The facial weak order in finite Coxeter groups

We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W . We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Björner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes. Résumé. Nous étudions une structure de poset qui étend l’ordre faible sur un groupe de Coxeter fini W à l’ensemble de toutes les faces du permutoèdre deW . Nous appellons cet ordre l’ordre faible facial. Nous montrons d’abord deux caractérisations alternatives de ce poset : une première, géométrique, qui généralise la notion d’ensemble d’inversion, et une seconde, combinatoire, qui utilise des comparaisons entre les représentants minimaux et maximaux des faces. Ces caractérisations sont ensuite utilisées pour montrer que l’ordre faible facial est en fait un treillis, généralisant ainsi un résultat bien connu de A. Björner pour l’ordre faible classique. Finalement, nous montrons que toute congruence de treillis de l’ordre faible classique induit une congruence de treillis de l’ordre faible facial, et nous donnons une interprétation géométrique de ses classes.

The (right) Cayley graph of a Coxeter system (W, S) is naturally oriented by the (right) weak order on W : an edge is oriented from w to ws if s ∈ S is such that (w) < (ws), see [BB05, Chapter 3] for details.A celebrated result of A. Björner [Bjö84] states that the weak order is a complete meet-semilattice and even a complete ortholattice in the case of a finite Coxeter system.The weak order is a very useful tool to study Coxeter groups as it encodes the combinatorics of reduced words associated to (W, S), and it underlines the connection between the words and the root system via the notion of inversion sets, see for instance [Dye11,HL16] and the references therein.
In the case of a finite Coxeter system, the Cayley graph of W is isomorphic to the 1-skeleton of the W -permutahedron.Then the weak order is given by an orientation of the 1-skeleton of the W -permutahedron associated to the choice of a linear form of the ambiant Euclidean space.This point of view was very useful in order to build generalized associahedra out of a W -permutahedron using N. Reading's Cambrian lattices, see [Rea12,HLT11,Hoh12].
We study a poset structure on all faces of the W -permutahedron that we call the facial weak order.This order was introduced by D. Krob, M. Latapy, J.-C.Novelli, H.-D. Phan and S. Schwer in [KLN + 01] for the symmetric group then extended by P. Palacios and M. Ronco in [PR06] for arbitrary finite Coxeter groups.Recall that the faces of the W -permutahedron are naturally parameterized by the Coxeter complex P W which consists of all standard parabolic cosets W/W I for I ⊆ S. The aims of this article are: 1. To give two alternative characterizations of the facial weak order (see Theorem 10): one in terms of root inversion sets of parabolic cosets which extend the notion of inversion sets of elements of W , and the other using weak order comparisons between the minimal and maximal representatives of the parabolic cosets.The advantage of these two definitions is that they give immediate global comparison, while the original definition of [PR06] uses cover relations.2. To show that the facial weak order is a lattice (see Theorem 13), whose restriction to the vertices of the permutahedron produces the weak order as a sublattice.This result was motivated by the special case of type A proved in [KLN + 01]. 3. To show that any lattice congruence ≡ of the weak order extends to a lattice congruence ≡ P of the facial weak order (see Theorem 18).This provides a complete description (see Theorem 21) of all cones of the simplicial fan F ≡ associated to the weak order congruence ≡ in [Rea05].The results of this paper are based on combinatorial properties of Coxeter groups, parabolic cosets, and reduced words.However, their motivation and intuition come from the geometry of the Coxeter arrangement and of the W -permutahedron.So we made a point to introduce enough of the geometrical material to make the geometric intuition clear.

Preliminaries
We start by fixing notations and classical definitions on finite Coxeter groups.Details can be found in textbooks by J. Humphreys [Hum90] and A. Björner and F. Brenti [BB05].The reader familiar with finite Coxeter groups and root systems is invited to proceed directly to Section 2.

