ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.08 / 305–315
https://doi.org/10.26493/1855-3974.2584.68d
(Also available at http://amc-journal.eu)
The antiprism of an abstract polytope*
Ian Gleason
Univeristy of California, Berkeley, United States
Isabel Hubard †
Instituto de Matemáticas, Universidad Nacional Autonoma de México,
Circuito Exterior, C.U. Coyoacán 04510, México D.F.
Received 21 March 2021, accepted 19 August 2021, published online 14 June 2022
Abstract
Antiprisms of polygons are classical convex vertex-transitive polyhedra. In this paper,
for any given (abstract) polytope, we define its antiprism. We then find the automorphism
group of the antiprism of P in terms of the extended group of P (the groups of auto-
morphisms and dualities) as well as some transitivity properties. We also give a relation
between some products of abstract polytopes and their antiprisms.
Keywords: Antiprism, abstact polytopes.
Math. Subj. Class. (2020): 51M20, 52B05
The antiprism is a classical convex polyhedron. The antiprism of a polygon can be
constructed by taking, in Euclidian 3-space, two identical copies of a regular n-gon in
parallel planes, in such a way that the vertices of one of the polygons are “aligned” with
the mid points of the edges of the other. By taking the convex hull of all the vertices, we
obtain the antiprism over an n-gon (see Figure 1).
For higher dimensions, the concept of a convex antiprism is not always defined (see [1,
2] and [3] for further discussion of the subject). In this paper we define the antiprism of any
abstract polytope and show that it is indeed again an abstract polytope. The given definition
generalizes the antiprism of a polygon and satisfies that the antiprism of a polytope and its
dual is the same.
The paper uses some of the ideas and notation of the products of polytopes described
in [4]. Moreover, we give relations between some products and their antiprisms. We then
use such relations to compute the automorphism group of an antiprism. These results are
summarized in the following theorem.
*The authors would like to thank Steve Wilson for suggesting studying the antiprism of an abstract polytope.
We also gratefully acknowledge financial support of the PAPIIT-DGAPA, under grant IN-109218 and of CONA-
CyT, under grant A1-S-21678.
†Corresponding author.
E-mail addresses: ianandreigf@gmail.com (Ian Gleason), isahubard@im.unam.mx (Isabel Hubard)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
306 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
Figure 1: Antiprism over a pentagon.
Theorem A. Let P and Q be two abstract polytopes and Ant(P),Ant(Q) be their an-
tiprisms.
(a) If ⋊⋉ and ⊕ denote the join product and the direct sum of abstract polytopes, respec-
tively, then Ant(P ⋊⋉ Q) ∼= Ant(P)⊕Ant(Q).
(b) If Γ̂ denotes the extended group of P and P = Qm11 ⋊⋉ Q
m2
2 ⋊⋉ · · · ⋊⋉ Qmrr , where
each Qi is a prime polytope with respect to the join product, then
Γ(Ant(P)) = Πri=1
(
(Γ̂(Qi))mi ⋊ Smi
)
.
In particular, if P is prime with respect to the join product, then Γ(Ant(P)) ∼= Γ(P)
whenever P is a not a self-dual polytope, while if P is self-dual, then Γ(P) has index
2 in Γ(Ant(P)) = Γ̂(P).
(c) Let P be a prime polytope with respect to the join product. If P is a k-orbit polytope
of rank n, then Ant(P) is either a 2nk-orbit polytope (if P is self-dual) or a 2n+1k-
orbit polytope (if P is not self-dual).
The paper is organized as follows. Section 1 reviews the basic notions about abstract
polytopes and their products. In Section 2 we define the antiprism and show that is always
an abstract polytope and analyse the flags of the antiprism in terms of the flags of the poly-
tope. Sections 3 and 4 deal with the interaction between some products and the antiprism,
and with the study of the automorphism group of an antiprism, respectively.
1 Abstract polytopes, their join product and direct sum
Abstract polytopes are combinatorial generalizations of the face lattice of convex poly-
topes. In this section we give the basic definitions from the theory of abstract polytopes, as
well as two of their products. For details on these subjects we refer the reader to [5] and
[4], respectively.
