The hull metric on Coxeter groups

We reinterpret an inequality, due originally to Sidorenko, for linear extensions of posets in terms of convex subsets of the symmetric group $\mathfrak{S}_n$. We conjecture that the analogous inequalities hold in arbitrary (not-necessarily-finite) Coxeter groups $W$, and prove this for the hyperoctahedral groups $B_n$ and all right-angled Coxeter groups. Our proof for $B_n$ (and new proof for $\mathfrak{S}_n$) use a combinatorial insertion map closely related to the well-studied promotion operator on linear extensions; this map may be of independent interest. We also note that the inequalities in question can be interpreted as a triangle inequalities, so that convex hulls can be used to define a new invariant metric on $W$ whenever our conjecture holds. Geometric properties of this metric are an interesting direction for future research.


Introduction
Let G be a connected undirected graph with vertices V (G) and let denote the shortest-path distance between vertices, clearly a metric on V (G). A subset C ⊆ V (G) is convex if no shortest path between two vertices from C leaves C: whenever u, v ∈ C and d(u, w) + d(w, v) = d(u, v) we must have w ∈ C. For X ⊆ V (G), the convex hull Conv(X) is the intersection (itself clearly convex) of all convex sets containing X.
The following definition, which to our knowledge has not been considered before, is central to this work. Definition 1.1. We say a graph G has the hull property if for every u, v, w ∈ V (G), (1) |Conv(u, v)| · |Conv(v, w)| ≥ |Conv(u, w)| .
Equivalently, G has the hull property if and only if log |Conv(−, −)| : is a metric on V (G). It has the strong hull property if for every u, v, w ∈ V (G), (2) |Conv(u, v)| · |Conv(v, w)| ≥ |Conv(u, v, w)| . In addition to the infinite families of groups for which we prove Conjecture 1.2 below, we have also verified the conjecture by computer for the finite Coxeter groups of types G 2 , H 3 , D 4 , and F 4 .
It would be an interesting direction for future research to study whether Conjecture 1.2 might be extended to arbitrary Tits buildings. To see the strength of Conjecture 1.2, consider the following very special case: Example 1.3. Let W = S n be the symmetric group, with the adjacent transpositions as simple generators. Let w ∈ S n be any permutation and w rev the reversed permutation (identifying permutations with their one-line notation w(1) . . . w(n)).
Then it follows from observations of Björner-Wachs [3] that |Conv(id, w)| = |[id, w] R | is the number of linear extensions of the corresponding 2-dimensional poset P w , while |Conv(id, w rev )| is the number of linear extensions of the complement P w = P w rev , and furthermore that any 2-dimensional poset is isomorphic to some P w . Since Conv(w, w rev ) = S n , the hull property implies that for any 2-dimensional poset P : (3) e(P )e(P ) ≥ n!
where e(P ) denotes the number of linear extensions, a result due originally to Sidorenko [16].
Sidorenko's original proof [16] of (3) relied on intricate max-flow min-cut arguments, but the inequality (3) has since been shown to be related to several different topics and techniques in convex geometry and combinatorics. Bollobás-Brightwell-Sidorenko [5] later gave a convex geometric proof using a known special case of the still-open Mahler Conjecture. In [10,11] the authors gave a strengthening and new proof related to the algebraic notion of generalized quotients [3] in Coxeter groups. Theorem 1.4 below extends Sidorenko's inequality to all pairs of elements (not just w, w rev ) and to the hyperoctahedral group (equivalently, to certain signed posets). Our proof relies on a new combinatorial insertion map for linear extensions (see Definition 3.1), closely related to promotion [15]. This combinatorial proof of Sidorenko's inequality gives a new solution to an open problem of Morales-Pak-Panova [13] different from that in [10]. In the case of the symmetric group, a combinatorial proof similar to our proof in Section 3 has been obtained independently by Chan-Pak-Panova [7]. Theorem 1.4. Conjecture 1.2 holds when W is the symmetric group S n or hyperoctahedral group B n .
