectively. The symmetric differenceP4P^{0} of these paths is a collection of cycles. Since G is chordful, we can replace in each of these cycles the subpath of P (resp. ofP^{0}) by a chord. We thus obtain a path from sto t which belongs toB∩B^{0}. It follows thatB∩B^{0} induces a connected subgraph of Gand thus that B∩B^{0} ∈BG.
Assume reciprocally that G has a cycle (si)_{i∈Z}_{`}, with a missing chord sxsy. Consider the subsetsB^{:}={six≤i≤y}andB^{0}^{:}={siy≤i≤x}, where the inequalities between labels inZ`
have to be understood cyclically. Clearly,B, B^{0} ∈BG whileB∩B^{0}={sx, sy}∈/ BG.
Example 9. The following sets are building sets:
• B^{ex0 :}= 2^{[4]}r{∅}is the graphical building set over the complete graphK_{4},
• B^{ex1 :}= 2^{[4]}r
∅,{1,3} is the graphical building set over the graphK4r {1,3} ,
• B^{ex2 :}= 2^{[4]}r
∅,{1,3},{1,4},{1,3,4} is the graphical building set overK4r
{1,3},{1,4} ,
• B^{ex3 :}=
{1},{2},{3},{4},{5},{1,2,3},{1,3,4,5},{1,2,3,4,5} is not graphical,
• B^{ex4 :}=
{1},{2},{3},{4},{5},{1,2,3,4},{3,4,5},{1,2,3,4,5} is not graphical.
The two building setsB^{ex0} andB^{ex2}are closed under intersection, while the other three are not.
In this paper, we focus on polytopal realizations of the nested complex of a building set, a simplicial complex defined below. Following [Zel06] and contrarily to [Pos09], we do not includeS in the definition ofBnested sets.
Definition 10. ABnested setNis a subset ofBr{S} such that (N1) for anyN, N^{0}∈N, either N⊆N^{0} or N^{0} ⊆N or N∩N^{0}=∅, and
(N2) for anyk≥2 pairwise disjoint setsN1, . . . , Nk ∈N, the unionN1∪ · · · ∪Nk is not inB.
The Bnested complexis the simplicial complex N(B)of allBnested sets.
Example 11. For a graphical building set BG, Conditions (N1) and (N2) in Definition10 can be replaced by the following: for anyN, N^{0} ∈N, eitherN ⊆N^{0} or N^{0} ⊆N or N∪N^{0} ∈/BG. In particular, theBGnested complex is a clique complex: a simplex belongs toBG if and only if all its edges belong toBG.
TheBnested sets can be represented by the inclusion poset of their elements. Since we only consider connected building sets, the Hasse diagrams of these posets are always trees. In the next definition, we consider rooted trees whose vertices are labeled by subsets of S. For any vertex v in a rooted tree T, we call descendant setof v in T the union desc(v,T) of the label sets of all descendants ofvin T, including the label set of the vertexv itself. TheBnested sets are then in bijection with the followingBtrees.
Definition 12. ABtreeis a rooted tree whose label sets partition S and such that (1) for any vertexv ofT, the descendant setdesc(v,T) belongs toB,
(2) for anyk≥2incomparable verticesv_{1}, . . . , v_{k} ∈T, the unionS
i∈[k]desc(v_{i},T)is not inB.
We denote byN(T)^{:}={desc(v,T)v vertex ofTdistinct from its root}theBnested set corre sponding to a BtreeT. Note that N(T) is a maximalBnested set if and only if all the vertices ofTare labeled by singletons ofS. We then identify a vertex ofTwith the element ofSlabeling it.
TheBnested sets and theBtrees naturally encode a geometric representation of theBnested complex as a complete simplicial fan. In the next definition, we define 11R:=P
r∈Rer, forR⊆S, and we denote by ¯11Rthe projection of 11R toHorthogonal to 11S. Moreover, we considerHas a linear space.
Definition 13. The Bnested fanF(B)is the complete simplicial fan of H^{:}=H^{=}(S)with a cone C(N)^{:}= cone{−11¯_{N} N ∈N}={x∈Hx_{r}≤x_{s} for each pathr→sin T} =^{:}C(T) for eachBnested set NandBtree Twith N=N(T).
TheBnested fan is the normal fan of various deformed permutahedra (see Definition2). We want to underline two relevant examples:
(i) the deformed permutahedronDefo(z) with right hand sidez^{:}= (z_{R})_{R⊆S}defined byz_{R}= 3^{R−2} ifR∈Bandz_{R}=∞otherwise, see [Dev09];
(ii) the Minkowski sumMink(11_{B}) of the faces of the standard simplex corresponding to all the elements of the building set B, see [Pos09, Section 7].
