Flag graphs and symmetry type graphs

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Flag graphs and symmetry type graphs

Mar´ıa del R´ıo Francos

IMate, UNAM

SCDO’16, Queenstown, NZ.

Mar´ıa del R´ıo Francos (IMate, UNAM) Flag graphs and symmetry type graphs SCDO’16, Queenstown, NZ. 1 / 27

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Content

Aim.

The aim of this work is to give a classification on the possible different symmetry type of maniplexes.

I. Maniplexes and symmetry type graphs.

II. Map operations.

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I(a). Maniplexes

Maniplexes were first introduced by S. Wilson (2012), aiming to unify the notion of maps and polytopes.

Given a set of flagsF(M) and a sequence (s₀,s₁, . . . ,s_n₋₁), where each s_i partitions F(M) into sets of size two and the partitions described bys_i and sj are disjoint wheni 6=j.

A maniplex Mof rankn−1 (or (n−1)-maniplex) is defined by a

connected graphGM which vertex set is F(M) and with edges of colouri corresponding to the matching s_i, to which we refer as the flag graphof the maniplex M.

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I(a). Maniplexes

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I(a). Maniplexes

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Examples of maniplexes

0-maniplex.

Graph with two vertices joined by an edge of colour 0.

1-maniplex.

It is associated to anl-gon, which graph contains 2l vertices joined by a perfect matching of colours 0 and 1 and each of sizel.

2-maniplex.

Can be considered as a map, as Lins and Vince defined a map (1982-1983), by a trivalent edge coloured graph.

0

0 0

0 1 1

1 1 0

0

0 0

0

0 0

1 1

1

1 1 1

1

1 1

1 2

2 2

2

2 2

Thus, maniplexes generalize the notion of maps to higher rank.

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Examples of maniplexes

0-maniplex.

Graph with two vertices joined by an edge of colour 0.

1-maniplex.

It is associated to anl-gon, which graph contains 2l vertices joined by a perfect matching of colours 0 and 1 and each of sizel.

2-maniplex.

Can be considered as a map, as Lins and Vince defined a map (1982-1983), by a trivalent edge coloured graph.

0

0 0

0 1 1

1 1 0

0

0 0

0

0 0

1 1

1

1 1 1

1

1 1

1 2

2 2

2

2 2

Thus, maniplexes generalize the notion of maps to higher rank.

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Monodromy (or connection) group of M

To each (n−1)-maniplex Mwe can associate a subgroup of the permutation group Sym(F(M)),

Mon(M) :=hs₀,s₁, . . . ,s_n₋₁i

known as themonodromy (or connection) groupof the maniplex M.

The action ofs0,s1, . . . ,sn−1 on any flag Φ∈ F(M) is defined by Φ·s_i = Φⁱ; i = 0,1, . . . ,n−1.

And satisfy the following

(i) Alls₀,s₁, . . . ,s_n−1 are fixed-point free involutions.

(ii) sisj =sjsi andsisj is fixed-point free, whenever |i −j| ≥2. (iii) The action ofMon(M) on F(M) is transitive.

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Monodromy (or connection) group of M

Mon(M) :=hs₀,s₁, . . . ,s_n₋₁i

known as themonodromy (or connection) groupof the maniplex M. The action ofs0,s1, . . . ,sn−1 on any flag Φ∈ F(M) is defined by

Φ·s_i = Φⁱ; i = 0,1, . . . ,n−1.

(i) Alls₀,s₁, . . . ,s_n−1 are fixed-point free involutions.

(ii) sisj =sjsi andsisj is fixed-point free, whenever |i −j| ≥2. (iii) The action ofMon(M) on F(M) is transitive.

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Monodromy (or connection) group of M

Mon(M) :=hs₀,s₁, . . . ,s_n₋₁i

known as themonodromy (or connection) groupof the maniplex M. The action ofs0,s1, . . . ,sn−1 on any flag Φ∈ F(M) is defined by

Φ·s_i = Φⁱ; i = 0,1, . . . ,n−1.

(i) Alls₀,s₁, . . . ,s_n₋₁ are fixed-point free involutions.

(ii) sisj =sjsi andsisj is fixed-point free, whenever |i −j| ≥2.

(iii) The action ofMon(M) on F(M) is transitive.

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Faces of rank i = 0, 1, . . . , n − 1 of M

The set of i-facesof an (n−1)-maniplex corresponds to the orbit of the flags in F(M) under the action of the group generated by the set

F_i :={s_j|i 6=j}.

