The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

TL;DR

This paper proves that the absolute orders on Coxeter groups $A_n$ and $B_n$ are strong Sperner, extending previous results about related orders on symmetric groups.

Contribution

It provides a concise and elegant proof that the absolute orders on $A_n$ and $B_n$ are strong Sperner, a significant extension of prior work on symmetric groups.

Findings

Absolute orders on $A_n$ and $B_n$ are strong Sperner

Extends previous results on symmetric groups

Provides a concise, elegant proof

Abstract

Over 50 years ago, Rota posted the following celebrated `Research Problem': prove or disprove that the partial order of partitions on an $n$-set (i.e., the refinement order) is Sperner. A counterexample was eventually discovered by Canfield in 1978. However, Harper and Kim recently proved that a closely related order --- i.e., the refinement order on the symmetric group --- is not only Sperner, but strong Sperner. Equivalently, the well-known absolute order on the symmetric group is strong Sperner. In this paper, we extend these results by giving a concise, elegant proof that the absolute orders on the Coxeter groups $A_n$ and $B_n$ are strong Sperner.

Equations

$$\pi \le \mu \text{ if and only if }{l}_T(\mu )=l_T(\pi )+l_T({\pi }^{-1}\mu )$$

$$w=r_n{r}_{n-1}\cdots r_2{r}_1$$

$$l_T\left(\prod _{i=0}^{n-1}{r}_{n-i}\right)=|\{i:r_i\ne e\}|.$$

$$\prod _{i=0}^{n-1}{C}_{d_{n-i}}=C_{d_n}\times \cdots \times C_{d_2}\times C_{d_1}$$

Full text

1. Introduction

In 1928, Sperner [7] proved that the poset of subsets of $[n]=\{1,2,\ldots,n\}$ has the property that none of its antichains (i.e., collections of pairwise incomparable vertices in the poset) has cardinality larger than the largest rank. In 1967, Rota [6] famously conjectured that the refinement order $\Pi_{n}$ (i.e., the poset of partitions of $[n]$) has this same property (which became known as the Sperner property). There were many attempts to prove Rota’s conjecture, but in 1978, Canfield [2] discovered a counterexample to Rota’s conjecture for $n$ larger than Avogadro’s number. Although the refinement order $\Pi_{n}$ is not Sperner, there is a closely related poset on the symmetric group $S_{n}$ (also called the refinement order) which Harper and Kim [5] recently proved is not only Sperner, but strong Sperner. The refinement order on $S_{n}$ is anti-isomorphic to a well-known (see, e.g., [1]) order on $S_{n}$ called the absolute order; i.e., $x\leq y$ in the refinement order if and only if $y\leq x$ in the absolute order. Hence an immediate corollary to [5] is that the absolute order on $S_{n}$ is strong Sperner.

The main result in this paper is Theorem 6.1, which states that the absolute orders on the Coxeter groups $A_{n}$ and $B_{n}$ are strong Sperner. The outline of the paper is as follows. In Section 2, we recall that the Coxeter groups $A_{n}$ and $B_{n}$ can be realized as symmetry groups for regular polytopes. This perspective is crucial to the proof of our main result. In Section 3, we recall basic definitions and facts regarding posets, the Sperner property, and absolute orders. Section 4 contains Proposition 4.2, which states that any element of $A_{n}$ or $B_{n}$ can be uniquely expressed as a product of certain reflective symmetries of a fixed flag. This proposition is a key ingredient in the proof of our main theorem, but is also an interesting result independently. Another key ingredient is Harper’s Product Theorem [4], which is used in Section 5 to show that certain spanning subposets of the absolute orders have the normalized flow property, a strengthening of the strong Sperner property. Finally, Section 6 contains the proof of the main theorem.