We also prove Conjecture 1.2 for the large class of right-angled Coxeter groups using very different methods. Heuristically, these groups are on the opposite extreme among Coxeter groups from the finite groups covered in Theorem 1.4, as the non-commuting products s i s j have infinite order, as opposed to the small, finite orders seen in finite Coxeter groups. In addition, right-angled Coxeter groups are a natural case in which to verify our conjectured metric because of their ubiquity in geometric group theory (see, for example, the survey [8] It is natural to ask when the inequality (1) is in fact an equality. In the case of a triple (id, w, ww 0 ) in a finite Weyl group W with longest element w 0 , equality is achieved when w is separable (see [10,11]) and conjecturally only in this case. For the symmetric group, this corresponds to the case where P w is a series-parallel poset.
1.2. Outline. Section 2 covers needed background on Coxeter groups, convex subsets thereof, and the weak order. Section 3 proves Conjecture 1.2 in the case W = S n using a new combinatorial insertion map for linear extensions; this argument is extended in Section 4 to prove the conjecture for the hyperoctahedral group B n . Section 5 proves some structural properties of inversion sets in right-angled Coxeter groups and uses these to give an explicit injective proof of Conjecture 1.2 when W is any right-angled Coxeter group.
Finally, in Section 6, we demonstrate that the (strong) hull property is a rather special property of graphs by proving that the graph of regions of a graphical hyperplane arrangement has the hull property if and only if it is in fact the reflection arrangement for S n , in which case the property follows from Theorem 1.4.

Background on Coxeter groups
The material in this section is well known and may be found, for example, in Björner-Brenti [2].
2.1. Coxeter groups and weak order. A Coxeter group is a group W together with a distinguished generating set S = {s 1 , . . . , s r } subject to relations s 2 i = id for i = 1, . . . , r and (s i s j ) mij = id for i = j ∈ {1, . . . , r} for some numbers m ij ∈ {2, 3, . . .} ⊔ {∞}. The r × r matrix with entries m ij is the Coxeter matrix. The elements of S are called simple reflections, while their W -conjugates are called reflections; the set of reflections is denoted T . Given an element w ∈ W , an expression w = s i1 · · · s i ℓ of minimal length is called a reduced word for w, and in this case ℓ = ℓ(w) is the length of w.
The (right) weak order (W, ≤ R ) has cover relations w ⋖ R ws whenever ℓ(ws) = ℓ(w) + 1 and s ∈ S. A reflection t ∈ T is a (left) inversion of w if ℓ(tw) < ℓ(w), and the set of inversions of w is denoted T L (w). The following facts are well known: Proposition 2.2. Let w = s i1 · · · s i ℓ be any reduced word for w ∈ W , then We denote the meet operation in (W, ≤ R ) by ∧ R .

2.2.
Convexity in Coxeter groups. We say a subset C ⊆ W is convex if it is convex in the Cayley graph Cay(W, S) (the Hasse diagram of weak order). For Proposition 2.5 (Björner and Wachs [3]). Let x 1 , . . . , x d ∈ W , then In particular, Conv(id, x) = [e, x] R for all x ∈ W .
2.3. Parabolic subgroups and quotients. For J ⊆ S, the parabolic subgroup W J is the subgroup of W generated by J (viewed as a Coxeter group with simple generators J).
Proposition 2.6. Let J ⊆ S, then: • Each left coset wW J of W J in W has a unique element of minimal length, denoted w J . The set {w J | w ∈ W } is called the parabolic quotient and is denoted W J .
3. The symmetric group 3.1. The insertion map. Recall that the symmetric group S n is a Coxeter group generated by S = {s 1 , . . . , s n−1 } where s i is the simple transposition (i i + 1). The set of reflections is In this section, let P be a poset on n elements {p 1 , . . . , p n }. A linear extension is an order-preserving bijection λ : P → [n] from the poset P to the n-element chain [n]. Denote the set of linear extensions of P by L(P ). Each λ ∈ L(P ) has a naturally associated permutation π λ where π λ (i) = j if λ(p j ) = i. Definition 3.1. We now define a map f P : S n → L(P ), called the insertion map, from the symmetric group S n to the set of linear extensions of any poset P on n elements. This map will be defined recursively, and will resemble promotion (see Section 3.20 of [17]).