Finite reflection groups and Coxeter systems
Let (V, • | • ) be an n-dimensional Euclidean vector space.For any vector v ∈ V {0}, we denote by s v the reflection interchanging v and −v while fixing the orthogonal hyperplane pointwise.Remember that ws v = s w(v) w for any vector v ∈ V {0} and any orthogonal transformation w of V .
We consider a finite reflection group W acting on V , that is, a finite group generated by reflections in the orthogonal group O(V ).The Coxeter arrangement of W is the collection of all reflecting hyperplanes.Its complement in V is a union of open polyhedral cones.Their closures are called chambers.The Coxeter fan is the polyhedral fan formed by the chambers together with all their faces.This fan is complete (its cones cover V ) and simplicial (all cones are simplicial), and we can assume without loss of generality that it is essential (the intersection of all chambers is reduced to the origin).We fix an arbitrary chamber C, called fundamental chamber.The n reflections orthogonal to the facet defining hyperplanes of C are called simple reflections.The set S of simple reflections generates W .The pair (W, S) forms a Coxeter system.

Roots and weights
We consider a root system Φ for W , i.e., a set of vectors invariant under the action of W and containing precisely two opposite roots orthogonal to each reflecting hyperplane of W .The simple roots ∆ are the roots orthogonal to the defining hyperplanes of C and pointing towards C.They form a linear basis of V .The root system Φ splits into the positive roots Φ + := Φ ∩ cone(∆) and the negative roots Φ − := Φ ∩ cone(−∆) = −Φ + , where cone(X) denotes the set of nonnegative linear combinations of vectors in X ⊆ V .In other words, the positive roots are the roots whose scalar product with any vector of the interior of the fundamental chamber C is positive, and the simple roots form the basis of the cone generated by Φ + .Each reflection hyperplane is orthogonal to one positive and one negative root.For a reflection s ∈ R, we set α s to be the unique positive root orthogonal to the reflection hyperplane of s.
We denote by α ∨ s := 2α s / α s | α s the coroot corresponding to α s ∈ ∆, and by ∆ ∨ := {α ∨ s | s ∈ S} the coroot basis.The vectors of its dual basis ∇ := {ω s | s ∈ S} are called fundamental weights.In other words, the fundamental weights of W are defined by α ∨ s | ω t = δ s=t for all s, t ∈ S. Geometrically, the fundamental weight ω s gives the direction of the ray of the fundamental chamber C not contained in the reflecting hyperplane of s.We let Ω := W (∇) = {w(ω s ) | w ∈ W, s ∈ S} denote the set of all weights of W , obtained as the orbit of the fundamental weights under W .

Length, reduced words and weak order
The length (w) of an element w ∈ W is the length of the smallest word for w as a product of generators in S.
For u, v ∈ W , the product uv is said to be reduced if the concatenation of a reduced word for u and of a reduced word for v is a reduced word for uv, i.e., if (uv) = (u) + (v).We say that u ∈ W is a prefix of v ∈ W if there is a reduced word for u that is the prefix of a reduced word for v, i.e., if The (right) weak order is the order on W defined equivalently by A. Björner shows in [Bjö84] that the weak order defines a lattice structure on W (finite Coxeter group), with minimal element e and maximal element w • (which sends all positive roots to negative ones and all positive simple roots to negative simple ones).The conjugation w → w • ww • defines a weak order automorphism while the left and right multiplications w → w • w and w → ww • define weak order anti-automorphisms.We refer the reader to [BB05, Chapter 3] for more details.The weak order encodes the combinatorics of reduced words and enjoys a useful geometric characterization within the root system, which we explain now.The (left) inversion set of w is the set N(w) := Φ + ∩ w(Φ − ) of positive roots sent to negative ones by w Moreover, the weak order is characterized in term of inversion sets by: for any u, v ∈ W .We refer for instance the reader to [HL16, Section 2] and the references therein.We say that a simple reflection s ∈ S is a left ascent of w ∈ W if (sw) = (w) + 1 and a left descent of w if (sw) = (w) − 1.We denote by D L (w) the set of left descents of w.Note that for s ∈ S and w ∈ W , we have s ∈ D L (w) ⇐⇒ α s ∈ N(w) ⇐⇒ s ≤ w.Similarly, s ∈ S is a right descent of w ∈ W if (ws) = (w) − 1, and we denote by D R (w) the set of right descents of w.

