Algebraic Theory of SKEW-MORPHISMS Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 November 4, 2014 Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
(Cayley) Regular Action Cayley: A finite group G is isomorphic to a subgroup of S G : σ : G → S G ; σ g ( h ) = gh , for all h ∈ G Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
(Cayley) Regular Action Cayley: A finite group G is isomorphic to a subgroup of S G : σ : G → S G ; σ g ( h ) = gh , for all h ∈ G G ∼ = G L left regular representation Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Given a left-regular representation of a finite group G L , Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Given a left-regular representation of a finite group G L , consider � G � the induced action on unordered pairs , and let E be a union of 2 � G � orbits of G L on . 2 Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Given a left-regular representation of a finite group G L , consider � G � the induced action on unordered pairs , and let E be a union of 2 � G � orbits of G L on . 2 Definition The graph Γ = ( G , E ) is called a Cayley graph , and G L ≤ Aut (Γ) Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Given a left-regular representation of a finite group G L , consider � G � the induced action on unordered pairs , and let E be a union of 2 � G � orbits of G L on . 2 Definition The graph Γ = ( G , E ) is called a Cayley graph , and G L ≤ Aut (Γ) Aut (Γ) may be bigger than G L : Aut (Γ) = G L · Stab Aut (Γ) ( u ) Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Equivalent definition: Let G be a finite group and X be a subset of G that does not contain the identity 1 G and is closed under taking inverses: The Cayley graph Γ = C ( G , X ) is the graph with vertex set V = G and edge set E = { { g , gx } | g ∈ G , x ∈ X } . Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Cayley Graphs Equivalent definition: Let G be a finite group and X be a subset of G that does not contain the identity 1 G and is closed under taking inverses: The Cayley graph Γ = C ( G , X ) is the graph with vertex set V = G and edge set E = { { g , gx } | g ∈ G , x ∈ X } . Stab Aut (Γ) (1 G ) preserves X . Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
◮ if G is abelian and X contains at least one non-involution, then ϕ : x → x − 1 is a (non-trivial) group automorphism of G that preserves X (since X is closed under inverses): ϕ ∈ Stab Aut (Γ) (1 G ) Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
◮ if G is abelian and X contains at least one non-involution, then ϕ : x → x − 1 is a (non-trivial) group automorphism of G that preserves X (since X is closed under inverses): ϕ ∈ Stab Aut (Γ) (1 G ) ◮ Cayley graphs C ( G , X ) for which Stab Aut (Γ) (1 G ) is non-trivial and G L is normal in Aut (Γ) are called normal Cayley graphs Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Graphical Regular Representation Definition A Cayley graph Γ = ( G , X ) is a graphical regular representation of the group G iff Aut (Γ) = G L Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Graphical Regular Representation Definition A Cayley graph Γ = ( G , X ) is a graphical regular representation of the group G iff Aut (Γ) = G L iff Stab Aut (Γ) (1 G ) = � id � . Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Graphical Regular Representation Definition A Cayley graph Γ = ( G , X ) is a graphical regular representation of the group G iff Aut (Γ) = G L iff Stab Aut (Γ) (1 G ) = � id � . Abelian groups of exponent greater than 2 do not admit GRR’s. Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Graphical Regular Representation Theorem (Godsil, Watkins, ...) Let G be a finite group that does not have a GRR, i.e., a finite group that does not admit a regular representation as the full automorphism group of a graph. Then G is an abelian group of exponent greater than 2 or G is a generalized dicyclic group a , x | a 2 n = 1 , x 2 = a n , x − 1 ax = a − 1 � � or G is isomorphic to one of the 13 groups : Z 2 2 , Z 3 2 , Z 4 2 , D 3 , D 4 , D 5 , A 4 , Q × Z 3 , Q × Z 4 , a , b , c | a 2 = b 2 = c 2 = 1 , abc = bca = cab � � , a , b | a 8 = b 2 = 1 , b − 1 ab = a 5 � � , a , b , c | a 3 = b 3 = c 2 = 1 , ab = ba , ( ac ) 2 = ( bc ) 2 = 1 � � , a , b , c | a 3 = b 3 = c 3 = 1 , ac = ca , bc = cb , b − 1 ab = ac � � . Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Proportion of GRR’s among Cayley graphs Conjecture: Almost all Cayley graphs are GRR’s. Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Proportion of GRR’s among Cayley graphs Conjecture: Almost all Cayley graphs are GRR’s. # of GRR’s of order ≤ n lim # of Cayley graphs of order ≤ n = 1 n →∞ Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Proportion of GRR’s among Cayley graphs Conjecture: Almost all Cayley graphs are GRR’s. # of GRR’s of order ≤ n lim # of Cayley graphs of order ≤ n = 1 n →∞ True for nilpotent groups. Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Orientable Maps ◮ an orientable map M is a 2-cell embedding of a graph in an orientable surface; an embedding in which every face is homeomorphic to the open disc Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Orientable Maps ◮ an orientable map M is a 2-cell embedding of a graph in an orientable surface; an embedding in which every face is homeomorphic to the open disc ◮ given a finite connected graph Γ = ( V , E ), an orientable embedding of Γ is determined by choosing a cyclic local permutation ρ v of arcs emanating from each vertex: ρ ∈ S D (Γ) : ρ = ∪ ρ v , v ∈ V Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Orientable Maps ◮ an orientable map M is a 2-cell embedding of a graph in an orientable surface; an embedding in which every face is homeomorphic to the open disc ◮ given a finite connected graph Γ = ( V , E ), an orientable embedding of Γ is determined by choosing a cyclic local permutation ρ v of arcs emanating from each vertex: ρ ∈ S D (Γ) : ρ = ∪ ρ v , v ∈ V ◮ if Γ = C ( G , X ) is a Cayley graph, ρ g can be thought of as a cyclic permutation of X : ρ g (( g , x )) = ( g , ρ g ( x )) Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Classical Examples - The Five Platonic Solids Four of the five platonic solids are embeddings of Cayley graphs Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Regular Maps Definition A map automorphism of an orientable map M is a permutation of the darts of the map that preserves adjacency and faces: ϕλ = λϕ, ϕρ = ρϕ Robert Jajcay Comenius University robert.jajcay@fmph.uniba.sk ............................. Sobolev Institute 2014 Algebraic Theory of SKEW-MORPHISMS
Recommend
More recommend
