Chains in shard lattices and BHZ posets
For every finite Coxeter group , we prove that the number of chains in the shard intersection lattice introduced by Reading on the one hand and in the BHZ poset introduced by Bergeron, Zabrocki and the third author on the other hand, are the same. We also show that these two partial orders are related by an equality between generating series for their Möbius numbers, and provide a dimension-preserving bijection between the order complex on the BHZ poset and the pulling triangulation of the permutahedron arising from the right weak order, analogous to the bijection defined by Reading between the order complex of the shard poset and the same triangulation of the permutahedron.
To study finite Coxeter groups, it has appeared to be very useful to introduce several partial orders on their elements. The most usual partial orders are the Bruhat order and the weak order (or permutahedron order). They are now very classical, and have been used in plenty of works.
Introduced more recently in [Rea09, Rea11], the shard intersection order (or shard order for short) has proved itself a convenient tool to study other aspects of Coxeter groups, in particular in relation to noncrossing partitions, -sortable elements and cluster combinatorics.
Another partial order has been introduced in [BHZ06], that will be called the BHZ partial order on the finite Coxeter group . It has the distinction of not being a lattice, having several maximal elements. This partial order is closely related in type to the structure of some Hopf algebras on permutations; see [BHZ06] for more details.
Our aim in this note is to explain an unexpected connection between the shard order and the BHZ order. The main result is the following one.
Let be a finite Coxeter group. For every subset of simple reflections and for every integer , the number of -chains whose maximal element has support is the same for the shard order on and for the BHZ order on .
We give two different proofs of the main theorem. The first is self-contained, and follows from a common recursion obtained in section 4. In this statement and from now on, a -chain in a poset means a weakly increasing sequence of elements
A sequence is called a strict -chain.
There is a general relationship in partial orders between chains and strict chains (see for example [Sta97, §3.11]). The correspondence is defined by seeing a chain as a pair (strict chain, sequence of multiplicities). This allows one to relate the enumeration of all strict chains and the enumeration of all chains.
In our context, one can restrict this correspondence to chains whose last element has a given support. This implies that the analogue of Theorem 1.1 for strict chains also holds. This has the following consequence, by summing over all subsets .
Let be a finite Coxeter group. The -vectors of the order complexes for the shard order on and for the BHZ order on are equal.
The starting point of this article was in fact a previous observation, relating some generating series of Möbius numbers for shard orders and for BHZ order. We prove a refined version of such a statement in Section 5.
An alternative approach to the main theorem is provided in Section 6. There, we give a dimension-preserving bijection between the order complex of the BHZ poset and a certain pulling triangulation of the permutahedron. Reading, in [Rea11, Theorem 1.5], gives a dimension-preserving bijection between the order complex of the shard lattice and an isomorphic triangulation of the permutahedron. This establishes the following proposition:
Let be a finite Coxeter group. For every integer , the number of -chains is the same for the shard order on and for the BHZ order on .
2 Preliminaries and first lemmas
Let be a finite Coxeter system, where is a Coxeter group and is the fixed set of simple reflections. The letter will always denote the rank of , which is the cardinality of . Let denote the length of an element with respect to the set of generators. The symbol will be the unit of , and the unique longest element. We refer the reader unfamiliar with the subject to [Hum90, GP00, BB05] for general references on (finite) Coxeter groups.
For an element in , we will use the following (more or less standard) notations. The set of right descents of is the set of elements such that . Let be the support of , that is, the set of elements such that appears in some (or equivalently any) reduced word for .
The following notations are needed for the definition of the shard order. Let be the subgroup of generated by for . This is a conjugate of a standard parabolic subgroup. Let be the set of (left) inversions of , defined as , where is the set of positive roots in a root system for .
We also need the standard properties of parabolic decomposition of elements of . Let us recall them briefly. For a subset , let be the (standard) parabolic subgroup of generated by the simple reflections in . Given a subset of , there is a factorisation , where is the parabolic subgroup and the set of minimal length coset representatives of classes in . Equivalently, every can be uniquely written where and the right descents of do not belong to . Moreover, the expression is reduced, meaning that . Following [BB05], we call the pair the parabolic components of . We denote by the longest element of .