Parabolic subgroups and cosets
The standard parabolic subgroup W I is the subgroup of W generated by I ⊆ S. It is also a Coxeter group with simple generators I, simple roots We denote by W I := {w ∈ W | (ws) > (w) for all s ∈ I} the set of elements of W with no right descents in I.For example, W ∅ = W while W S = {e}.Observe that for any x ∈ W I , we have x(∆ I ) ⊆ Φ + and thus x(Φ + I ) ⊆ Φ + .We will use this property repeatedly in this paper.Any element w ∈ W admits a unique factorization w = w I • w I with w I ∈ W I and w I ∈ W I , and moreover, (w) = (w I ) + (w I ) (see e.g., [BB05, Prop.2.4.4]).Therefore, W I is the set of minimal length coset representatives of the cosets W/W I .Throughout the paper, we will always implicitly assume that x ∈ W I when writing that xW I is a standard parabolic coset.Note that any standard parabolic coset xW I = [x, xw •,I ] is an interval in the weak order.The Coxeter complex P W is the abstract simplicial complex whose faces are all standard parabolic cosets of W : We will also need Deodhar's Lemma: for s ∈ S, I ⊆ S and x ∈ W I , either sx ∈ W I or sx = xr for some r ∈ I. See e.g., [GP00, Lemma 2.1.2]where it is stated for the cosets W I \W instead of W/W I .

Permutahedron
Remember that a polytope P is the convex hull of finitely many points of V , or equivalently a bounded intersection of finitely many affine halfspaces of V .The faces of P are the intersections of P with its supporting hyperplanes and the face lattice of P is the lattice of its faces ordered by inclusion.The inner primal cone of a face F of P is the cone generated by {u − v | u ∈ P, v ∈ F }.The outer normal cone of a face F of P is the cone generated by the outer normal vectors of the facets of P containing F .Note that these two cones are polar to each other.The normal fan is the complete polyhedral fan formed by the outer normal cones of all faces of P .We refer to [Zie95] for details on polytopes, cones, and fans.
The W -permutahedron Perm p (W ) is the convex hull of the orbit under W of a generic point p ∈ V (not located on any reflection hyperplane of W ). Its vertex and facet descriptions are given by We often write Perm(W ) instead of Perm p (W ) as the combinatorics of the W -permutahedron is independent of the choice of p and is encoded by the Coxeter complex P W .More precisely, each standard parabolic coset xW I corresponds to a face F(xW I ) of Perm p (W ) given by Therefore, the k-dimensional faces of Perm p (W ) correspond to the cosets xW I with |I| = k and the face lattice of Perm p (W ) is isomorphic to the inclusion poset (P W , ⊆).The normal fan of Perm p (W ) is the Coxeter fan.The graph of the permutahedron Perm p (W ) is isomorphic to the Cayley graph of the Coxeter system (W, S).Moreover, when oriented in the linear direction w • (p) − p, it coincides with the Hasse diagram of the (right) weak order on W .We refer the reader to [Hoh12] for more details on the W -permutahedron.

Facial weak order on the Coxeter complex
In this section we study an analogue of the weak order on standard parabolic cosets, which we call the facial weak order.It was defined in type A by D. Krob (1) (1) (2) (2) (1) (1) (2) (2) (2) (2) Fig. 1: The facial weak order on the standard parabolic cosets of the Coxeter group of types A2 and B2.Edges are labelled with the cover relations of type (1) or (2) as in Definition 1.
Remark 2 1.These cover relations translate to the following geometric conditions on faces of the permutahedron Perm(W ): a face F is covered by a face G if and only if either F is a facet of G with the same weak order minimum, or G is a facet of F with the same weak order maximum.This paper gives two convenient equivalent definitions for the facial weak order (see Section 2.2).The first uses sets of roots (see Section 2.1) to generalize the geometric characterization of the weak order with inversion sets.The second one uses weak order comparisons on the minimal and maximal representatives of the cosets.The advantage of these definitions is that they give immediate global comparison, whereas the original definition of [PR06] uses cover relations.We use these new characterizations of the facial weak order to prove that this poset is in fact a lattice (see Section 2.3).