An (abstract) polytope is a partially ordered set (poset) P , whose elements are called
faces, such that it has a minimal and a maximal element and is ranked: all its maximal
chains, called flags, have the same number of elements. This endows the poset with a rank
function r satisfying that if F,G ∈ P with F ≤ G, then r(F ) ≤ r(G), and if r(F ) = r(G),
then F and G are either equal or they are not incident in P . We say that the minimal face
has rank −1, and if the range of the rank function is {−1, 0, . . . , n}, then we say that P
has rank n or is a n-polytope. A face of rank i is said to be an i-face and the 0-, 1- and
n− 1-faces are the vertices, edges and facets of P , respectively. The minimal and maximal
faces are the improper faces of P , and all other faces are proper. We also require that P
satisfies the diamond condition, meaning that whenever F,G ∈ P are two incident faces
I. Gleason and I. Hubard: The antiprism of an abstract polytope 307
such that their ranks differ by 2, then there are exactly two faces H,H ′ of P satisfying that
F < H,H ′ < G. Finally, we ask that P be strongly connected in the sense that the poset is
connected and each of its open intervals with more than two elements is connected as well.
A section of P is a closed interval of P . Every section of P is a polytope in its own
right. The diamond condition is equivalent to saying that all sections of rank 1 have exactly
4 faces. This condition also implies that for each i ∈ {0, 1, . . . , n − 1} and every flag Φ,
there is a unique i-adjacent flag to Φ that differs from Φ only in the element of rank i. We
shall denote the set of all flags of P by F(P), and the i-adjacent flag of Φ by Φi.
The dual of a polytope P is the poset that has the same elements as P , but with the
reverse order. If a polytope is isomorphic to its dual, it is said to be self-dual.
An automorphism of P is an order preserving bijection. The group of all automor-
phisms of P is its automorphism group and it shall be denoted by Γ(P). A duality of a
self-dual polytope is an order reversing bijection. The composition of two dualities of a
self-dual polytope is not a duality, but an automorphim. Thus, the extended group of P
is the group that contains all automorphisms and dualities of P and it will be denoted by
Γ̂(P). Note that Γ̂(P) has Γ(P) as a subgroup of index at most 2; the groups coincide
whenever P is not self-dual.
Given two polytopes P and Q, their join product, P ⋊⋉ Q, is the polytope whose
elements are the pairs (F,G), with F ∈ P and G ∈ Q. Two elements (F,G) and (F ′, G′)
are incident in P ⋊⋉ Q if and only if F ≤P F ′ and G ≤Q G′. The rank of (F,G) is
rankP(F ) + rankQ(G) + 1. A polytope P is said to be prime with respect to the join
product if it cannot be decomposed as the join product of two polytopes of ranks at least 0.
The direct sum of the polytopes P and Q, with maximum elements Fn andGm, respec-
tively is P⊕Q = {(F,G) ∈ P ⋊⋉ Q | F ̸= Fn, G ̸= Gm}∪{(Fn, Gm)}. The order of the
direct sum is given by (F,G) ≤P⊕Q (F ′, G′) if and only if F ≤P F ′ and G ≤Q G′, and
the rank of the face (F,G) is rankP(F )+rankQ(G), which implies that the rank of P⊕Q
is n +m. A polytope P is said to be prime with respect to the direct sum if it cannot be
decomposed as the join product of two polytopes of ranks at least 1. The following lemma
falls straightforward from the definitions.
Lemma 1.1. Let P and Q be two polytopes, and let (F,G) be a proper face of P ⊕ Q.
Then the section {(H,K) ∈ P ⊕ Q | (H,K) ≤P⊕Q (F,G)} is isomorphic to the join
product of the sections {H ∈ P | H ≤P F} and {K ∈ Q | K ≤Q G}.
In [4] the authors study the automorphism group of a product in terms of the automor-
phisms groups of the factors. In particular we have the following result.