Let w ∈ S n , its image λ = f P (w) is determined as follows: At step k, for k = 1, 2, . . . , n, we start with λ(p w(k) ) = k. If at any time we have λ(p a ) = k where p a is not a maximal element of the induced subposet P on p w(1) , . . . , p w(k) , find the smallest value λ(p b ) where p b > P p a and λ(p b ) has already been defined, and swap the values of p a and p b . Continue this procedure to move the value k up the poset, until λ −1 (k) is a maximal element of the induced subposet P on p w(1) , . . . , p w(k) .
Example 3.2. We show an example of the insertion map defined above. Let P be as in Figure 1 and w = 45312. Figure 1 shows how the insertion map works at each step, where the final result λ = f P (w) is provided at the end whose corresponding permutation is π λ = 52134. After step n − 1 of our construction of f P (w), only one element p i has not been assigned a value and we have a linear extension λ ′ of P \ {p i }. Let us write ϕ pi : L(P \ {p i }) → L(P ) for the map used in the last step: assign value n to p i , if there exists p j covering p i , find such p j with the smallest λ ′ (p j ) and swap the values at p i and p j , and continue this process until the value n reaches a maximal element in P , giving us a linear extension.
Recall that a promotion chain (see [17]) of a linear extension λ ∈ L(P ) is a saturated chain t 1 > t 2 > · · · > t ℓ from a maximal element to a minimal element in P such that Proof. Take λ ∈ L(P ) and let t 1 > t 2 > · · · > t ℓ be its promotion chain. In order for λ to come from some λ ′ ∈ L(P \ {p i }), we invert the process of swapping values: as λ(t 1 ) = n, erase the value at t 1 , find the element covered by t 1 with the largest λ-value, which is t 2 , erase the value at t 2 and give λ(t 2 ) to t 1 , continue this process until p i has its value erased. This inverse process can succeed if and only if p i is part of the promotion chain of λ.

Inversion posets.
Definition 3.4. The inversion poset P u of a permutation u ∈ S n is a poset on n elements p 1 , . . . , p n such that p i < p j if i < j and u −1 (i) < u −1 (j).
The following observation of Björner-Wachs [4] is the fundamental link between the notions of convex subsets of the symmetric group and linear extensions of posets, and it is in analogy with this proposition that definitions related to posets can be extended to other Coxeter groups. This perspective on extending definitions and results about posets to other Coxeter groups was studied by Reiner [14] and more recently by the authors [9]. Proposition 3.5 (Björner and Wachs [4]). The interval [id, w] R in the weak order is exactly {π λ | λ ∈ L(P w )}.
We will also be using the permuted inversion poset wP u , where w ∈ S n acts on P u by permuting the labels. In wP u , p w(i) < p w(j) if and only if p i < p j in P u . Note that P u is always naturally labeled, meaning that p i → i is a linear extension, but wP u is not necessarily naturally labeled.
The inversion poset can be generalized as follows.
Definition 3.6. Let U ⊂ S n , then the poset P U is defined on n elements p 1 , . . . , p n such that Therefore, the inversion poset P u is just P {id,u} and the permuted inversion poset wP u is P {w,wu} . The following proposition resembles Proposition 3.5, we include the proof for convenience.
Proof. By Proposition 2.5, Conv(U ) consists of those w ∈ S n such that The set u∈U T L (u) consists of those t i,j where i < j and u −1 (i) > u −1 (j) for all u ∈ U ; likewise, the set u∈U T L (u) does not contain any t i,j where i < j and u −1 (i) < u −1 (j) for all u ∈ U . So w ∈ Conv(U ) if and only if w −1 (i) < w −1 (j) for all i, j such that u −1 (i) < u −1 (j) for all u ∈ U . This corresponds exactly to how P U is constructed.