Root and weight inversion sets of standard parabolic cosets
We now define a collection of roots and a collection of weights associated to each standard parabolic coset.The notion of root inversion sets of standard parabolic cosets generalizes the inversion sets of elements of W (see Proposition 6).We will use root inversion sets extensively for our study of the facial weak order.In contrast, weight inversion sets are not as essential and could be ignored for a first reading.We define them here as they are polar to the root inversion sets and appear naturally in our geometric intuition of the W -Coxeter arrangement and of the W -permutahedron (see Proposition 4).The following statement gives the precise connection to the geometry of the W -permutahedron and is illustrated on Figure 2  It is well-known that the map N, sending an element w ∈ W to its inversion set N(w) = Φ + ∩ w(Φ − ) is injective, see e.g., [HL16, Section 2].The following corollary is the analog for the maps R and W.
Corollary 5 The maps R and W are both injective.
Our next statement connects the root inversion set R(xW ∅ ) to the inversion set and reduced words of x ∈ W .For brevity we write R(x) instead of R(xW ∅ ).
Proposition 6 For any x ∈ W , we have: The next statement gives a characterization of the weak order on W in terms of root inversion sets, which generalizes the characterization of the weak order in term of inversion sets.
Corollary 7 For x, y ∈ W , we have Finally, we observe that the root and weight inversion sets of a parabolic coset xW I can be computed from that of its minimal and maximal length representatives x and xw •,I .
Proposition 8 The root and weight inversion sets of xW I can be computed from those of x and xw •,I by

Two alternative characterizations of the facial weak order
Using the root inversion sets defined in the previous section, we now give two equivalent characterizations of the facial weak order defined by P. Palacios and M. Ronco in [PR06] (see Definition 1).In type A, the equivalence (i) ⇐⇒ (ii) below is stated in [KLN + 01, Theorem 5] in terms of half-inversion tables.We have illustrated the facial weak order by the means of root inversion sets in Figure 3.
Remark 9 In Figure 3, each face is labelled by its root inversion set.To visualize the roots, we consider the affine plane P passing through the simple roots {α, β, γ}.
A positive (resp.negative) root ρ is then seen as a red upward (resp.blue downward) triangle placed at the intersection of Rρ with the plane P .For instance, the root set Finally, for the implication (ii)⇒(i) we consider two standard parabolic cosets xW I and yW J which satisfy (ii) and construct a path of cover relations as in Definition 1 between them.We proceed by induction on the cardinality | R(xW We consider only the first case, the other one being symmetric.To proceed by induction, we find a new coset zW K so that: • xW I < • zW K is one of the cover relations of Definition 1, • zW K and yW J still satisfy (ii), and The new coset is constructed by adding or deleting at least one root from R(xW I ).We first show that there exists s ∈ S such that −x(α s ) / ∈ R(yW J ).We then fix such an s and look at two cases, on whether or not s ∈ I.Each case produces one of the two cover relations in P. Palacios and M. Ronco's Definition 1.As each case produces a cover relation we see that by induction we have a path from xW I to yW J as we already have a path from zW K to yW J and a cover between xW I and zW K . 2 Using our characterization (ii) of Theorem 10 together with Corollary 7, we obtain that the (right) facial weak order and the (right) weak order coincide on the elements of W .Note that this is not obvious with the cover relations from Definition 1.
Corollary 11 For any x, y ∈ W , we have x ≤ y in (right) weak order if and only if xW ∅ ≤ yW ∅ in (right) facial weak order.
The weak order anti-automorphisms x → xw • and x → w • x and the automorphism x → w • xw • carry out on standard parabolic cosets.The following statement gives the precise definitions of these maps.