Theorem 1.2 ([4]). Let P = Qm11 ⊕Q
m2
2 ⊕· · ·⊕Qmrr , where each Qi is a prime polytope
with respect to the direct sum. Then
Γ(P) = Πri=1
(
(Γ(Qi))mi ⋊ Smi
)
.
2 The antiprism
The antiprism of a polygon is a convex polyhedron in ordinary 3-space. Its faces are two
regular n-gons and 2n equilateral triangles. When n = 3, we obtain the regular octahedron.
Otherwise, the antiprism over an n-gon is an Archemidian solid, as their faces are all regular
polygons and its group of symmetries acts transitively on the vertices. In this section, for
each polytope P , we give a construction of a new polytope Ant(P) which generalizes the
construction of the antiprism of a polygon.
308 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
Let P be an n-polytope. To formally define the antiprism of P , Ant(P), we let P be a
symbol, and define
Ant(P) := {(F,G) | F,G ∈ P, F ≤P G} ∪ {P},
where the order is given by
(F,G) ≤ (H,K) if and only if F ≤P H ≤P K ≤P G; (2.1)
(F,G) ≤ P for every F,G ∈ P. (2.2)
Throughout this section, when we say that an ordered pair of elements of P is an element
of Ant(P), we shall be referring to an element of Ant(P) different than P .
Note then that P is the maximum element of Ant(P) and that, if F−1 and Fn denote
the minimum and maximum elements of P , respectively, then (F−1, Fn) is in fact the
minimum element of Ant(P). Moreover, for H,F,G ∈ P , with H ≤P F ≤P G, we have
that (H,F ) ≤ (F, F ) and (F,G) ≤ (F, F ), but the only element of Ant(P) greater than
(F, F ) is P .
Suppose that P has rank n and its rank function is rankP . Define
rankAnt(P)(F,G) := rank(F,G) = n+ rankP(F )− rankP(G), (2.3)
rankAnt(P)(P ) := rank(P ) = n+ 1.
Note that for every (F,G) ∈ Ant(P), we have that 0 ≤ rankP(G) − rankP(F ) ≤ n +
1, implying that rank(F,G) ∈ {−1, . . . , n} and therefore rank: Ant(P) → {−1, . . . ,
n+ 1}.
Moreover, if rank(F,G) = −1, then
n+ rankP(F )− rankP(G) = −1.
This is equivalent to have that
rankP(G) = n+ 1 + rankP(F ).
But rankP(G) ≤ n which implies that F should have rank −1, and thus G has rank n; in
other words, rank(F,G) = −1 if and only if F = F−1 and G = Fn. We can further see
that rank(F,G) = n if and only if F = G. Hence, the facets of Ant(P) are the elements
(F, F ), with F ∈ P .
There are other faces of Ant(P) that are easy to identify. For example, if (F,G) is a
vertex, it should satisfy that rank(F,G) = n+ rankP(F )− rankP(G) = 0. That is,
rankP(G) = n+ rankP(F ).
Again, since rankP(G) ≤ n we have two options: either rankP(G) = n and rankP(F ) =
0, or rankP(G) = n− 1 and rankP(F ) = F−1. This implies that the vertices of Ant(P)
are either of the form (v, Fn), where v is a vertex of P , or of the form (F−1, f), where f is
a facet of P .
Before showing that Ant(P) is a polytope, let us analyze the case when P is 2-
polytope. Let P be a 2-polytope with vertices {v1, . . . , vp} and edges {e1, . . . ep} in such
a way that for every i = 1, . . . , p− 1, vi, vi+1 ≤ ei, and v1, vp ≤ ep. Let m and M be the
I. Gleason and I. Hubard: The antiprism of an abstract polytope 309
minimum and maximum elements of P , respectively. We already know that Ant(P) has a
unique minimum (m,M) and a unique maximum P , that there are 2p vertices, namely:
(v1,M), . . . , (vp,M), (m, e1), . . . , (m, ep),
and that the facets, 2-faces in this case, are of the form (F, F ), where F is any element of
P . Thus there are 2p+ 2 facets. Finally, the 1-faces are the elements
(e1,M), . . . , (ep,M),
(m, v1), . . . , (m, vp),
(v1, ep), (v1, e1), (v2, e1), . . . , (vp, ep),
and there are 4p of them.