Given subsets of , the more general notation stands for the set of elements of the parabolic subgroup that have no right descents in . Then there is a unique factorisation , similar to the previous one and denoted in the same way; see [GP00, Chapter 3] or [BBHT92] for more details on these decompositions.
The following lemma is proved by Möbius inversion on the boolean lattice of subsets of .
The number of elements of with support is
Finally, we discuss a second lemma that is crucial for the proof of Theorem 1.1. Fix and . We introduce two sets attached to the pair . To distinguish between the two orders considered in this article, we will use the symbols and as markers standing for shard and for BHZ respectively. The first one is
The second one is
Both these sets are subsets of .
The map defines a bijection from to , with inverse map .
First, let and decompose in parabolic components: with and . Since , and is a reduced expression (i.e., the concatenation of reduced words for and for gives a reduced word for ), we have . Note also that, since , we have . Indeed, since for all , we have by definition of that for all . Since this forces
By uniqueness of the parabolic components, this defines a map , defined by .
Now, we show that the map , defined by , is well-defined. This will imply, using (4), that and therefore our claim.
Since , and the fact that the expression is reduced, we have . Since for all , we have for all . Hence . Therefore, and the claim is proven. ∎
3 Shard lattice and BHZ posets
In this section, we will recall the definitions and main properties of the shard and BHZ partial orders. For proofs and details, the reader should refer to the articles [Rea09, Rea11, STW15] and [BHZ06].
Recall that, in order to distinguish between the two orders considered in this article, we will use the symbols and as markers standing for shards and for BHZ respectively.
3.1 Shard intersection order
The shard order on can be defined as follows. Let and in . Then if and only if and . Recall that is the set of (left) inversions of and is the subgroup defined in section 2. This definition is not the original one from [Rea09, Rea11], but comes from the reformulation explained in [STW15, §4.7].
Here are some basic properties. The unique minimal element is , the unique maximal element is . The partial order is ranked, and the rank of an element is the cardinality of the descent set .
Moreover, for every , the interval is isomorphic to the shard order for the parabolic subgroup associated to the set of right descents .
By [Rea09, Th. 4.3], the Möbius function satisfies
3.2 The BHZ order
The BHZ partial order on is defined as follows: if and only if .
The partial order is ranked and the rank of an element is the cardinality of the support . The unique minimal element is , and the maximal elements are the elements of of support .
The Möbius function is described completely by [BHZ06, Th. 4]:
Let us now describe the principal upper ideal of an element. Let be an element with support .
The principal upper ideal of in the BHZ poset is in bijection with by the map that sends to .
This is just a direct consequence of the definition of the BHZ order.
Recall the set defined in (2).
The set of elements such that and is in bijection with .
This follows from the previous proposition, together with the uniqueness of the decomposition .
4 Counting chains and proof of Theorem 1.1
In this section, we obtain recursions (on the length ) for the numbers of -chains for both shard order and BHZ order.
4.1 Counting chains in shards
Let be the set of -chains in the shard order such that the top element of the chain has support . Recall the set defined in (3).
For , the numbers of -chains in the shard order satisfy
The cardinality of is the sum
where is the set of -chains ending with .
Now -chains with top element are in bijection with -chains in the shard order for a parabolic subgroup of type . Their number only depends on the set and is the sum
Therefore one gets
Exchanging the summations, one obtains
The inner sum is exactly the cardinality of as defined in (3). ∎
4.2 Counting chains in BHZ posets
Let be the set of -chains in the BHZ order such that the top element of the chain has support . Recall the set defined in (2).
For , these numbers of -chains in the BHZ order satisfy
There is a bijection between -chains and triples (element , -chain with top element , element greater than or equal to ).
By proposition 3.2, given an element of support , the number of possible with and is the cardinality of .