The facial weak order is a lattice
In this section, we show that the facial weak order on standard parabolic cosets is actually a lattice.It generalizes the result for the symmetric group due to D. Krob, M. Latapy, J.-C.Novelli, H.-D. Phan and S. Schwer [KLN + 01] to the facial weak order on arbitrary finite Coxeter groups introduced by P. Palacios and M. Ronco [PR06].The characterizations of the facial weak order given in Theorem 10 are key here.
Theorem 13 The facial weak order (P W , ≤) is a lattice.The meet and join of two standard parabolic cosets xW I and yW J are given by: Note that in the second point of the previous statement, the minimal representative of the coset z ∨ W K∨ is in fact z ∨ w •,K∨ , not z ∨ .Unlike in the rest of the paper, we take the liberty to use another coset representative than the minimal one to underline the symmetry between meet and join in facial weak order.
Example 14 Before proving the above statement, we give an example of a computation of the meet in the facial weak order.Consider the Coxeter system a, b, c a 2 = b 2 = c 2 = (ab) 3 = (bc) 3 = (ac) 2 = 1 of type A 3 .To find the meet of cbaW bc and acbW ∅ , we compute: Thus we have that cbaW bc ∧ acbW ∅ = z ∧ W K∧ = cW a .
Proof of Theorem 13: Throughout the proof we use the characterization of the facial weak order given in Theorem 10 (iii): xW I ≤ yW J ⇐⇒ x ≤ y and xw •,I ≤ yw •,J .We first prove the existence of the meet, then use Proposition 12 to deduce the existence and formula for the join.
Existence of meet.For any s ∈ K ∧ , we have Therefore, we have z ∧ ∈ W K∧ . Since It remains to show that z ∧ W K∧ is the greatest lower bound.Consider a standard parabolic coset zW K such that zW K ≤ xW I and zW K ≤ yW J .We want to show that zW K ≤ z ∧ W K∧ , that is, z ≤ z ∧ and zw •,K ≤ z ∧ w •,K∧ .The first inequality is immediate since z ≤ x and z ≤ y so that z ≤ x ∧ y = z ∧ .For the second one, we consider the reduced words x = zx , y = zy , and z ∧ = zz ∧ where z ∧ = x ∧ y .Since zw •,K ≤ xw •,I and zw •,K ≤ yw •,J , we have Thus w •,K ≤ x w •,I ∧ y w •,J , since all words are reduced here.Therefore K ⊆ D L (x w •,I ∧ y w •,J ).
We now claim that D L (x w •,I ∧y w •,J ) ⊆ D L (z ∧ w •,K∧ ).To see it, consider s ∈ D L (x w •,I ∧ y w •,J ) and assume by contradiction that s / ∈ D L (z ∧ w •,K∧ ).Then s does not belong to D L (z ∧ ), since the expression z ∧ w •,K∧ is reduced.By Deodhar's Lemma (see Section 1.4) we obtain that either sz In the first case we obtain In the second case, we get To conclude the proof, we get from Existence of join.The existence and the formula for the join follow from that of the meet, using the anti-automorphism Ψ : Remark 15 It is well-known that the map x → xw • is an orthocomplementation of the weak order: it is involutive, order-reversing and satisfies xw • ∧ x = e and xw • ∨ x = w • .In other words, it endows the weak order with a structure of ortholattice; see for instance [BB05, Corollary 3.2.2].This is not anymore the case for the facial weak order: the map xW I −→ w • xw •,I W I is indeed involutive and order-reversing, but is not an orthocomplementation: for a (counter-)example, consider x = e and I = S.
3 Lattice congruences of the facial weak order In this section we observe that any lattice congruence ≡ of the weak order extends to a lattice congruence ≡ P of the facial weak order.Let us first recall the definition of lattice congruences, see e.g., [Rea05].
Definition 16 An order congruence is an equivalence relation ≡ on a poset P such that: (i) Every equivalence class under ≡ is an interval of P .
(ii) The projection π ↑ : P → P (resp.π ↓ : P → P ), which maps an element of P to the maximal (resp.minimal) element of its equivalence class, is order preserving.The quotient P/≡ is a poset on the equivalence classes of ≡, where the order relation is defined by X ≤ Y in P/≡ if and only if there exists representatives x ∈ X and y ∈ Y such that x ≤ y in P .If P is moreover a finite lattice, then ≡ is automatically a lattice congruence, i.e., for any x ≡ x and y ≡ y , we have x ∧ y ≡ x ∧ y and x ∨ y ≡ x ∨ y .The poset quotient P/≡ then inherits a lattice structure.
Consider a lattice congruence ≡ of the weak order on W whose up and down projections are denoted by π ↑ and π ↓ respectively.We want to extend ≡ to a lattice congruence ≡ P of the facial weak order on P W .We need the following technical statement.(iii) Π ↑ and Π ↓ are order preserving.Therefore, the fibers of Π ↑ and Π ↓ coincide and define a lattice congruence ≡ P of the facial weak order.
N. Reading proved in [Rea05] that ≡ naturally defines a complete simplicial fan which coarsens the Coxeter fan.Namely, for each congruence class γ of ≡, consider the cone C γ obtained by glueing the maximal chambers cone(x(∇)) of the Coxeter fan corresponding to the elements x in γ.It turns out that each of these cones C γ is convex and that the collection of cones {C γ | γ ∈ W/≡}, together with all their faces, form a complete simplicial fan which we denote by F ≡ .We now use the congruence ≡ P of the facial weak order to describe all cones of F ≡ (not only the maximal ones).This shows that the lattice structure on the maximal faces of F ≡ extends to a lattice structure on all cones of the fan F ≡ .As in Section 2.1, we first introduce the root and weight inversion sets of the congruence classes of ≡ P .
, M. Latapy, J.-C.Novelli, H.-D. Phan and S. Schwer in [KLN + 01], then extended for arbitrary finite types by P. Palacios and M. Ronco in [PR06].Definition 1 ([PR06]) The (right) facial weak order is the order ≤ on the Coxeter complex P W defined by cover relations of two types: for I ⊆ S and x ∈ W I , (1) 2. Under the inclusion x → xW ∅ from W to P W , we will see in Corollary 11 that the restriction of the (right) facial weak order to the vertices of P W is the (right) weak order.It is not obvious at first sight from Definition 1. 3. It is known that for I ⊆ S the set of minimal length coset representatives W I has a maximal length element w • w •,I .The element w •,I w •,I {s} is therefore the maximal length element of the set W I {s} I = W I ∩ W I {s} , which is the set of minimal coset representatives of the cosets W I /W I {s} , see [GP00, Section 2.2] for more details.