We note that the facets (m,m) and (M,M) are p-gons, as their vertices are of the form
(m, ei) and (vi,M), respectively. In contrast, the facets of type (vi, vi) and (ei, ei) are
triangles, as their only vertices are either of the form (vi,M), (m, ei−1), (m, ei) or of the
form (vi,M), (vi+1,M), (m, ei). It is not too difficult now to see that Ant(P) is in fact
isomorphic to the classical antiprism.
Given an abstract polytope P , we should say that Ant(P) is the antiprism of P .
In order to show that the antiprism of any polytope is again a polytope, we shall start
by analyzing the sections of Ant(P). As we noted before, the only elements of rank
n are of the type (F, F ), where F ∈ P . Let us take a look into the sections QF :=
(F, F )/(F−1, Fn) where, as before, F−1, Fn are the minimum and maximum faces of the
n-polytope P , respectively.
Let us fix a face F of P . If (H,G) ∈ QF , then (H,G) ≤ (F, F ), which implies that
H ≤ F ≤ F ≤ G. In other words, F is a face of the section G/H of P . On the other
hand, if H,G ∈ P are such that H ≤ F and F ≤ G, then (H,G) ∈ QF . That means that
the faces of the section QF are in one to one correspondence with the order pairs (H,G)
of elements of P such that H ≤ F ≤ G.
Since P is a polytope, then P−F := F/F−1 and P
+
F := Fn/F are also polytopes. Let
δ : P+F → (P
+
F )
∗ be a duality mapping P+F to its dual. Now, H ∈ P
−
F if and only if
H ≤ F ; on the other hand, F ≤ G if and only if Gδ ∈ (P+F )∗. Consider now the join
product of P−F with (P
+
F )
∗. We have that
ψ : P−F ⋊⋉ (P
+
F )
∗ → QF (2.4)
(H,Gδ) 7→ (H,G)
is a well-defined bijection between P−F ⋊⋉ (P
+
F )
∗ and QF . Furthermore, note that
(H,Gδ) ≤⋊⋉ (H ′, G′δ) if and only if H ≤ H ′ and Gδ ≤ G′δ, which is equivalent to
have H ≤ H ′ and G′ ≤ G. That is, (H,Gδ) ≤⋊⋉ (H ′, G′δ) if and only if H ≤ H ′ ≤
F ≤ G′ ≤ G, which is equivalent to have that (H,G) ≤QF (H ′, G′). Thus, ψ is an
isomorphism between P−F ⋊⋉ (P
+
F )
∗ and QF . This implies that all the facets of Ant(P)
are abstract polytopes. In particular we note that QF−1 ∼= P∗, while QFn ∼= P .
We turn now our attention to the co-faces P/(F,G) of Ant(P). We observe that
P/(F,G) = {(H,K) ∈ Ant(P) | (F,G) ≤Ant(P) (H,K)} ∪ {P}
∼= {(H,K) ∈ P ∗ P | F ≤P H ≤P K ≤P G)} ∪ {P}
∼= {(H,K) ∈ P ∗ P | H,K ∈ G/F,H ≤G/F K} ∪ {P}
∼= Ant(G/F ).
310 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
This says that all the co-faces of Ant(P) are antiprisms of polytopes of smaller rank than
that of P .
Theorem 2.1. Let P be an n-polytope, then Ant(P) is an n+ 1 polytope.
Proof. The function given in (2.3) is the desired rank function, with range {−1, . . . , n+1},
and it is clear from the definition that Ant(P) has a minimum and a maximum face.
We now proceed by induction over n.
Let P = {F−1, F0} be a 0-polytope. Then Ant(P) = {(F−1, F0), (F−1, F−1),
(F0, F0), P}, where (F−1, F0) ≤ (F−1, F−1), (F0, F0) ≤ P . Hence, Ant(P) is iso-
morphic to an edge, that is, Ant(P) is a 1-polytope.