Therefore the number of -chains with last element of support and next-to-last element only depends on the support of . The formula follows by gathering -chains according to the support of their next-to-last element . ∎
It is clear that Proposition 1.3 is a weakening of Theorem 1.1. We wish to show, conversely, that Theorem 1.1 can be deduced from Proposition 1.3. In section 6, we will then give an independent proof of Proposition 1.3, which, together with the current argument, constitute a separate proof of Theorem 1.1.
Suppose that we know Proposition 1.3 holds for any finite Coxeter group. Write for the number of -chains in with respect to .
Because there are natural inclusions of posets for subsets , one can show that
By inclusion-exclusion, this can be expressed as:
5 An equality of characteristic polynomials
In this section, we will compare two generating series, the first one built from Möbius numbers from the bottom in the shard order, the second one from Möbius numbers to the top elements in the BHZ order. They turn out to be equal, up to sign.
5.1 Characteristic polynomials for shards
Let us consider the following polynomial in one variable:
This is exactly the characteristic polynomial of the shard poset, as defined usually for graded posets with a unique minimum. We will instead compute the more refined generating series
with variables indexed by subsets of . This reduces to by sending to .
Using the known value (5) for the Möbius function in the shard order, one finds
Introducing to represent and exchanging the summations, this becomes
5.2 Characteristic polynomials for BHZ posets
Let us consider the following polynomial
This is something like a characteristic polynomial of the opposite of the BHZ poset, except that there are several maximal elements, so there is an additional summation over them.
In fact, let us instead compute the more refined generating series
with variables indexed by subsets of .
By the known value of the Möbius function (6) in the BHZ order, one obtains
Let us use to denote . Then for fixed, is determined from by the relation , by definition of the BHZ partial order. One can therefore replace the summation over and by a summation over and . One obtains
where the condition that has been removed because it is implied by the other conditions. The inner sum can be factorised into the product of
where we used the definition of .
At the end, one finds
Note that by using the involution , the inner sum is the same as the sum over such that . This proves the following result.
For every finite Coxeter group , there is an equality .
6 A bijection between strict chains in the BHZ poset and the faces of a pulling triangulation of the permutahedron
As before, is a finite Coxeter system. In [Rea11, Theorem 1.5], the author exhibits a bijection between strict -chains in the shard lattice and -faces (faces of dimension ) of a pulling triangulation of the -permutahedron. We exhibit here a bijection between strict -chains in the BHZ poset and -faces of an isomorphic pulling triangulation of the -permutahedron.
Recall that the -permutahedron is the convex hull of the -orbit of a generic point in the space on which acts as a finite reflection group. The faces of are naturally indexed by the cosets for all : the face is of dimension for any and can be identified with ; see for instance [Hoh12]. Moreover, the vertices of a face of are intervals in the right weak order. The right weak order on is defined by if a reduced word for is a prefix of a reduced word for , i.e., ; see [BB05, §3]. Then the set of vertices of the face is the set .
For a polytope, with a fixed total order on its vertices, the corresponding pulling triangulation of is defined as follows (see [Lee97]). Let be the initial (minimum) vertex. Inductively, determine the pulling triangulation of each facet which does not include vertex (with respect to the total order on its vertices defined by restriction). Then, cone each of the simplices from these triangulations over the vertex . Remark that it is not actually necessary to start with a total order on the vertices: it is sufficient if there is a unique minimum vertex for each face of the polytope. We can therefore define to be the pulling triangulation of with respect to right weak order.
Reading in [Rea11, Theorem 1.5] gave a bijection from the faces of the order complex of the shard lattice to the pulling triangulation defined with respect to the reverse of weak order. These two triangulations are equivalent under the linear transformation defined by multiplication on the left by .
We first define our map for strict -chains containing . To in the BHZ poset, , we associate the following subset of :
where , or equivalently, . Observe that is the maximal element of the input chain, while .