Definition 3
The root inversion set R(xW I ) and weight inversion set W(xW I ) of a standard parabolic coset xW I are respectively defined by R(xW I ) := x Φ − ∪ Φ + I ⊆ Φ and W(xW I ) := x ∇ S I ⊆ Ω.

•Fig. 2 :
Fig. 2: The root inversion sets (left) and weight inversion sets (right) of the A2 standard parabolic cosets.Note that positive roots point downwards.
−α − β} is represented on the right.The star in the middle represents both α + β + γ and −α − β − γ.Theorem 10 The following conditions are equivalent for two standard parabolic cosets xW I , yW J ∈ P W : (i) xW I ≤ yW J in facial weak order, (ii) R(xW I ) R(yW J ) ⊆ Φ − and R(yW J ) R(xW I ) ⊆ Φ + , (iii) x ≤ y and xw •,I ≤ yw •,J in weak order.Proof of Theorem 10 : We give a brief outline of the proof.The implication (i)⇒(iii) is immediate by looking at each cover relation separately.Combining Corollary 7 and Proposition 8 gives the implication (iii)⇒(ii).

Fig. 3 :
Fig.3: The facial weak order on the standard parabolic cosets of the Coxeter group of type A3.Each coset xWI is replaced by its root inversion set R(xWI ), represented as follows: down blue triangles stand for negative roots while up red triangles stand for positive roots, and the position of each triangle is given by the barycentric coordinates of the corresponding root with respect to the three simple roots (α1 on bottom left, α2 on top, and α3 on bottom right);see Remark 9 for a more detailed discussion.

Proposition 12
The maps xW I −→ w • xw •,I W I and xW I −→ xw •,I w • W w•Iw• are anti-automorphisms of the weak order on parabolic cosets of W . Consequently, the map xW I −→ w • xw • W w•Iw• is an automorphism of the weak order on parabolic cosets of W .

Definition 19
The root inversion set R(Γ) and the weight inversion set W(Γ) of a congruence class Γ of ≡ P are defined by R(Γ)= xW I ∈Γ R(xW I ) and W(Γ) = xW I ∈Γ W(xW I ).Proposition 20 For any two congruence classes Γ, Γ of ≡ P , we have Γ ≤ Γ in the quotient of the facial weak order by≡ P if and only if R(Γ) R(Γ ) ⊆ Φ − and R(Γ ) R(Γ) ⊆ Φ + .Theorem 21 The collection of cones {cone(W(Γ)) | Γ ∈ P W /≡ P } forms the complete simplicial fan F ≡ .