Suppose now that the antiprism of any polytope of rank (n − 1) is a polytope and let
P be an n-polytope. Since the facets of Ant(P) are a join product of polytopes, then they
are polytopes. In particular, every flag of Ant(P), when taking away the maximum face,
can be seen to be contained flag of a facet of Ant(P). Since the flags of the facets have all
n+ 2 elements, every flag of Ant(P) has exactly n+ 3 elements.
The diamond condition is satisfied and all the proper sections of Ant(P) are connected:
this is straightforward to see as a proper section of Ant(P) is contained either in a facet or
in a vertex figure of Ant(P). The facets of Ant(P) are joins of polytopes (hence polytopes)
and the vertex figures are anitprisms over proper sections of P , which by hypothesis of
induction are also polytopes.
We only have to see that Ant(P) itself is connected. Let (F,G), (H,K) be two proper
faces of Ant(P). We divide the analysis in several cases, depending on whether or not
F,G,H and K are proper or improper faces of P . Note that F and G (resp. H and
K) cannot be improper face of P simultaneously, unless they are equal. Without loss of
generality, we may assume that rankP(G) ≤ rankP(K).
• If G,K ̸= Fn, then (F,G), (F−1, G), (F−1, F−1), (F−1,K), (H,K) is a sequence
of incident proper faces of Ant(P).
• If F,H ̸= F−1, then (F,G), (F, Fn), (Fn, Fn), (H,Fn), (H,K) is a sequence of
incident proper faces of Ant(P).
• If K = Fn, F = F−1 and H is a proper face of P , then H ̸= F−1, Fn and G ̸=
Fn. Since P is connected, then there exists a sequence G = G1, G2, . . . , Gh = H
of incident faces of P all of which, except perhaps for G, are proper faces. Then
(F,G) = (F−1, G1), (F−1, G2), . . . , (F−1, Gh) = (F−1, H), (H,H), (H,Fn) =
(H,K) is a sequence of incident proper faces of Ant(P).
• If K = H = Fn, F = F−1 and G is a proper face of P , then (F−1, G), (G,G),
(G,Fn), (Fn, Fn) is a sequence of incident proper faces of Ant(P).
• If K = H = Fn, F = G = F−1, then let J ∈ P be any proper face of P (exists as
we are assuming n > 0). Hence (F−1, F−1), (F−1, J), (J, J), (J, Fn), (Fn, Fn) is
a sequence of incident proper faces of Ant(P).
Hence Ant(P) is connected and therefore it is a polytope.
Note that given a polytope P and its dual P∗, there is a duality δ : P → P∗. We know
that δ is a bijection that reverses the order, and hence every element of P∗ can be written
as Fδ, where F is a face of P . Hence, there is a natural bijection between the faces of
I. Gleason and I. Hubard: The antiprism of an abstract polytope 311
the antirpism of P and the faces of the antiprism of P∗. In fact, we have the following
proposition.
Proposition 2.2. For any polytope P , Ant(P) ∼= Ant(P∗), where P∗ denotes the dual
of P .
Proof. Let P and P ∗ be the maximum elements of Ant(P) and Ant(P∗), respectively,
and let δ : P → P∗ be a duality. Let ψ : Ant(P) → Ant(P∗) be given by:
(F,G) 7→ (Gδ, Fδ),
P 7→ P ∗.
Then clearly ψ is a well-defined bijection between Ant(P) and Ant(P∗). Furthermore
(F,G) ≤Ant(P) (H,K) if and only if F ≤P H ≤P K ≤P G if and only if Gδ ≤P∗
Kδ ≤P∗ Hδ ≤P∗ Fδ if and only if (Gδ, Fδ) ≤P∗ (Kδ,Hδ) which is equivalent to
(F,G)ψ ≤P∗ (H,K)ψ. Since it is now straightforward to see that δ−1 also induces a
bijection that preserves the order and is the inverse of ψ. This settles the proposition.
