Graph Dynamical Systems and Coxeter Groups Matthew Macauley - PowerPoint PPT Presentation

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Graph Dynamical Systems and Coxeter Groups Matthew Macauley Department of Mathematical Sciences Clemson University ADM Seminar Clemson University January 15th, 2009

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References 1 Acyclic Orientations Equivalences Enumeration of equivalence classes Neutral networks for equivalence Poset structure of κ -equivalence classes Enumeration problems 2 Sequential Dynamical Systems Functional equivalence Dynamical equivalence Cycle equivalence Aut( Y )-actions 3 Coxeter Groups Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups 4 Summary Connections to other areas of mathematics Future research

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems A recursion for enumerating acyclic orientations Let Y be an undirected graph. For e ∈ e [ Y ], let Y ′ e and Y ′′ denote the graphs formed e from Y by deleting and contracting e , respectively. ◮ For any e ∈ e [ Y ], there is a bijection → Acyc( Y ′ e ) ∪ Acyc( Y ′′ β e : Acyc( Y ) − e ) defined by 8 O ρ ( e ) O Y ′ , �∈ Acyc( Y ) , > Y < O ρ ( e ) O Y �− → O Y ′ , ∈ Acyc( Y ) and O Y ( e ) = ( v , w ) , Y > O ρ ( e ) : O Y ′′ , ∈ Acyc( Y ) and O Y ( e ) = ( w , v ) . Y ◮ Thus, the function α ( Y ) := | Acyc( Y ) | satisfies the recurrence α ( Y ) = α ( Y ′ e ) + α ( Y ′′ e ) .

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Acyclic orientations as posets Let S Y be the set of total orderings (or permutations) of v [ Y ]. An element O Y ∈ Acyc( Y ) defines a partial ordering on the vertex set v [ Y ] by i ≤ O Y j if there is a directed path from i to j in O Y . This induces a well-defined map f α ( π ) = O π f α : S Y − → Acyc( Y ) , Y , where π is a linear extension of O π Y . f α 1 12435 14235 41235 5 2 12453 14253 41253 4 3 14523 41523 Figure: An element of Acyc(Circ 5 ), and its 8 linear extensions arranged in a graph. Each edge { i , j } is oriented ( i , j ) iff i appears before j in the extensions. ◮ We get an equivalence relation ∼ α on S Y , and a bijection f ∗ f ∗ Y ([ π ] α ) = O π Y : S Y / ∼ α − → Acyc( Y ) , Y .

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Equivalences on Acyc( Y ) The cyclic group C n = � σ � acts on the set S Y of orderings of v [ Y ]: σ π 1 π 2 · · · π n − 1 π n �− → π 2 · · · π n − 1 π n π 1 . Via the function f α : S Y → Acyc( Y ), this corresponds to converting a source of O Y into a sink. ◮ This source-to-sink operation (or a “click”) puts an equivalence relation on Acyc( Y ), denoted ∼ κ . Figure: Source-to-sink operations

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems The dihedral group D n = � σ, ρ � acts on the set S Y of orderings of v [ Y ]: σ ρ π 1 π 2 · · · π n − 1 π n �− → π 2 · · · π n − 1 π n π 1 , π 1 π 2 · · · π n − 1 π n �− → π n π n − 1 · · · π 2 π 1 . ◮ Via f ∗ α , the reflection ρ reverses all edge orientations of O Y . Source-to-sink oper- ations with reversals together put a coarser equivalence relation on Acyc( Y ), denoted ∼ δ . Define the functions: κ ( Y ) = | Acyc( Y ) / ∼ κ | , δ ( Y ) = | Acyc( Y ) / ∼ δ | . ◮ The group Aut( Y ) acts on Acyc( Y ), Acyc( Y ) / ∼ κ , and Acyc( Y ) / ∼ δ , yielding equivalence relations ∼ ¯ α , ∼ ¯ κ , and ∼ ¯ δ . ◮ In conclusion, via f ∗ α , there are equivalence on Acyc( Y ), coming from actions of C n , D n , and Aut( Y ) on S Y : C n D n Acyc( Y ) Acyc( Y ) / ∼ κ Acyc( Y ) / ∼ δ Aut( Y ) Acyc( Y ) / ∼ ¯ Acyc( Y ) / ∼ ¯ Acyc( Y ) / ∼ ¯ α κ δ