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Automorphism group of M , Aut( M )

Every automorphism α ofMinduces a bijection on the flags.

The action ofAut(M) on the set of flags is semi-regular.

The action ofAut(M) on the set of flags is transitive only if Mis regular.

Aut(M) partitions the setF(M) into orbits of the same size.

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Automorphism group of M , Aut( M )

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Automorphism group of M , Aut( M )

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Automorphism group of M , Aut( M )

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Automorphism group of M , Aut( M )

Aut(M) is isomorphic to the edge-colour preserving automorphism group ofGM.

The action of the elements in Aut(M) commutes with the elements of Mon(M).

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Automorphism group of M , Aut( M )

Aut(M) is isomorphic to the edge-colour preserving automorphism group ofGM.

The action of the elements in Aut(M) commutes with the elements of Mon(M).

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k -orbit maniplex

We say that the maniplexMis a k-orbit maniplex whenever the automorphism group Aut(M) has exactlyk orbits on F(M).

A1-orbit maniplex is known asregular (orreflexible).

A2-orbit maniplex, with adjacent flags belonging to different orbits, is known as a chiralmaniplex.

It can be seen that there are 2ⁿ−1different possible types of 2-orbit (n−1)-maniplexes.

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k -orbit maniplex

A1-orbitmaniplex is known as regular (orreflexible).

A2-orbit maniplex, with adjacent flags belonging to different orbits, is known as a chiralmaniplex.

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k -orbit maniplex

A2-orbitmaniplex, with adjacent flags belonging to different orbits, is known as a chiralmaniplex.

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k -orbit maniplex

A2-orbitmaniplex, with adjacent flags belonging to different orbits, is known as a chiralmaniplex.

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I(b). Symmetry type graph of M , T ( M )

Definition.

The symmetry type graphT(M) of a maniplex Mis a quotient graph of the flag graph GM obtained from the action of the groupAut(M) on the flags of M.

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Symmetry type graph of M , T ( M )

Thus, the symmetry type graph of a k-orbit map has k-vertices

Given two flag orbitsOΦ andOΨ, as vertices ofT(M), there is an edge of colour i = 0,1, . . . ,n−1 between them if and only if there exists flags Φ⁰ ∈ OΦ and Ψ⁰∈ OΨ such that Φ⁰ and Ψ⁰ arei-adjacent inM.

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Counting symmetry types

The number of types of k-orbit maniplexes depends on the number of n-valent pre-graphs on k vertices that can be properly edge coloured with n colours and that the connected components of the 2-factor with colours i and j, with |i−j| ≥2 are always as the following.

i

j j

j

j j

i j

i i

i

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The symmetry type graph of areflexible maniplexconsist of one vertex and n semi-edges.

There are 2ⁿ−1 different possible symmetry type graphs on 2 vertices.

There are 2n−3 different possible symmetry type graphs on 3 vertices.

n−1 0 1

i j

I⊂ {0,1, . . . , n−1}, J={0,1, . . . , n−1} \I

I J I

J J

J

j−1 j

J J

J

j+ 1 j

J J

j j+ 1 j−1

J

j+ 1 j−1

J={0,1, . . . , n−1} \ {j−1, j, j+ 1} j+ 1

j−1

j−1 j+ 1 j+ 1

j

j−1 j

j+ 1 j−1

j

u v w u v w

u v w

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Face transitivity

Definition.

An (n−1)-maniplex Mis i -face-transitive ifAut(M) is transitive on the faces of ranki = 0,1, . . . ,n−1.

Definition.

An (n−1)-maniplex Mis fully-face-transitiveif it isi-face-transitive for every i = 0,1, . . . ,n−1.

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Face transitivity

Definition.

An (n−1)-maniplex Mis i -face-transitive ifAut(M) is transitive on the faces of ranki = 0,1, . . . ,n−1.

Definition.

An (n−1)-maniplex Mis fully-face-transitiveif it isi-face-transitive for every i = 0,1, . . . ,n−1.

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Highly symmetric maniplexes

Given the symmetry type graph of a maniplex one can read from the appropriate coloured subgraphs the different types of face-transitivities that the maniplex has.

Theorem. (Number of face-orbits of M)

Let Mbe an(n−1)-maniplex with symmetry type graph T(M). Then, the number of connected components in the (n−1)-factor of T(M) of colours {0,1, . . . ,n−1} \ {i}, determine the number of orbits of the i -faces of M, where i ∈ {0,1, . . . ,n−1}.