However, these two realizations are not always removahedra. In general, the support functions realizing the normal fan F(B) can be characterized by local conditions [Zel06, Proposition 6.3], but it is difficult to use these conditions to characterize which nested complexes can be realized as removahedra. In this paper, we adopt a different approach.
2.3. Results. The objective of this paper is to discuss necessary and sufficient conditions for the Bnested fan to be the normal fan of a removahedron (see Definition5), and to study some prop erties of the resulting removahedra (vertex description, Minkowski sum decomposition). We thus consider the removahedron Remo(B) described by the facet defining inequalities of the permuta hedronPerm(S) whose normal vectors are rays of theBnested fan,i.e.
Remo(B)^{:}=H∩ \
B∈B
H^{≥}(B) =
x∈H
X
s∈B
xs≥
B+ 1 2
for allB∈B
.
Example 14. Figure1represents the removahedraRemo(B^{ex0}),Remo(B^{ex1}) andRemo(B^{ex2}) cor responding to the graphical building setsB^{ex0},B^{ex1}andB^{ex2}of Example9. Observe thatRemo(B^{ex0}) and Remo(B^{ex2}) realize the corresponding nested complexes, whereas Remo(B^{ex1}) is not even a simple polytope.
1234 1234
3124 3412
1234 1234
3124 3412
1342 1423
3^{}4^{}1^{}2 4^{}3^{}1^{}2
4321 3421
3142 3241
3^{}2^{}1^{}4 1342
1^{}4^{}3^{}2 1423
1243 1^{}2^{}3^{}4 2134 1324 4123
4132
2314 3124
1234 1234
1324 3124 3412
1342 1423
Figure 1. The 3dimensional permutahedron Perm([4]) =Remo(B^{ex0}) and the removahedraRemo(B^{ex1}) andRemo(B^{ex2}).
We want to understand when doesRemo(B) realize the nested complexN(B). The following statement provides a general sufficient condition.
Theorem 15. If B is a connected building set closed under intersection, then the normal fan of the removahedronRemo(B)is theBnested fanF(B). In particular,Remo(B)is a simple polytope.
We provide two different complementary proofs of this result, which both show relevant by product properties of these removahedra:
(i) In Section 3, we apply a result from [HLT11] which characterizes the valid right hand sides to realize a complete simplicial fan as the normal fan of a convex polytope. For this, we first compute for each maximal BtreeT the intersection point a(T) of all facet defining hyper planes ofRemo(B) normal to the rays of the coneC(T). This provides along the way relevant formulas for the vertex coordinates of the corresponding vertex of the removahedronRemo(B).
We then show that the vector joining the pointsa(T) anda(T^{0}) corresponding to two adja cent cones C(T) andC(T^{0}) indeed points in the right direction for Remo(B) to realize the nested fanF(B).
(ii) In Section4, we show that certain Minkowski sums of dilated faces of the standard simplex realize the Bnested fan as soon as all Bpaths appear as summands. We then find the appropriate dilation factors for the resulting polytope to be a removahedron.
Relevant examples of application of Theorem15arise from graphical building sets of chordful graphs. If we restrict to graphical building sets, Lemma24shows that chordfulness of G is also a necessary condition for theBGnested fanF(BG) to be the normal fan ofRemo(BG). We therefore obtain the following characterization of the graphical building sets whose nested fan is the normal fan of a removahedron. This characterization is illustrated by Example14.
Theorem 16. The BGnested fan F(BG) is the normal fan of the removahedron Remo(BG)if and only if the graphG is chordful.
Example 17. Specific families of chordful graphs provide relevant examples of graph associahedra realized by removahedra,e.g.:
• the path associahedra,aka.classical associahedra [SS93,Lod04],
• the star associahedra,aka.stellohedra [PRW08],
• the tree associahedra [Pil13],
• the complete graph associahedra,aka.classical permutahedra.
In contrast, for the cycle associahedra,aka.cyclohedra, the nested fanF(BO_{n}) is not the normal fan of a removahedron since the cycle On is not chordful forn≥4.
2 3 1
5 4
2 5 1 3 4
2 3 4 1
5
1 2 4 3
5 2
5 3 1 4
2 4 5 1 3
2 5 4 1 3 3 2 1
5 4
3 2 1
4 5
2 3 1
4 3 5
1 2
4 5
3 1 2
5 4
Figure 2. The 4dimensional removahedronRemo(B^{ex3}) realizes theB^{ex3}nested fan, althoughB^{ex3}is not closed under intersection.