Theorem 3.8. For any w, u ∈ S n , we have an injection Proof. Use induction on n, the base case n = 1 being trivial. Let P = P {id,w,wu} and let A be the set of maximal elements of P , which is an antichain in P . We have the following partition Fix p a ∈ A and consider the restriction of w, viewed as a permutation on n − 1 elements, by keeping the relative order of indices. In this way, the inversion poset P w ′ is P w \ {p a }. Also consider the restriction For λ with λ(p a ) = n, the insertion map f Pw can be identified with the composition ϕ pa • f P w ′ . As a result, the map λ → f Pw (π λ ), f wPu (π λ ) of interest of this theorem can be written as where the middle injection is given by the induction hypothesis on w ′ , u ′ . It suffices to show that the last map provided by ϕ is an injection.
Recall from Lemma 3.3 that since ϕ pa (α) = λ, p a must be in the promotion chain of λ. Similarly, the promotion chains of λ and τ both contain a and a ′ . In particular, this means that p a and p a ′ are comparable in P w and that p w −1 (a) and p w −1 (a ′ ) are comparable in P u . As a < a ′ , we must have w −1 (a) < w −1 (a ′ ).
Corollary 3.9. The Cayley graph Cay(S n ) has the strong hull property.

The hyperoctahedral group
In this section, we work with the family of finite Coxeter groups of type B, called the hyperoctahedral groups. We will extend the constructions and results from Section 3 to this setting in order to complete the proof of Theorem 1.4.
The Coxeter group B n has simple generators {t i,i+1 |1 ≤ i ≤ n−1}∪{t 1,1 } where t i,j is the involution (i j)(īj) if i = ±j and (i j) if i =j. The set of reflections is   The set of type B linear extensions is denoted B(P ). Any λ ∈ B(P ) naturally corresponds to an element π λ ∈ B n where π λ (i) = j if λ(p j ) = i for all i ∈ [n] (see Proposition 4.9 for an analog of Proposition 3.7.) We now define an insertion map for type B posets, which requires more careful treatment than the insertion map for posets as in Section 3. The map ϕ P p k : B(P \ {p k , pk}) → B(P ) is defined as follows: • For λ ′ ∈ B(P \{p k , pk}), we construct λ : P → [n] by first setting λ(p k ) = n, λ(pk) = −n, and λ(p j ) = λ ′ (p j ) if j = ±k. • If λ(p a ) = n but p a is not a maximal element of P , find the p b greater than p a with the smallest value for λ(p b ) and swap the λ-values of p a and p b and simultaneously swap those of pā and pb; iterate this process until λ −1 (n) is a maximal element of P .
Remark. When we have λ(p a ) = n and p a is not a maximal element of P , it is possible that the p b covering a with the smallest λ-value happens to be b = −a, in which case swapping the values of p a and p b and of pā and pb are done with just one swap.
We will write ϕ p k for ϕ P p k for simplicity when there is no confusion. Note that the two maps ϕ P p k and ϕ P p −k from B(P \ {p k , pk}) to B(P ) do not coincide. Definition 4.4. For λ ∈ B(P ), its promotion chain p t1 > p t2 > · · · > p t ℓ is a saturated chain from a maximal element of P to a minimal element of P constructed as follows: • First, p t1 = λ −1 (n) is the element with the largest λ-value; • Now suppose that p ti has been assigned value n, let p ti+1 be the element covered by p ti with the largest λ-value and swap the λ-values of p ti and p ti+1 and of p −ti and p −ti+1 .
Note that as the chain p t1 > p t2 > · · · > p t ℓ records the movement of the value n while p −t1 < p −t2 < · · · < p −t ℓ records the movement of the value −n.
The following lemma follows immediately from the above construction and is analogous to Lemma 3.3 so we omit the proof here.