2.1 The flags of a polytope and the flags of its antiprism
In this section we study the relation between the flags of Ant(P) and the flags of P .
Let P be an n-polytope (with maximum element Fn and minimum element F−1) and
consider V to be the set of all ordered (n + 1)-tuples with entries 0 and 1. We are going
to see that there is a bijection between F(Ant(P)) and F(P) × V . For this, consider a
flag of Ant(P), {A−1, A0, . . . An+1}, where rankAi = i. Then An+1 = P and for each
i = −1, 0 . . . n, there exist F i, Gi ∈ P such that Ai = (F i, Gi). It is straightforward to
see that F−1 = F−1, G−1 = Fn and Fn = Gn := F , for some F ∈ P . Furthermore,
observe that
F−1 ≤ F 0 ≤ F 1 ≤ · · · ≤ Fn = F = Gn ≤ Gn−1 ≤ · · · ≤ G0 ≤ G−1 = Fn, (2.5)
is a sequence of faces of P in which, of course, many of the elements might repeat. For
example, the sequence could be such that F 0 = F 1 = F 2 = · · · = Fn = F−1.
On one hand, note that for a given i ∈ {0, . . . n}, we have that either
rank(F i) = rank(F i+1) and rank(Gi) = rank(Gi+1) + 1
or
rank(F i) + 1 = rank(F i+1) and rank(Gi) = rank(Gi+1).
In particular, either rank(F 0) = −1 and rank(G0) = n − 1 or rank(F 0) = 0 and
rank(G0) = n. Hence, we can regard the sequence in (2.5) as a sequence of incident
faces of P that has exactly one element of each rank. That is, a flag of P . In other words,
each flag of Ant(P) induces a flag of P in a natural way.
On the other hand, the sequence in (2.5) also defines an element of V in the following
way. For each i ∈ {0, . . . , n}, let ai = 0 if rank(F i−1) = rank(F i) and ai = 1 otherwise.
It should be clear that (a0, . . . an) is an element of V .
The above assignment is a bijection. To see this, take Φ ∈ F(P) and v ∈ V . Denote
by Φi the i-face of Φ and by vi the i-th element of v, i.e. v = (v0, v1, . . . , vn). We define
the flag {A−1, A0, . . . An+1} of Ant(P), where eachAi = (F i, Gi), in the following way.
312 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
First, An+1 = P and A−1 = (F−1, Fn) = (Φ−1,Φn). Now, we define inductively the
elements F i and Gi. Suppose F i−1 is defined as the j-face Φj , then we define F i :=
Φj+vi . (For example, if v0 = 0, then F
0 = Φ−1+v0 = Φ−1 = F−1, and if v0 = 1,
then F 0 = Φ−1+v0 = Φ0.) Similarly, we first define G
n := Fn and, inductively, suppose
that Gi+1 is defined as the k-face Φk, then we define Gi := Φk+1−vi+1 . Thus, | {vj ∈
v | vj = 1} |= m + 1, for some −1 ≤ m ≤ n and hence Gn = Fn = Φm and
| {vj ∈ v | vj = 0} |= n−m. This implies that
G−1 = Φm+(1−vn)+(1−vn−1+···+(1−v0)) = Φm+(n−m) = Φn.
It should not be difficult to see that this assignment of a flag of Ant(P), given a pair
(Φ, v) ∈ F(P)×V is inverse to the above description, where each flag of Ant(P) induces
a flag of P and an element of V . We have therefore established that
Lemma 2.3. Let P be an n-polytope. Then the flags of Ant(P) are in one-to-one corre-
spondence with the set F(P)× V , where F(P) denotes the set of flags of P and V the set
of all ordered (n+ 1)-tuples with entries 0 and 1.
3 Products and the antiprism
In the next section we will study the automorphism group of an antiprism. We shall see that
computing it for polytopes that are prime with respect to the join product is straighforward.
To completely determine the automorphism group of any antiprism, we need some of the
results given in this section.