To conclude, we observe that a general building setB does not need to be closed under inter section for the removahedron Remo(B) to realize the Bnested fanF(B). In fact, our first proof of Theorem 15 shows the following refinement. We say that two building blocks B, B^{0} ∈ B are exchangeableif there exists two maximalBnested setsN,N^{0} such thatNr{B}=N^{0}r{B^{0}}.
Theorem 18. If the intersection of any two exchangeable building blocks ofB also belongs toB, then the normal fan of the removahedronRemo(B)is theBnested fan F(B).
This result is illustrated by the building setB^{ex3}of Example9and its removahedronRemo(B^{ex3}) represented in Figure2. However, the condition of Theorem18 is still not necessary for the re movahedronRemo(B) to realize the nested complexN(B). For example, we invite the reader to check that the removahedronRemo(B^{ex4}) of the building setB^{ex4}of Example9realizes the corre sponding nested complex, even if{1,2,3,4}∩{3,4,5}={3,4}∈/B^{ex4}while{1,2,3,4}and{3,4,5}
are exchangeable. Corollary22gives a necessary and sufficient, thought unpractical, condition for the removahedronRemo(B) of an arbitrary building setBto realize the nested fan F(B).
3. Counting paths in maximalBtrees
Our first approach to Theorem 15 is the following characterization of the valid right hand sides to realize a complete simplicial fan as the normal fan of a convex polytope. A proof of this statement can be founde.g.in [HLT11, Theorem 4.1].
Theorem 19([HLT11, Theorem 4.1]). Given a complete simplicial fanFinR^{d}, consider for each rayρofF a halfspaceH^{≥}_{ρ} ofR^{d}containing the origin and defined by a hyperplaneH^{=}_{ρ} orthogonal to ρ. For each maximal cone C of F, let a(C) ∈ R^{d} be the intersection of the hyperplanes H^{=}_{ρ} forρ∈C. Then the following assertions are equivalent:
(i) The vector a(C^{0})−a(C) points from C to C^{0} for any two adjacent maximal cones C, C^{0} ofF.
(ii) The polytopes
conv{a(C)C maximal cone ofF } and \
ρray ofF
H^{≥}_{ρ}
coincide and their normal fan isF.
Since we are given a complete simplicial fan F(B) and we want to prescribe the right hand sides of the inequalities to describe a polytope realizing it, we are precisely in the situation of Theorem19. Our first step is to compute the intersection points of the hyperplanes normal to the rays of a maximal cone ofF(B). We associate to any maximalBtreeTa pointa(T)∈R^{S}whose coordinatea(T)sis defined as the number of pathsπin Tsuch thatsis the topmost vertex∧(π) of π in T. Note that all coordinates of a(T) are strictly positive integers since we always count the trivial path reduced to the vertex sof T. The following lemma ensures that the point a(T) lies on all hyperplanes ofRemo(B) normal to the rays of the coneC(T).
Lemma 20. For any maximal Btree T and any element s ∈ S, the point a(T) lies on the hyperplane H^{=}(desc(s,T)).
The proof of this lemma is inspired from similar statements in [LP13, Proposition 6] and [Pil13, Proposition 60]. Although the latter covers the present result, we provide a simpler and self contained proof for the convenience of the reader.
Proof of Lemma 20. Consider aBtree T, and let Π be the set of all paths inT. For anyπ∈Π, the topmost vertex∧(π) ofπinTis a descendant ofsinTif and only if both endpoints ofπare descendants ofsinT. It follows that
X
r∈desc(s,T)
a(T)_{r}= X
r∈desc(s,T)
X
π∈Π
11_{∧(π) =}_{r}=X
π∈Π
11_{∧(π)}_{∈}_{desc(s,T)}=
desc(s,T)+ 1 2
since the number of paths π ∈ Π such that ∧(π) ∈ desc(s,T) is just the number of pairs of endpoints in desc(s,T), with possible repetition. We therefore havea(T)∈H^{=}(desc(s,T)).