Lemma 4.5. The map ϕ p k : B(P \ {p k , pk}) → B(P ) is injective, with image consisting of those λ ∈ B(P ) whose promotion chain passes through p k . Definition 4.6. Let P be a type B poset on 2n elements. Then the insertion map f P : where P (k) is the type B poset on 2k elements obtained from P by restricting to elements p w(1) , p w(1) , . . . , p w(k) , p w(k) .
Example 4.7. We show an example of the insertion map f P (Definition 4.6), where w = 4213 and P is as the type B poset in Figure 2. Notice that in step ϕ p1 , the value 3 is first assigned to p 1 which is a minimal element in P , and then moved all the way up to p1. Note that if λ is a type B linear extension of P and we insert π λ to P , then we obtain λ back with no swaps applied. Definition 4.8. Let U ⊂ B n . Then the poset P U is defined on p 1 , . . . , p n , p1, . . . , pn such that p i < p j if u −1 (i) < u −1 (j) for all u ∈ U .
The proof of Proposition 4.9 is identical to that of Proposition 3.7. Proposition 4.9. For U ⊂ B n , the convex set Conv(U ) is exactly {π λ |λ ∈ B(P U )}.
As in Section 3, for w ∈ B n , we also write P w = P {id,w} and write wP u = P w,wu which is P u with its indices permuted according to w. Theorem 4.10. For any w, u ∈ B n , we have an injection: Proof. We use induction on n. Write P = P {id,w,wu} for simplicity. The base case is n = 1 where B 1 consists of two elements: id, which is 1 in one-line notation, and t 1,1 , which is1 in one-line notation. If w = id, then P = P u so the above map is an isomorphism onto the second factor and if w =1, then P = P w so the above map is an isomorphism onto the first factor. In both cases, we obtain an injection. Now assume that the claim is true for n − 1.
Let A be the set of maximal elements of P . For λ ∈ B(P ), λ −1 (n) ∈ A so this provides us with a natural partition of B(P ): Notice that unlike in type A, we now need to take out p a and pā simultaneously, where λ(pā) =n if λ(p a ) = n. It is possible that both a andā belong to A, in which case B(P \ {p a , pā}) contributes twice in the above partition. However, these two parts will be treated differently.
Assume to the contrary that (λ, τ ) ∈ B(P w ) × B(wP u ) lies in the image of both ϕ pa and ϕ p a ′ . By Lemma 4.5, ϕ pa is injective so a = a ′ . Moreover, both p a and p a ′ are in the promotion chains of both λ and τ so in particular they are comparable in both P w and wP u . Assume without loss of generality that a < a ′ . Then we must have w −1 (a) < w −1 (a ′ ) and further (wu) −1 (a) < (wu) −1 (a ′ ) so p a < P p a ′ in P , contradicting the maximality of p a .
The hull property then follows from Theorem 4.10 and Proposition 4.9 via the same argument given in the proof of Corollary 3.9.
Corollary 4.11. The Cayley graph Cay(B n ) has the strong hull property.
Remark. Our framework for signed permutations will not deduce that the type D Cayley graph has the strong hull property. Since in type D n we no longer have inversions of the form t i,ī , it is possible that p a and pā are comparable in both P w and wP u , but not comparable in P {id,w,wu} .
Writing e B (P ) for the number of type B linear extensions of the type B poset P , we can deduce a direct analog of Sidorenko's inequality (3). Proof. By Proposition 4.9 we have that e B (P w ) = | Conv(id, w)| and similarly that e B (P ww0 ) = | Conv(id, ww 0 )|. The hull property thus gives that the product is at least | Conv(id, w, ww 0 )| = |B n | = 2 n n!.

Right-angled Coxeter groups
A Coxeter group W is called right-angled if all entries m ij of the corresponding Coxeter matrix lie in {1, 2, ∞}.