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Edge-transitive maps

1

2₁₂ 2₀₁

2 2₀ 2₂ 2₁

4G_d

4F 4G

4H

4Gp

4H_d 4H_p

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Fully-transitivity on k -orbit maniplexes (k = 2, 3, 4)

Hubard showed that there are2ⁿ−n−3classes of fully-transitive 2-orbit (n−1)-maniplexes.

We showed that 3-orbitmaniplexes arenever fully-transitive, but they are i-face-transitive.

Also, that if a4-orbitmaniplex is not fully-transitivethen it is i-face-transitive for alli but at most three ranks.

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Generators of Aut( M ) given T ( M )

Let Mbe a k-orbit (n−1)-maniplex and let T(M) its symmetry type graph.

Suppose that v1,e1,v2,e2. . . ,eq−1,vq is a distinguished walk that visits every vertex ofT(M), with the edgee_i having colour a_i, for eachi = 1, . . .q−1.

v1

v2 v5

v4

v3, v6

v7

v9

v10

v11

v8, v12

a1

a2 a3

a4

a5

a6

a7

a8

a11

a10

a9

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Generators of Aut( M ) given T ( M )

Let Si ⊂ {0, . . . ,n−1} be such thatvi has a semi-edge of colours if and only ifs ∈S_i.

Let Bi,j ⊂ {0, . . . ,n−1} be the set of colours of the edges between the vertices v_i andv_j (withi <j) that are not in the distinguished walk

v1

v2 v5

v4

v3, v6

v7

v9

v10

v11

v8, v12

a1

a2 a3

a4

a5

a6

a7

a8

a11

a10

a9

b

s

Let Φ∈ F(M) be a base flag ofMsuch that Φ projects to v1 in T(M).

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Generators of Aut( M ) given T ( M )

Theorem.

The automorphism group of Mis generated by the union of the sets

{αa1,a2,...,ai,s,ai,ai−1,...,a1 |i = 1, . . . ,k−1,s ∈Si}, and

{α_a₁_,a₂_,...,a_i_,b,a_j_,a_j−1_,...,a₁ |i,j ∈ {1, . . . ,k−1},i <j,b∈B_i,j}.

v1

v2 v5

v4

v3, v6

v7

v9

v10

v11

v8, v12

a1

a2 a3

a4

a5

a6

a7

a8

a11

a10

a9

b

s

αa1,a2,a3,s,a3,a2,a1 αa1,a2,b,a7,a6,a5,a4,a3,a2,a1

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II. Map operations

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Theorem. [Orbani´c, Pellicer, Weiss]

Let Mbe a k-orbit map. Then the medial mapMe(M) is a k-orbit or a 2k-orbit map, depending on whether or notMis a self-dual map.

Theorem. [Orbani´c, Pellicer, Weiss]

Let Mbe a k-orbit map. Then the truncation map Tr(M)is a k-orbit,

3k

2-orbit or a 3k-orbit map.

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Theorem.

Each of the 14 edge-transitive symmetry type graphs is the symmetry type graph of a medial map.

Proposition.

Let Mbe a k-orbit map. Then Me(Me(M))is a k-orbit map ifMis a map on the torus of type {4,4}, or is a map on the Klein Bottle of type {4,4}|m,n|, where n is odd.

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Theorem.

Let Mbe a k-orbit map and Cham_t(M) the t-times chamfering map of Mhaving s flag-orbits. Then one of the following holds.

1 s = 4^tk,2^tk or k.

2 If s 6= 4^tk, thenχ(M) = 0 (Mis on the torus or on the Klein bottle) andMis of type {6,3}.

3 IfMis a the torus of type {6,3}then s =k and k = 1,2,3,4.

4 IfMis on the Klein bottle of type{6,3} then s= 2^tk and3|k.

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Conclusion

We extended the classification of all possible symmetry types of k-orbit 2-maniplexes

self-dual, properly and improperly, k-orbit maps with k ≤7.

with the operations medial and truncation on maps, up tok ≤6.

Also, we determined all possible symmetry types of maps that result from other maps after applying the chamfering operation and give the number of possible flag-orbits that has the chamfering map of a k-orbit map.

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Thank you

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Remarks

In order to characterize the symmetry types of k-orbit maniplexes, as well it was done in this thesis for 2-maniplexes, we lead to the open problem of study different operations on maniplexes and the symmetry types of maniplexes that are obtained from applying such operations on a maniplex.