Guided by Theorem19, we now compute the differencea(T^{0})−a(T) for two adjacent maximal Btrees T and T^{0}. Let s, s^{0} ∈ S be such that the cones C(T) and C(T^{0}) are separated by the hyperplane of equation xs=xs^{0}, and moreover xs≤xs^{0} in C(T) whilexs≥xs^{0} inC(T^{0}). Let ¯T denote the tree obtained by contracting the arcs→s^{0} inTor, equivalently, the arcs^{0} →sinT^{0}. Since bothTandT^{0} contract to ¯T, the children of the node of ¯Tlabeled by{s, s^{0}}are all children of sor s^{0} in both T andT^{0}. We denote byS (resp. byS^{0}) the elements ofS which are children of s (resp. of s^{0}) in both T and T^{0}. In contrast, we let R denote the elements of S which are children ofsinTand ofs^{0} inT^{0}, andR^{0} those which are children ofs^{0} in Tand ofsinT^{0}. These notations are summarized on Figure3. Forr∈S∪S^{0}∪R∪R^{0}, we denote the set of descendants ofrby desc(r)^{:}= desc(r,T) = desc(r,T^{0}) = desc(r,T).¯
Lemma 21. LetBbe a building set andT,T^{0}be two adjacent maximalBtrees. Using the notation just introduced, we set
δ_{X}^{:}= X
x∈X
desc(x) and π_{X}^{:}= X
x6=x^{0}∈X
desc(x) · desc(x^{0})
S' R S R'
S' R' O s
S R
s' s s'
O
Figure 3. Two adjacent maximal BtreesT (left) andT^{0} (right).
forX ∈ {S, S^{0}, R, R^{0}}. Then the difference a(T^{0})−a(T)is given by a(T^{0})−a(T) = ∆(T,T^{0})·(es−es^{0}), where the coefficient∆(T,T^{0})is defined by
∆(T,T^{0})^{:}= (δ_{S}+ 1)(δ_{S}^{0}+ 1) +δ_{R}^{0}(δ_{S}+δ_{S}^{0}+δ_{R}+ 2) +π_{R}^{0}−π_{R}. Proof. By definition of the coordinates ofa(T), we compute
a(T)s= (δS+ 1)(δR+ 1) +πS+πR,
a(T)s^{0} = (δS^{0}+ 1)(δR^{0}+ 1) + (δS^{0}+δR^{0})(δR+δS+ 1) + 1 +δS+δR+πS^{0}+πR^{0}, and
a(T^{0})_{s}= (δ_{S}+ 1)(δ_{R}^{0}+ 1) + (δ_{S}+δ_{R}^{0})(δ_{R}+δ_{S}^{0}+ 1) + 1 +δ_{S}^{0}+δ_{R}+π_{S}+π_{R}^{0}, a(T^{0})s^{0} = (δS^{0}+ 1)(δR+ 1) +πS^{0}+πR.
Moreover, the coordinates a(T)r anda(T^{0})r coincide ifr∈Sr{s, s^{0}}since the flip fromT toT^{0} did not affect the children of the noder. The result immediately follows.
Combining Theorem19and Lemmas20and21, we thus obtain the following characterization.
Corollary 22. The Bnested fan F(B) is the normal fan of the removahedron Remo(B) if and only if∆(T,T^{0})>0 for any pair of adjacent maximalBtrees T,T^{0}.
The following lemma gives a sufficient condition for this property to hold.
Lemma 23. For any two adjacent maximalBtreesT,T^{0}as in Lemma21, ifdesc(s,T)∩desc(s^{0},T^{0}) belongs toB∪ {∅}, then∆(T,T^{0})>0.
Proof. By assumption, the set [
r∈R
desc(r) = desc(s,T)∩desc(s^{0},T^{0})
either belongs to the building setBor is empty. Since desc(r)∩desc(r^{0}) =∅forr6=r^{0}∈Rand desc(r)∈Bforr∈R, we conclude thatR ≤1 by Condition(N2)in Definition10. ThusπR= 0,
δS≥0 andδS^{0} ≥0. The statement follows.
It follows that if B is closed under intersection of exchangeable elements, then the Bnested fan F(B) is the normal fan of the removahedron Remo(B). This concludes our first proof of Theorems 15 and 18. For graphical building sets, we conclude from Lemma 8 that Remo(BG) realizesF(B) as soon as G is chordful. Conversely, the following lemma shows that the condition of Corollary22is never satisfied for building sets of non chordful graphs, thus concluding the proof of the characterization of Theorem16.
Lemma 24. Let G be a connected graph that is not chordful. Then there exist two adjacent maximalBGtreesT,T^{0} such that∆(T,T^{0})≤0
Proof. Consider a cycle O in G not inducing a clique and choose two verticesa, b∈O not connected by a chord. As a andb are not connected, {{a},{b}} is a Bnested set. We complete it to a B nested set ¯N formed by subsets of O all containing either a or b, such that ¯N be maximal for this property. LetBa andBb denote the maximal elements of ¯Ncontainingaandb, respectively.