All our results here deal with the interaction of the join product and the direct sum with
the anitprism.
Proposition 3.1. Let P and Q be two polytopes. Then Ant(P ⋊⋉ Q) ∼= Ant(P)⊕Ant(Q).
Proof. Let ψ : Ant(P ⋊⋉ Q) → Ant(P)⊕Ant(Q) be such that(
(F,G), (H,K))ψ =
(
(F,H), (G,K)
)
and, if P, PP and PQ are the maximum elements of Ant(P ⋊⋉ Q),Ant(P) and Ant(Q),
respectively, then Pψ = (PP , PQ). We shall show that ψ is an isomorphism.
First note that
(
(F,G), (H,K)
)
∈ Ant(P ⋊⋉ Q) implies that (F,G), (H,K) ∈ P ⋊⋉ Q
and that (F,G) ≤P⋊⋉Q (H,K). Hence, we have that F,H ∈ P with F ≤P H , and that
G,K ∈ Q with G ≤Q K; that is, (F,H) ∈ Ant(P) and (G,K) ∈ Ant(Q). Moreover,
(F,G) is not the maximum element of Ant(P), and (H,K) is not the maximum element of
Ant(Q), which implies that
(
(F,H), (G,K)
)
∈ Ant(P)⊕Ant(Q). Furthermore, observe
that different elements of Ant(P ⋊⋉ Q) go to different elements of Ant(P)⊕Ant(Q) under
ψ and therefore ψ is a well-defined function from Ant(P ⋊⋉ Q) to Ant(P)⊕Ant(Q).
Similarly, let ϕ : Ant(P)⊕Ant(Q) → Ant(P ⋊⋉ Q) be such that(
(F,H), (G,K)
)
7→
(
(F,G), (H,K)
)
.
A similar argument as the one above shows that ϕ is also a well-defined function. Note that
both ψϕ and ϕψ are the identity map, which implies that both functions are bijections, and
one is the inverse of the other.
I. Gleason and I. Hubard: The antiprism of an abstract polytope 313
We need to show that these two functions preserve the orders. Let
(
(F0, G0), (H0,K0)
)
,(
(F1, G1), (H1,K1)
)
∈ Ant(P ⋊⋉ Q), then(
(F0, G0), (H0,K0)
)
≤Ant(P⋊⋉Q)
(
(F1, G1), (H1,K1)
)
⇔ (F0, G0) ≤P⋊⋉Q (F1, G1) ≤P⋊⋉Q (H1,K1) ≤P⋊⋉Q (H0,K0)
⇔ F0 ≤P F1 ≤P H1 ≤P H0 and G0 ≤Q G1 ≤Q K1 ≤Q K0
⇔ (F0, H0) ≤Ant(P) (F1, H1) and (G0,K0) ≤Ant(Q) (G1,K1)
⇔
(
(F0, H0), (G0,K0)
)
≤Ant(P)⊕Ant(Q)
(
(F1, H1), (G1,K1)
)
.
Therefore both ψ and ϕ preserve the orders and hence ψ is an isomorphism.
Lemma 3.2. If P is a prime polytope with respect to the join product, then Ant(P) is a
prime polytope with respect to the direct sum.
Proof. Suppose otherwise. Then there exists a polytope P that is prime with respect to
the join product, but such that Ant(P) is not prime with respect to the direct sum. Let
Ant(P) = Q⊕K, where Q and K are polytopes of rank at least 1.
Note that Ant(P) contains a facet that is isomorphic to P . In fact, if Fn denotes the
maximum element of P , then (Fn, Fn) ∈ Ant(P) has rank n and if (F,G) ≤ (Fn, Fn),
then G = Fn (since F ≤P Fn ≤P Fn ≤P G). That means that the section
(Fn, Fn)/(F−1, Fn) of Ant(P) is isomorphic to P .
But by Lemma 1.1, a facet of the direct product Q ⊕ K is isomorphic to a non-trivial
join product. Hence P is not prime with respect to the join product, which contradicts our
hypothesis.