Proposition 5.1. Let W be a right-angled Coxeter group and u, v ∈ W , then Proof. It suffices to prove the case when u ∧ R v = id (so T L (u ∧ R v) = ∅). In this case, suppose for the sake of contradiction that there is some t ∈ T L (u) ∩ T L (v); by Proposition 2.2 this means that there are reduced expressions u = s i1 · · · s i k and v = s j1 · · · s j ℓ and indices 1 ≤ k ′ ≤ k, 1 ≤ ℓ ′ ≤ ℓ with where a = s i1 · · · s i k ′ −1 and b = s j1 · · · s j ℓ ′ −1 . Since W is right-angled, no two distinct simple reflections are conjugate (see, e.g., page 23 of [2]) so we must have Now, let C(s) denote the centralizer of s in W ; it follows from [6] that C(s) = W J , where J consists of those simple reflections commuting with s. At most one of a, b lies in C(s), otherwise we would have s ≤ R u, v, contradicting the fact that u ∧ R v = id, so suppose without loss of generality that b ∈ C(s) = W J . Since b −1 a commutes with s, we have a ∈ bW J , and thus a ≥ R b J = id. But this means that Theorem 5.2. Let W be a right-angled Coxeter group, then for any u, v ∈ W we have an injection Proof. By Proposition 2.5 we have and by Proposition 5.1 we know that ∧ R is given by intersection of (left) inversion sets. Thus the injectivity of φ follows from the obvious fact that a subset X ⊆ U ∪V may be uniquely reconstructed given X ∩ U and X ∩ V for any sets U, V . Corollary 5.3. Let W be right-angled, then Cay(W ) has the strong hull property.

Graphical arrangements
An acyclic orientation of a simple undirected graph G is a choice of direction for each edge of G such that there are no directed cycles. We write AC(G) for the set of acyclic orientations of G, and endow AC(G) itself with a graph structure by taking o, o ′ ∈ AC(G) to be adjacent if the orientations differ on exactly one edge of G.
Proposition 6.1. Let K n denote the complete graph on n vertices, then Proof. Acyclic orientations o of K n correspond to regions R of the braid arrangement H, the hyperplane arrangement with hyperplanes x i = x j for 1 ≤ i < j ≤ n, with the orientation of the edge ij in o indicating on which side of the hyperplane x i = x j the region R lies. Furthermore, two regions R, R ′ are adjacent if and only if the corresponding orientations o, o ′ are adjacent in AC(K n ). Now, it is well known (see, e.g. [12]) that the graph of regions of a reflection arrangement is naturally identified with the Cayley graph of the reflection group, in this case S n , with its system of simple generators.
In light of Corollary 3.9, which establishes the hull property for Cay(S n ) ∼ = AC(K n ), it is natural to ask: for which graphs G does AC(G) have the hull property? Equivalently, for which graphical hyperplane arrangements does the graph of regions have the hull property? Theorem 6.2 shows that there are essentially no new cases; this is meant to illustrate that the hull property is very special.   Therefore it suffices to classify good 2-connected graphs H, and in particular to show that any 2-connected graph H which does not contain an induced copy of G 4 or C n (n ≥ 4) is a complete graph, and thus good by Theorem 3.8.
Let H be a 2-connected graph H which does not contain an induced copy of G 4 or C n (n ≥ 4). Any 2-connected graph on fewer than four vertices is a complete graph, so suppose H has N ≥ 4 vertices. If H is not a complete graph, then there exist two vertices v, v ′ whose path distance from each other is exactly two. Since H is 2-connected, there is a cycle C on at least four elements containing both v, v ′ , and such that v, v ′ share a common neighbor u ∈ C. As H avoids induced copies of C n for n ≥ 4, there must exist chords of C in H which triangulate C. Since vv ′ is not a edge, one of these chords must yield a 4-cycle v, u, v ′ , w for some w adjacent to v, v ′ , and furthermore the diagonal uw must be an edge of H. But this creates an induced G 4 in H, a contradiction.
Remark. It would be interesting to investigate the prevalence of the hull property within other families of graphs. The anonymous referee intriguingly notes that among the antipodal isometric subgraphs of hypercubes that they were able to check, the only examples which satisfied the hull property were Cayley graphs of Coxeter groups. Another referee also suggests the case of (properly defined) acyclic orientations of signed graphs, which could generalize our type B result.