By maximality of ¯N, all remaining vertices in Or(B_{a}∪B_{b}) are connected to bothB_{a} and B_{b}. Moreover, there are at least two such verticess, s^{0}. Consider two maximalBnested setsNandN^{0} both containing ¯NandB_{a}∪B_{b}∪ {s, s^{0}}, and such thatNcontainsB_{a}∪B_{b}∪ {s}andN^{0} contains B_{a}∪B_{b}∪ {s^{0}}. The correspondingBtrees T and T^{0} are such that node s^{0} coverss in T ands coverss^{0} in T^{0}. Moreover, using the notations introduced earlier in this section,R^{0}=S=S^{0}=∅ while R ≥2 as R contains one vertex of Ba and one of Bb. Thus δS =δS^{0} = δR^{0} =πR^{0} = 0
andπR≥1, which implies ∆(T,T^{0})≤0.
As already observed earlier, for general building sets, the condition of Lemma23is not necessary forRemo(BG) to realizeF(B). For example, in the building setB^{ex4}of Example 9, the building blocks{1,2,3,4}and{3,4,5}are exchangeable but{1,2,3,4}∩{3,4,5}={3,4}∈/B^{ex4}. However, {1,2,3,4} and {3,4,5} are the only two intersecting exchangeable building blocks of B^{ex4}, and for any two maximal B^{ex4}trees T,T such that N(T)r
{1,2,3,4}} = N(T^{0})r
{3,4,5}}, we have ∆(T,T^{0}) = 1. Therefore, Corollary22ensures thatRemo(B^{ex4}) realizes F(B^{ex4}).
Remark 25. The arguments of this first proof of Theorems15and18can be used to show that any nestohedron can be realized as a skew removahedron. A skew permutohedron is the convex hullPerm^{p}(S)^{:}= convP
s∈Spσ(s)es
σ∈SS of the orbit of a generic pointp∈R^{S} (i.e.ps6=p^{0}_{s} fors6=s^{0}∈S) under the action of the symmetric groupS_{S} onR^{S}by permutation of coordinates.
Equivalently, a skew permutahedron is the deformed permutahedron Perm^{φ}(S)^{:}=Defo(z^{φ}) for a right hand side z^{φ}^{:}= (z^{φ}_{R})R⊆S∈R^{2}
S defined byz_{R}^{φ}^{:}=φ(R) for some functionφ:N→R>0. For example, the classical permutahedronPerm(S) is the permutahedronPerm^{φ}(S) forφ(n) = ^{n+1}_{2}
. Askew removahedronis a polytope
Remo^{p}(B) =Remo^{φ}(B)^{:}=
x∈H
X
s∈B
xs≥φ(B) for all B∈B
,
obtained by removing inequalities from the facet description of a skew permutahedronPerm^{p}(S) = Perm^{φ}(S). Note that even if skew removahedra have much more freedom than classical removahe dra, they do not contain all deformed permutahedra.
A consequence of the realization of [Dev09] is that all graph associahedra can be realized as skew removahedra, namely by removing facets of the skew permutahedronPerm^{φ}(S) forφ(n) =γ^{n} with γ >2. The arguments presented in this Section provide an alternative proof of this result and extend it to all nestohedra. Let us quickly give the proof here.
For aBtreeT, consider the pointa^{γ}(T)∈R^{S} defined by X
r∈desc(s,T)
a^{γ}(T)r=γdesc(s,T) for alls∈S.
We do not need to compute explicitly the coordinates ofa^{γ}(T). We will only use that for anys∈S, a^{γ}(T)s=γdesc(s,T)− X
r∈desc(s,T)
γdesc(r,T).
Consider now two adjacent maximalBtrees T,T^{0} with the same notations as in Figure3. For a subsetX ∈ {S, S^{0}, R, R^{0}}, define
δX:= X
x∈X
desc(x) and ΓX:= X
x∈X
γ^{desc(x)}.
Using these notations, we compute:
a^{γ}(T)s=γ^{1+δ}^{S}^{+δ}^{R}−ΓS−ΓR,
a^{γ}(T)_{s}^{0} =γ^{2+δ}^{S}^{+δ}^{R}^{0}^{+δ}^{S}^{0}^{+δ}^{R}−γ^{1+δ}^{S}^{+δ}^{R}−Γ_{S}^{0}−Γ_{R}^{0}, and
a^{γ}(T^{0})_{s}=γ^{2+δ}^{S}^{+δ}^{R}^{0}^{+δ}^{S}^{0}^{+δ}^{R}−Γ_{S}−Γ_{R}^{0}−γ^{1+δ}^{S}^{0}^{+δ}^{R}, a^{γ}(T^{0})s^{0}=γ^{1+δ}^{S}^{0}^{+δ}^{R}−ΓS^{0}−ΓR.