4 Automorphism groups
We now turn our attention to the study of the automorphism group of the antiprism of P .
It is not difficult to see that every automorphism of P induces an automorphism of
Ant(P). In fact, given γ ∈ Γ(P) the mapping γ̂ : Ant(P) → Ant(P) given by (F,G)γ̂ :=
(Fγ,Gγ), for (F,G) ∈ Ant(P), and P γ̂ := P is clearly an automorphism of Ant(P).
Similarly, if P is a self-dual polytope and δ is a duality of P , then δ̂ : (F,G) 7→ (Gδ, Fδ)
(and P δ̂ = P ) is also an automorphism of Ant(P). In other words, we have the following
lemma. Keep in mind that we have defined the extended group of a non-self-dual polytope
simply as its automorphism group.
Lemma 4.1. Let P be a polytope and let Γ̂(P) denote its extended group. Then, Γ̂(P) is
(isomorphic to) a subgroup of G(Ant(P)).
It is not difficult to see that if ψ : F(Ant(P)) → F(P) × V is the bijection from
Lemma 2.3, γ ∈ Γ̂(P) and γ̃ is the automorphism of Ant(P) induced by γ, then for every
flag Φ ∈ F(P) and every (n+1)-tuple v ∈ V , we have that (Φ, v)ψ−1γ̃ψ = (Φγ, v). This
implies that if P is a self-dual polytope, dualities of P induce automorphisms of Ant(P).
The above observation, together with Lemmas 4.1 and 2.3 imply the following result.
Proposition 4.2. Let P an n-polytope and let Ant(P) be its antiprism. If P is a k-orbit
polytope, and Ant(P) is an m-orbit polytope, then:
• if P is self-dual, then m ≤ k · 2n,
314 Ars Math. Contemp. 22 (2022) #P2.08 / 305–315
• if P is not self-dual, then m ≤ k · 2n+1.
Observe that, by the isomorphism given in (2.4), the facets of Ant(P) can be seen as
the join product of sections of P . This means that, maybe with the exception of the facets
(F−1, F−1) ∼= P∗ and (Fn, Fn) ∼= P , the facets of Ant(P) are not prime with respect to
the join product. Whenever P is a prime polytope with respect to the join product, we can
obtain a lot of information about Γ(Ant(P).
Proposition 4.3. Let P be a prime polytope with respect to the join product. Then,
Γ(Ant(P)) ∼= Γ̂(P).
Proof. By Lemma 4.1 we only need to show that any automorphism of Ant(P) is in fact
induced by either an automorphism or a duality (if P is self-dual) of P . In this proof we
abuse notation and refer to a polytope that is prime with respect to the join product simply
as a prime polytope.
As pointed out above, when P is a prime polytope, the only two facets of Ant(P)
that are also prime are (F−1, F−1) and (Fn, Fn). This means that any automorphism α ∈
Γ(Ant(P)) either fixes both such faces or interchanges them (as they cannot be permuted
with any other, or they would not be prime). It is then easy to see that if α fixes them, then
it induces an automorphism of P and that if interchanges them, then P is self-dual and α
induces a duality.
Propositions 4.2 and 4.3 immediately imply the following result.
Corollary 4.4. Let P an n-polytope that is prime with respect to the join product and let
Ant(P) be its antiprism. If P is a k-orbit polytope, and Ant(P) is an m-orbit polytope,
then:
• if P is self-dual, then m = k · 2n,
• if P is not self-dual, then m = k · 2n+1.
Lemma 3.2, together with Propositions 3.1 and 4.3 and Theorem 1.2, give us all the
necessary tools to compute the automorphism of the antiprism of any polytope.
Theorem 4.5. Let P = Qm11 ⋊⋉ Q
m2
2 ⋊⋉ · · · ⋊⋉ Qmrr , where each Qi is a prime polytope
with respect to the join product. Then
Γ(Ant(P)) = Πri=1
(
(Γ̂(Qi))mi ⋊ Smi
)
.
ORCID iDs
Isabel Hubard https://orcid.org/0000-0002-0960-3671
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