For a strict -chain not containing , we add in , and apply the above map on strict -chains containing , and remove from the resulting set.
The map is a dimension-preserving bijection from the order complex of the poset to .
Notice that the above theorem, together with [Rea11, Theorem 1.5], proves Proposition 1.3. As we showed in Section 4, Theorem 1.1 follows. This provides a second (not self-contained) proof of the main theorem.
Beware that the bijection is however not an isomorphism of simplicial complexes, as it does not preserve the incidence relation of simplices.
By its very construction, is given essentially by applying the same map twice, one time mapping chains containing to simplices containing , and another time mapping chains not containing to simplices not containing . Adding or removing is a bijective process that allows one to go from one case to the other. Therefore proving the bijectivity of can be done by looking only at the case where is present.
Before proving the theorem, we give some properties of the map .
If is a -chain in the BHZ poset starting with , and , then .
By [BHZ06, Proposition 6(3)], since , we have that . Since is a reduced factorization for each , it follows that . ∎
Note that this lemma implies that, although the image of is, by definition, just a set of elements in , in fact, we can determine the numbering of the elements by looking at their relative order with respect to .
The map is injective.
It suffices to consider applied to strict -chains containing . Suppose and . As discussed above, from , we can reconstruct the numbering of the elements. In particular, this allows us to determine , and then we can reconstruct . Since we can reconstruct the on the basis of their image under , it must be that is injective. ∎
Let be a strict -chain in the BHZ poset, and as in (21). Then:
For , we have .
is a subset of the vertices of the face of for which is the smallest vertex.
Let . Observe that . Since , we have that . Since , while , it follows that the parabolic factorization of with respect to the parabolic subgroup is . This establishes the first point. The second point is just a rephrasing of the first point using the description of the faces of given at the beginning of this section. ∎
We also need the following standard facts on pulling triangulations.
Let be a convex polytope with a fixed order on its vertices.
The pulling triangulation of restricts to the pulling triangulation of each face of .
Each face of the pulling triangulation is contained in a smallest face of , and contains the minimal vertex of that face (with respect to the order on the vertices).
The first point is established by induction on the dimension of . Every face of the pulling triangulation is contained in a smallest face of because the faces of form a lattice under the inclusion order. By the first point, the pulling triangulation of restricts to the pulling triangulation of . By the definition of the pulling triangulation of , the faces of it which do not include the minimal vertex of are each contained in a facet of , and therefore do not have as the smallest face in which they are contained. ∎
Proof of Theorem 6.2.
The map is injective by Lemma 6.4.
We now show that the image of a strict -chain is a -face of . The proof is by induction on . The base case is trivial. Also, it suffices to consider the case that , and .
For , define . Now . The chain lies in whose rank is less than , so by induction, is a face of the pulling triangulation of .
For , define . Note that for , we have , which can be rewritten as . Left-multiplication by is an order-preserving bijection from to . This map takes the pulling triangulation of to the pulling triangulation of the face of .
This set forms the vertices of a simplex with one vertex at , and the others forming a face of the pulling triangulation of . Since , the face of does not contain . Thus, is a -face of the pulling triangulation of .
Conversely, suppose we have a -face in . We may assume . Write . is also a face of . By Lemma 6.6(2), lies in a well-defined smallest face of . That face can be uniquely written as with and . By Lemma 6.6(2), contains the smallest vertex of this face, which is . Now defines a -face of containing . By induction, it is the image under of a chain . Since is not contained in for any , supp .
Define . Observe that . Thus .
We therefore have constructed a chain in the BHZ poset. We now claim that . The necessary calculation follows exactly the same logic as the proof that the image of a -chain is a -face of . ∎
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Institut de Recherche Mathématique Avancée, CNRS UMR 7501, Université de Strasbourg, F-67084 Strasbourg Cedex, France
Institut de Recherche Mathématique Avancée, CNRS UMR 7501, Université de Strasbourg, F-67084 Strasbourg Cedex, France
LaCIM et Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada
LaCIM et Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, Canada