Moreover, the coordinatesa(T)_{r}anda(T^{0})_{r}coincide ifr∈Sr{s, s^{0}}since the flip fromTtoT^{0}did not affect the children of the noder. We therefore obtain thata^{γ}(T^{0})−a^{γ}(T) = ∆(T,T^{0})·(e_{s}−es^{0}), where
∆(T,T^{0}) =γ^{2+δ}^{S}^{+δ}^{R}^{0}^{+δ}^{S}^{0}^{+δ}^{R}−γ^{1+δ}^{S}^{0}^{+δ}^{R}−γ^{1+δ}^{S}^{+δ}^{R}+ Γ_{R}−Γ_{R}0
≥γ^{δ}^{R} γ^{1+δ}^{S} −1
γ^{1+δ}^{S}^{0} −1
−1
+γ^{2+δ}^{S}^{+δ}^{S}^{0}^{+δ}^{R}(γ^{δ}^{R}^{0} −1)−ΓR^{0}. Since γ > 2, we have γ^{1+δ}^{S} −1
γ^{1+δ}^{S}^{0} −1
−1 > 0 and γ^{2+δ}^{S}^{+δ}^{S}^{0}^{+δ}^{R}(γ^{δ}^{R}^{0} −1)−ΓR^{0} ≥ γ^{δ}^{R}^{0}^{+1}−γ−ΓR^{0} ≥0 sinceγ^{n+m}≥γ^{n}+γ^{m}for 1≤n, m. We therefore obtain that ∆(T,T^{0})>0 and we conclude by Theorem19.
4. Minkowski sums
In this section, we provide an alternative proof that any building set closed under intersection can be realized as a removahedron. The approach of this section is complementary to the previous one since it focusses on Minkowski sums. As illustrated by the following statement observed independently by A. Postnikov [Pos09, Section 7] and E.M. Feichtner and B. Sturmfels [FS05], Minkowski sums provide a powerful tool to realize nested complexes.
Theorem 26 ([Pos09, Section 7], [FS05]). For any building set B, theBnested fan F(B) is the normal fan of the Minkowski sum
Mink[B] =X
B∈B
y_{B}4B,
where(yB)B∈Bare arbitrary strictly positive real numbers, and4Bdenotes the face of the standard simplex corresponding to B.
Remember from Remark4that the normal fan, and thus the combinatorics, of the Minkowski sumMink[B] only depends onB, not on the values of the dilation factors (yB)_{B∈B} (as soon as all these values are strictly positive). These Minkowski sumsMink[B] are deformed permutahedra, but not necessarily removahedra. In this section, we relax Theorem26to give a sufficient condition for a subsum ofMink[B] to keep the same normal fan, and prove that a wellchosen subsum ofMink[B]
is indeed a removahedron.
4.1. Generating sets and building paths. We say that a subset C of B is generating if for eachB ∈ Band for eachb ∈B, the setB is the union of the sets C ∈Csuch thatb ∈C ⊆B.
The following statement can be seen as a relaxation of Theorem26: it shows that the Minkowski sumMink[C] of the faces of the standard simplex over a generating subsetCofBstill has the same normal fan as the Minkowski sumMink[B] itself. Note that we do not make here any particular assumption on the building set B. The proof, adapted from that of [Pos09, Theorem 7.4], is delayed to the next section.
Theorem 27. IfCis a generating subset of a connected building setB, then theBnested fanF(B) is the normal fan of the Minkowski sum
Mink[C] = X
C∈C
yC4C,
where(y_{C})_{C∈C} are arbitrary strictly positive real numbers.
Example 28. Given a connected graph G, the setCG of all vertex sets of induced subpaths of G is a generating subset of the graphical building setBG. Set hereyC = 1 for allC∈CG. Then
(1) if G = P is a path, then BP andCP coincide andMink[BP] = Mink[CP] is precisely the associahedron of [SS93,Lod04];
(2) if G = T is a tree, then Mink[CT] is the (unsigned) tree associahedron of [Pil13];
(3) if G =K_{S}is the complete graph, thenBK_{S}= 2^{S}r{∅}whileCK_{S}={R⊆S1≤ R ≤2}.