2. The regular $n$-simplex and $n$-cube and their symmetries

We begin by recalling some basic facts about the regular $n$-simplex and $n$-cube and their symmetries. The regular $n$-simplex $\Delta_{n}$ is the convex hull of the standard basis $\{e_{1},e_{2},\ldots,e_{n+1}\}$ for $\mathbb{R}^{n+1}$. The symmetry group of $\Delta_{n}$ is the finite irreducible Coxeter group $A_{n}$. There is a bijective correspondence between faces of $\Delta_{n}$ and subsets of $[n+1]=\{1,2,\ldots,n+1\}$ determined by mapping the [math]-dimensional face $\{e_{i}\}$ in $\Delta_{n}$ to the singleton $\{i+1\}$ for each $i$. In this correspondence, the $i$-dimensional faces (or $i$-faces) in $\Delta_{n}$ correspond to subsets of $[n+1]$ of size $i+1$, and the facets (i.e., the $(n-1)$-faces) correspond to subsets of size $n$. There is an isomorphism between $A_{n}$ and $S_{n+1}$ such that the action of $A_{n}$ on $\Delta_{n}$ corresponds to the natural action of $S_{n+1}$ on $[n+1]$. The reflections in $A_{n}$ correspond to the transpositions in $S_{n+1}$; in particular, the reflection swapping $e_{i}$ and $e_{j}$ corresponds to the transposition $(i\text{ }j)$.

The $n$-cube $\square_{n}$ is the convex hull in $\mathbb{R}^{n}$ of the Cartesian product $\{-1,1\}^{n}\subset\mathbb{R}^{n}$. The dual polytope to the $n$-cube is the $n$-cross-polytope $\diamondsuit_{n}$, which is the convex hull of $\{\pm e_{1},\pm e_{2},\ldots,\pm e_{n}\}\subset\mathbb{R}^{n}$. The map $e_{i}\mapsto i$ induces a bijective correspondence between the faces of $\diamondsuit_{n}$ and subsets of $\{\pm 1,\pm 2,\ldots,\pm n\}$ satisfying a special property. Specifically, the $i$-faces of $\diamondsuit_{n}$ correspond to subsets $S\subset\{\pm j\}_{j=1}^{n}$ of size $i$, with the property that $k\in S$ implies $-k\notin S$ for all $k\in\{\pm j\}_{j=1}^{n}$. The symmetry group for each of the dual polytopes $\square_{n}$ and $\diamondsuit_{n}$ is the hyperoctahedral group $B_{n}$. The hyperoctahedral group is isomorphic to the signed permutation group — i.e., the group of permutations $w$ of the set $\{\pm j\}_{j=1}^{n}$ with the property that $w(-i)=-w(i)$ for all $i$ — and there is an obvious choice of isomorphism between these two groups such that their respective actions on $\diamondsuit_{n}$ and $\{\pm j\}_{j=1}^{n}$ correspond. Following [3, Section 2.2], we denote the signed permutation with cycle form $(a_{1}\text{ }a_{2}\text{ }\cdots\text{ }a_{k})(-a_{1}\text{ }-a_{2}\text{ }\cdots\text{ }-a_{k})$ by $(\!(a_{1},a_{2},\ldots,a_{k})\!)$, and $(a_{1}\text{ }a_{2}\text{ }\cdots\text{ }a_{k}\text{ }-a_{1}\text{ }-a_{2}\text{ }\cdots\text{ }-a_{k})$ by $[a_{1},a_{2},\ldots,a_{k}]$. The set of reflections in $B_{n}$ corresponds to the union of $\{[i]\}_{i=1}^{n}$ and $\{(\!(i,j)\!),(\!(i,-j)\!)\}_{1\leq i<j\leq n}$; in particular, the reflection of $\diamondsuit_{n}$ swapping $e_{i}$ and $-e_{i}$ corresponds to $[i]$, and the reflection swapping $e_{i}$ and $\pm e_{j}$ (for $i\neq j$) corresponds to $(i\text{ }\pm j)$.

Lemma 2.1**.**

For any pair $(C,C^{\prime})$ of distinct facets in $\Delta_{n}$ (resp. $\square_{n}$), there is a unique reflection in $A_{n}$ (resp. $B_{n}$) mapping $C$ to $C^{\prime}$.

Proof.

Let $C\neq C^{\prime}$ be facets in $\Delta_{n}$. Recall that facets in $\Delta_{n}$ correspond to subsets of size $n$ in $[n+1]$, and reflective symmetries of $\Delta_{n}$ correspond to transpositions in $S_{n+1}$. Since $C\neq C^{\prime}$, it follows that $C-C^{\prime}=\{i\}$ and $C^{\prime}-C=\{j\}$ for some $i\neq j$. The unique reflection mapping $C$ to $C^{\prime}$ is $(i\text{ }j)$.