Thus,Mink[CK_{S}] is the classical permutahedron, whileMink[BK_{S}] is a dilated copy of it.
In fact, the notion of paths can be extended from these graphical examples to the more general setting of connected building sets closed under intersection. Throughout the end of this section, we consider a building setB on the ground set S, closed under intersection. For any R⊆S, we define the Bhull of R to be the smallest element ofB containingR (it exists since B is closed under intersection). In particular, for anys, t ∈S, we define theBpath π(s, t) to be the Bhull of{s, t}. We denote by Π(B) the set of allBpaths.
Example 29. For a graphical building setBG, theBGpaths are precisely the induced subpaths of G,i.e.with our notations Π(BG) =CG.
Lemma 30. The setΠ(B)of allBpaths is a generating subset ofB.
Proof. LetB ∈Band b∈B. For anyb^{0} ∈B, the pathπ(b, b^{0}) contains band is contained inB (indeed, B contains both b and b^{0} and thus π(b, b^{0}) by minimality of the latter). Therefore, b^{0} belongs to the union of the setsC∈Π(B) such thatb∈C⊆B. The lemma follows by definition
of generating subsets.
In fact, the set of paths Π(B) is the minimal generating subset ofB, in the following sense.
Lemma 31. Any generating subset ofB containsΠ(B).
Proof. Consider a generating subsetCofB, ands, t∈S. TheBpathπ(s, t) is the smallest building block containing {s, t}. Therefore, π(s, t) has to be inC, since otherwise t would not belong to
the union of the setsC such thats∈C⊆π(s, t).
From Theorem27and Lemma30, we obtain that any Minkowski sumMink[Π(B)] is a realization of the Bnested complex. We now have to choose properly the dilation coefficients to obtain a removahedron. ForS⊆S, define the coefficient
¯ yS:=
s, t∈S^{2}π(s, t) =S .
Lemma 32. The dilation coefficients (¯yB)_{B∈B} satisfy the following properties:
(i) y¯S >0for all S∈Π(B), andy¯S = 0otherwise.
(ii) For allB∈B,
X
S⊆B
¯ yS =
B+ 1 2
.
Proof. Point (i) is clear by definition of the coefficients ¯yS. We prove Point (ii) by double counting:
for any B ∈ B and any b, b^{0} ∈ B (distinct or not), the path π(b, b^{0}) is included in B. We can therefore group pairs of elements ofB according to theirBpaths:
B+ 1 2
={b, b^{0} ∈B}= X
S⊆B

b, b^{0}∈B π(b, b^{0}) =S = X
S⊆B
¯
yS.
Combining Theorem27with Lemmas30and32, we obtain an alternative proof of Theorem15.
Corollary 33. For a building set B closed under intersection, the removahedron Remo(B)coin cides with the Minkowski sumP
B∈By¯B4B, and its normal fan is the Bnested fan.
Remark 34. For a graphical building set BG of a chordful graph G, the coefficients ¯yC are all equal to 1 for all subpaths C ∈ CG. It is not anymore true for arbitrary building sets closed under intersection. For example, consider the building set B^{ex5 :}=
{1},{2},{3},{1,2},{1,2,3} , for which we obtain
Remo(B^{ex5}) = 24_{{1,2,3}}+4_{{1,2}}+4_{{1}}+4_{{2}}+4_{{3}}. This Minkowski decomposition is illustrated in Figure4.
4.2. Proof of Theorem 27. This section is devoted to the proof of Theorem27. We start with the following technical lemma on the affine dimension of Minkowski sums.
Lemma 35. (i) Let(Pi)_{i∈I} be polytopes lying in orthogonal subspaces ofR^{n}. Then dimX
Pi=X dimPi. (ii) IfI⊆2^{S} is such thatT
I6=∅, thendimP
I∈I4I =S I −1.
Proof. Point (i) is immediate as the union of bases of the linear spaces generated by the poly topes P_{i} is a basis of the linear space generated by PP_{i}. For Point (ii), fix x ∈ T
I and an arbitrary orderI_{1}, . . . , I_{p} onI. DefineI_{j}^{0}^{:}=I_{j}r {x} ∪S
k<jI_{k}
. We then have dimX
I∈I
4_{I} ≥dim X
j∈[p]
I_{j}^{0}6=∅
4_{I}^{0}
j∪{x}≥ X
j∈[p]
I_{j}^{0}6=∅
dim4_{I}^{0}
j∪{x}= X
j∈[p]
I_{j}^{0}6=∅
I_{j}^{0}=
[ I
−1,
where the first inequality holds since4I_{j}^{0}∪{x} is a face of4I_{j}, the second one is a consequence of Point (i), and the last equality holds since we have the partition
[ I
r{x}= G
j∈[p]
I_{j}^{0}6=∅
I_{j}^{0}.