Now let $C\neq C^{\prime}$ be facets in $\square_{n}$. The facets of $\square_{n}$ correspond to the vertices of $\diamondsuit_{n}$, which in turn correspond to elements of $\{\pm 1,\pm 2,\ldots,\pm n\}$. Suppose without loss of generality that $C$ corresponds to $1$. Either $C^{\prime}$ corresponds to $-1$, $j$ for some $j\neq 1$, or $-j$ for some $j\neq 1$. In any case, there is a unique reflection in $B_{n}$ mapping $C$ to $C^{\prime}$ (specifically, the reflections $[1]$, $(\!(1,j)\!)$, and $(\!(1,-j)\!)$, respectively). ∎

Define a (complete) flag $\mathscr{F}=(\mathcal{P}_{i})_{i=0}^{n}$ in an $n$-dimensional regular polytope $\mathcal{P}$ to be a sequence of faces in $\mathcal{P}$, ordered by containment, with $\dim(\mathcal{P}_{i})=i$. The action of $A_{n}$ (resp. $B_{n}$) on $\Delta_{n}$ (resp. $\square_{n}$) induces a simply transitive action on the associated set of flags. Hence if we designate some flag in $\Delta_{n}$ or $\square_{n}$ — call it the standard flag $\mathscr{F}^{\text{std}}=(\mathcal{P}_{i}^{\text{std}})_{i=0}^{n}$ — then a correspondence between elements of its symmetry group and its set of flags can be defined via $w\mapsto w\cdot\mathscr{F}^{\text{std}}$. Note that, for all $i\in[0,n]$, the $i$-faces for the $n$-simplex (resp. the $n$-cube) are $i$-simplices (resp. $i$-cubes).

3. Posets, the Sperner property, and the absolute orders

Let $P$ be a finite graded poset with rank decomposition $P=\bigsqcup_{i=0}^{r}P_{i}$. In this paper, the term “poset” is always used to mean finite graded poset. A $k$-family in $P$ is a subset of $P$ containing no chain of size $k+1$. The poset $P$ is defined to be $k$-Sperner if the union of the $k$ largest rank levels $P_{i}$ is a $k$-family of maximal size; strong Sperner if $P$ is $k$-Sperner for all $k\in[1,r+1]$; and rank unimodal if $|P_{0}|\leq|P_{1}|\leq\cdots\leq|P_{j-1}|\leq|P_{j}|\geq|P_{j+1}|\geq\cdots\geq|P_{r}|$ for some $j$. Note that the 1-Sperner property is otherwise known as the Sperner property, and a 1-family is otherwise known as an antichain. A spanning subposet for a poset $P$ is any subposet with the same vertex set and rank function as $P$.

Lemma 3.1**.**

Suppose that $P$ is a spanning subposet of $P^{\prime}$. If $P$ is rank unimodal and strong Sperner, then so is $P^{\prime}$.

Proof.

Since $P$ is rank unimodal, its largest $k$ rank levels can be chosen so that their ranks are consecutive. Their union is a $k$-family in both $P$ and $P^{\prime}$. Since $P$ is $k$-Sperner, this union is a $k$-family in $P$ of maximal size, and therefore a $k$-family in $P^{\prime}$ of maximal size. ∎

We briefly recall some generalities about absolute orders; see, e.g., [1] for details. Let $W$ be a finite Coxeter group with set of reflections $T$. The absolute length $l_{T}$ on $W$ is the word length with respect to $T$. The absolute order on $W$ is defined by

[TABLE]

for all $\pi,\mu\in W$. Equivalently, the absolute order is the partial order on $W$ generated by the covering relations $w\rightarrow tw$, where $w\in W$, $t\in T$, and $l_{T}(w)<l_{T}(tw)$. This order is graded with rank function $l_{T}$. The absolute length generating function $\mathrm{P}_{W}(q)=\sum_{w\in W}q^{l_{T}(w)}$ satifies $\mathrm{P}_{W}(q)=\prod_{i=1}^{n}(1+(d_{i}-1)q)$, where $(d_{i})_{i=1}^{n}$ is the degree sequence for $W$ (and $n=\operatorname{\mathrm{rank}}(W)$) [1, p. 35]. It follows that $|T|=|l_{T}^{-1}(1)|=\sum_{i=1}^{n}(d_{i}-1)$. Moreover, the rank sequence $\left(|l^{-1}_{T}(i)|\right)_{i=0}^{n}$ for any absolute order is strictly log-concave by [8, Theorem 4.5.2], and thus all of the absolute orders are rank unimodal.

4. Factoring elements of $A_{n}$ and $B_{n}$

For all that follows, $\mathcal{P}$ denotes the regular $n$-simplex or $n$-cube, and $W$ denotes the corresponding symmetry group. Any reflective symmetry of an $i$-face $\mathcal{P}_{i}$ of $\mathcal{P}$ uniquely extends to a reflective symmetry of $\mathcal{P}$. Define $T_{\mathcal{P}_{i}}$ to be the embedding of the set of reflections of $\mathcal{P}_{i}$ into $W$.

Lemma 4.1**.**

Let $\mathcal{P}$ be the $n$-simplex or $n$-cube, and let $W$ be the corresponding group of symmetries with degree sequence $(d_{i})_{i=1}^{n}$. Fix a standard flag $(\mathcal{P}_{i}^{\text{std}})_{i=0}^{n}$ in $\mathcal{P}$, and set $T_{i}=T_{{\mathcal{P}_{i}}^{\text{std}}}$. It follows that, for all $i\in[1,n]$, $|T_{i}-T_{i-1}|=d_{i}-1$.

Proof.

The $n$-simplex (resp. the $n$-cube) has the property that, for each $i$, each of its $i$-faces is an $i$-simplex (resp. $i$-cube). Hence the symmetry group for any of its $i$-faces is $A_{i}$ (resp. $B_{i}$). If the degree sequence for the $n$-simplex (resp. $n$-cube) is $(d_{j})_{j=1}^{n}$, then the degree sequence associated to an $i$-face is $(d_{j})_{j=1}^{i}$. It follows that $|T_{i}-T_{i-1}|=|T_{i}|-|T_{i-1}|=\sum_{j=1}^{i}(d_{j}-1)-\sum_{j=1}^{i-1}(d_{j}-1)=d_{i}-1$. ∎

It is easily verified that the relation between a regular polytope and the degree sequence of its symmetry group described in Lemma 4.1 is satisfied by precisely the $n$-simplices, $n$-cubes, and $m$-gons (and none of the other regular polytopes). For ease of reference, we note here that the degree sequence $(d_{i})_{i=1}^{n}$ for $A_{n}$ is defined by $d_{i}=i+1$, and for $B_{n}$ is defined by $d_{i}=2i$.

Proposition 4.2**.**

Let $\mathcal{P}$ be the $n$-simplex or $n$-cube, and let $W$ be the associated symmetry group. Fix a standard flag $\mathscr{F}^{\text{std}}=(\mathcal{P}_{i}^{\text{std}})_{i=0}^{n}$ in $\mathcal{P}$, and set $T_{i}=T_{\mathcal{P}_{i}^{\text{std}}}$.

1. (1)

Any element $w\in W$ has a unique factorization of the form

[TABLE]

with $r_{i}\in(T_{i}-T_{i-1})\sqcup\{e\}$ for each $i$, where $e$ is the identity in $W$. 2. (2)

Given such a factorization, the length can be computed via

[TABLE] 3. (3)

Finally, $\prod_{i=0}^{n-1}r_{n-i}$ covers $\prod_{i=0}^{n-1}r^{\prime}_{n-i}$ if there exists $k$ such that $r_{k}\neq r^{\prime}_{k}=e$ and $r_{j}=r_{j}^{\prime}$ for all $j\neq k$.

Proof.

We begin by proving (1). The claim is clearly true for $n=1$. Now let $n>1$ be arbitrary, and suppose the claim is true for $n-1$. Let $w\in W$, with corresponding flag $\mathscr{F}=(\mathcal{P}_{i})_{i=0}^{n}$. If $(\mathcal{P}_{i})_{i=0}^{n-1}$ is a flag in the “standard facet” $\mathcal{P}_{n-1}^{\text{std}}$, then the claim follows by the inductive hypothesis. Suppose instead that $(\mathcal{P}_{i})_{i=0}^{n-1}$ is a flag in some other facet $C$. Lemma 2.1 implies that there is a unique reflection $r_{n}\in(T_{n}-T_{n-1})-\{e\}$ mapping $C$ to $\mathcal{P}_{n-1}^{\text{std}}$. By the inductive hypothesis, it follows that $r_{n}\cdot\mathscr{F}=(r_{n-1}\cdots r_{2}r_{1})\cdot\mathscr{F}^{\text{std}}$ with $r_{i}\in(T_{i}-T_{i-1})\sqcup\{e\}$ for all $i\in[1,n-1]$. Therefore $w\cdot\mathscr{F}^{\text{std}}=\mathscr{F}=r_{n}(r_{n-1}\cdots r_{2}r_{1})\cdot\mathscr{F}^{\text{std}}$, and the claim follows.