Proof of Theorem 27. LetCbe a generating subset of a connected building setB, lety^{:}= (y_{C})_{C∈C} be strictly positive real numbers, letz^{:}= (z_{R})_{R⊆S} be defined byz_{R}=P
C⊆Ry_{C}, and consider the polytopeMink(y) =Defo(z).
LetTbe aBtree andN^{:}=N(T) be the correspondingBnested set. ForN ∈N, let XN:=Nr
[
N^{0}∈N N^{0}(N
N^{0}
denote the label corresponding toN in theBtreeT, so that (XN)N∈N partitionsS.
ForC∈C, we defineN(C) to be the inclusion maximal elementN ofNsuch thatC∩X_{N} 6=∅. Note that this element is unique: otherwise, the union of the maximal elements N ∈ N such that C∩X_{N} 6=∅would be contained in B, thus contradicting Condition (N2) in Definition10.
Observe also thatN(C) is the inclusion minimal elementN ofNsuch thatC⊆N. We now define
FN:= X
C∈C
yC4_{C∩X}_{N(C)}.
We will show below thatF_{N} is a face of Mink(y) whose normal cone is precisely the cone C(N).
The map N→ F_{N} thus defines an antiisomorphism from the nested complex N(B) to the face lattice ofMink(y).
_{4}
1 1
_{1}
4 1
_{2}
1 3
_{1}
2 3
_{2}
0 0
_{0}
2 0
_{0}
0 2
_{1}
0 0
_{0}
1 0
_{1}
1 1
Remo(B^{ex5}) = 24_{{1,2,3}} + 4_{{1,2}} +
4_{{1}}+ 4_{{2}}+4_{{3}}
Figure 4. The Minkowski decomposition of the 2dimensional removahe dronRemo(B^{ex5}) into faces of the standard simplex.
Consider any vectorf^{:}= (f_{s})_{s∈S}in the relative interior of the coneC(N). This implies thatf_{s}=f_{N} is constant on eachN ∈N, andf_{N} < f_{N}^{0} forN, N^{0} ∈Nwith N (N^{0}. Let f :R^{S} →R be the linear functional defined by f(x) = hfxi = P
s∈Sf_{s}x_{s}. Since N(C) is the inclusion maximal elementN ofNsuch thatC∩N6=∅andN →f_{N} is increasing, the face of4_{C} maximizing f is precisely4_{C∩N}_{(C)}. It follows thatF_{N} is the face maximizingf onMink(y), since the face maxi mizingf on a Minkowski sum is the Minkowski sum of the faces maximizingf on each summand.
We conclude thatFNis a face ofMink(y) whose normal cone contains at leastC(N), and therefore that the mapN→FNis a poset antihomomorphism.
To conclude, it is now sufficient to prove that the dimension of FN is indeed S − N. The inequality dimFN ≤ S − N is clear since the normal cone of FN contains the cone C(N). To obtain the reverse inequality, observe that
dimF_{N}= dim
X
C∈C
y_{C}4_{C∩X}_{N(C)}
= dim
X
N∈N
X
C∈C N(C)=N
4_{C∩X}_{N}
≥ X
N∈N
dim
X
C∈C N(C)=N
4_{C∩X}_{N}
≥ X
N∈N
(XN −1) =S − N.
The first inequality holds by Lemma 35(i) since the X_{N} are disjoints. The second inequality follows from the assumption that Cis a generating subset of B. Indeed, fixN ∈N and pick an element x∈ X_{N}. Observe that if C ∈C is such thatx∈ C ⊆N, then N(C) = N. Moreover, sinceN is the union of the elementsC∈Csuch thatx∈C⊆N, we obtain thatX_{N} is the union of the setsC∩XN over the elementsC∈Csuch thatx∈C⊆N. By Lemma35(ii), this implies that
dim
X
C∈C N(C)=N
4_{C∩X}_{N}
≥dim
X
x∈C⊆NC∈C
4_{C∩X}_{N}
≥ X_{N} −1.
This concludes the proof that the dimension of F_{N} is given by S − N, and thus that the
mapN→F_{N}is an antiisomorphism.
Acknoledgments
I thank Carsten Lange for helpful discussions on the content and presentation of this paper. I am also grateful to an anonymous referee for relevant comments and suggestions.
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