To prove (2), we first let $\mathcal{P}$ be the $n$-simplex and let $W$ be its symmetry group. Assume without loss of generality that $\mathscr{F}^{\text{std}}=\left([i+1]\right)_{i=0}^{n}$. Then $T_{i}-T_{i-1}$ consists of all transpositions $(j\text{ }(i+1))$ with $j\in[1,i]$. If $w$ is a product of elements in $T_{i-1}\sqcup\{e\}$, then $w$ is a permutation of $[i]$. Hence $l_{T}(r_{i}w)>l_{T}(w)$, which implies that $l_{T}(r_{i}w)=l_{T}(w)+1$. The claim follows from a straight-forward induction on $n$. Now let $\mathcal{P}$ be the $n$-cube and $W$ its symmetry group. Assume without loss of generality $\mathscr{F}^{\text{std}}=(\mathcal{P}_{i}^{\text{std}})_{i=0}^{n}$ is chosen so that the symmetries of $\mathcal{P}^{\text{std}}_{i}$ correspond to symmetries of $\{\pm 1,\pm 2,\ldots,\pm i\}$. Then $T_{i}$ consists of all reflections of the form $[j]$, $(\!(j,k)\!)$, and $(\!(-j,k)\!)$ with $j,k\in[1,i]$ and $j\neq k$, and $T_{i}-T_{i-1}$ consists of all reflections of the form $[i]$, $(\!(j,i)\!)$, and $(\!(-j,i)\!)$ with $j\in[1,i-1]$. Similar to the case above, the product $r_{i}w$ of $r_{i}$ in $T_{i}-T_{i-1}$ with a product $w$ of reflections in $T_{i-1}$ has $l_{T}(r_{i}w)>l_{T}(w)$. Hence $l_{T}(r_{i}w)=l_{T}(w)+1$, and the claim follows by induction.

Finally, to prove (3), let $w$ and $w^{\prime}$ be elements of $W$ with the property that their expansions $w=\prod_{i=0}^{n-1}r_{i}$ and $w^{\prime}=\prod_{i=0}^{n-1}r^{\prime}_{i}$ satisfy $r_{k}\neq r^{\prime}_{k}=e$ for some $k$ and $r_{j}=r_{j}^{\prime}$ for all $j\neq k$. By Proposition 4.2.2, it follows that $l_{T}(w^{\prime})+1=l_{T}(w)$. Set $\sigma=\prod_{i=0}^{k-1}r_{i}$ and $\tau=\prod_{i=k+1}^{n-1}r_{i}$, so that $w=\sigma r_{k}\tau$ and $w^{\prime}=\sigma\tau$. Then $l_{T}((w^{\prime})^{-1}w)=l_{T}(\tau^{-1}\sigma^{-1}\sigma r_{k}\tau)=l_{T}(\tau^{-1}r_{k}\tau)=1$. Since $l_{T}(w^{\prime})+l_{T}((w^{\prime})^{-1}w)=l_{T}(w)$, it follows that $w$ covers $w^{\prime}$.∎

5. The Sperner property for products

It is not the case that a product of Sperner (or even strong Sperner) posets is necessarily Sperner. However, if two posets satisfy a strengthening of the strong Sperner property called the normalized flow property (abbreviated NFP), as well as an additional easily verified hypothesis, then their product has the NFP (and hence is strong Sperner) by Harper’s Product Theorem [4]. As the NFP is only referenced in this paper for the purpose of proving Lemma 5.1, we do not recall the fundamental definitions and theory regarding flows on posets here. The curious reader may refer to, e.g., [4].

Define a $k$-claw $C_{k}=\bigsqcup_{l=0}^{1}(C_{k})_{l}$ to be the graded poset with $|(C_{k})_{0}|=1$, $|C_{k}|=k$, and whose underlying graph is complete bipartite.

Lemma 5.1**.**

Let $\{k_{i}\}_{i=1}^{n}\subset\mathbb{Z}_{+}$. The product poset $\prod_{i=1}^{n}C_{k_{i}}$ is strong Sperner.

Proof.

Suppose that the weight $\nu(x)$ of any vertex $x$ in a claw equals 1. Each claw $C_{k_{i}}$ has a normalized flow (e.g., the flow $f$ defined by setting $f(e)=1/(k_{i}-1)$ for all edges $e$) and thus has the NFP. Moreover, the sequence $\left(\nu((C_{k_{i}})_{l})\right)_{l=0}^{1}$ is clearly log-concave (or 2-positive using the terminology of [4]) for each $i$. By [4, Product Theorem], it follows that any product of claws $C_{k_{i}}\times C_{k_{j}}$ has NFP and satisfies the property that the sequence $\left(\nu((C_{k_{i}}\times C_{k_{j}})_{l})\right)_{l=0}^{2}$ is log-concave. The claim follows by induction. ∎

6. Main result

Theorem 6.1**.**

The absolute orders on $A_{n}$ and $B_{n}$ are strong Sperner.

Proof.

Let $\mathcal{P}$ be the $n$-simplex or $n$-cube, and let $W$ be the associated symmetry group. Fix a standard flag $\mathscr{F}^{\text{std}}=(\mathcal{P}_{i}^{\text{std}})_{i=0}^{n}$ in $\mathcal{P}$, and set $T_{i}=T_{\mathcal{P}_{i}^{\text{std}}}$. Let $(d_{i})_{i=1}^{n}$ be the degree sequence for $W$. Consider the product poset

[TABLE]

of claws $C_{d_{i}}$. For each $i$, define a bijective correspondence between the vertices of the claw $C_{d_{i}}$ and the elements of $(T_{i}-T_{i-1})\sqcup\{e\}$ by mapping the $d_{i}-1$ vertices in $(C_{d_{i}})_{1}$ bijectively onto $T_{i}-T_{i-1}$ (such a bijection exists by Lemma 4.1) and the rank [math] vertex in $C_{d_{i}}$ to $e$. These bijective correspondences between claws and sets of reflections induce a bijective correspondence $\phi(r_{n},\ldots,r_{2},r_{1})=r_{n}\cdots r_{2}r_{1}$ between the vertices of the product poset $\prod_{i=0}^{n-1}C_{d_{n-i}}$ and the vertices of the absolute order $W$ by Proposition 4.2(1).

We claim that $\prod_{i=0}^{n-1}C_{d_{n-i}}$ can be viewed as a spanning subposet of $W$ via the above bijection between of the vertex sets. It suffices to prove that if $y$ covers $x$ in $\prod_{i=0}^{n-1}C_{d_{n-i}}$, then $\phi(y)$ covers $\phi(x)$ in $W$. Suppose that $(r_{n},\ldots,r_{2},r_{1})$ covers $(r^{\prime}_{n},\ldots,r^{\prime}_{2},r^{\prime}_{1})$ in the product of claws. Then there exists $k$ for which $r_{k}\neq r_{k}^{\prime}=e$ and $r_{j}=r_{j}^{\prime}$ for all $j\neq k$. By Proposition 4.2(3), the claim immediately follows. By Lemma 5.1, $\prod_{i=0}^{n-1}C_{d_{n-i}}$ is strong Sperner. Since $\prod_{i=0}^{n-1}C_{d_{n-i}}$ is a spanning subposet of $W$, it follows by Lemma 3.1 that $W$ is strong Sperner. ∎

Remark 6.2*.*

The remaining finite irreducible Coxeter groups (see, e.g., [1, p. 2]) are the other regular polytope symmetry groups $I_{2}(m)$ (for $m\geq 5$), $H_{3}$, $H_{4}$, and $F_{4}$, and the Weyl groups $D_{n}$ (for $n\geq 4$), $E_{6}$, $E_{7}$, and $E_{8}$. It is straight-forward to verify that Lemma 2.1, Lemma 4.1, Proposition 4.2, and Theorem 6.1 extend to the dihedral groups $I_{2}(m)$ for all $m$. It follows that the absolute orders on the dihedral groups are strong Sperner.