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Enumeration problems α ( Y ) := | Acyc( Y ) | = T Y (2 , 0) satisfies α ( Y ) = α ( Y / e ) + α ( Y \ e ) for any edge e 1 X α ( Y ) := | Acyc( Y ) / ∼ ¯ ¯ α | = α ( � γ � \ Y ) Aut( Y ) γ ∈ Aut( Y ) κ ( Y ) := | Acyc( Y ) / ∼ κ | = T Y (1 , 0) satisfies κ ( Y ) = κ ( Y / e ) + κ ( Y \ e ) for any cycle edge e δ ( Y ) := | Acyc( Y ) / ∼ δ | = ⌈ κ ( Y ) / 2 ⌉ . 1 X ¯ κ ( Y ) := | Acyc( Y ) / ∼ ¯ κ | = | Fix( γ ) | Aut( Y ) γ ∈ Aut( Y ) But what is | Fix( γ ) | ???

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Update graphs Definition The update graph U ( Y ) has vertex set S Y . The edge { π, σ } is present iff: π and σ differ by exactly an adjacent transposition ( i , i + 1), { π i , π i +1 } �∈ e [ Y ]. Example . Let Y = Circ 4 , the circular graph on 4 vertices. 1234 2341 1243 1423 1324 1342 3412 4123 3241 3421 3124 3142 1432 2143 2134 2314 2413 2431 3214 4321 4132 4312 4213 4231 Figure: The update graph U (Circ 4 ). Each connected component corresponds with a unique element of Acyc(Circ 4 ) .

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Constructing U ( Y ) from hyperplane arrangements The n-permutahedron Π n is the convex hull of all permutations of the points (1 , 2 , . . . , n ) ∈ R n . It is an ( n − 1)-dimensional polytope. The vertices and edges of Π n can be labeled as follows: Two vertices are adjacent if they differ by swapping two coordinates in adjacent position. An edge is labeled with a transposition ( x i , x j ) of the values of the two entries that are swapped. ◮ Π n is the update graph of E n (the graph with n vertices and no edges). ◮ Each transposition ( i j ) ∈ S n corresponds with a complete set of parallel edges of Π n . ◮ The update graph U ( Y ) can be constructed by “cutting” Π n with the normal central hyperplane H n i , j for every edge { i , j } ∈ e [ Y ]. This is the graphic hyperplane arrangement of Y .

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems An example 1234 2134 1243 2143 2314 1423 2413 1432 ( 1 2) 2341 4123 1 2 2431 (1 3) 4213 3241 4132 (2 3) 4231 3421 4312 3 4 4321 (a) Y (b) Constructing U ( Y ) Figure: Hyperplanes cuts corresponding with the edges { 1 , 2 } , { 2 , 3 } , and { 1 , 3 } in Y < K 4 .

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Let C ( Y ) and D ( Y ) be the graphs defined by ˘ ¯ v [ C ( Y )] = S Y / ∼ α , e [ C ( Y )] = { [ π ] α , [shift( π )] α } | π ∈ S Y , ¯ ∪ e [ C ( Y )] . e [ D ( Y )] = ˘ { [ π ] α , [refl( π )] α } | π ∈ S Y v [ D ( Y )] = S Y / ∼ α , ◮ By construction, there are a bijections between: Vertices of C ( Y ) (or D ( Y )) ← → Acyc( Y ) Connected components of C ( Y ) ← → Acyc( Y ) / ∼ κ Connected components of D ( Y ) ← → Acyc( Y ) / ∼ δ Example : Y = Circ 4 . 1243 4123 3412 3241 1234 2341 1324 2413 3214 2143 4132 2314 4321 1432 Figure: The graphs C (Circ 4 ) and D (Circ 4 ). The dashed lines are edges in D (Circ 4 ) but not in C (Circ 4 ). Clearly, κ (Circ 4 ) = 3 and δ (Circ 4 ) = 2.

Acyclic Orientations Equivalences Sequential Dynamical Systems Enumeration of equivalence classes Coxeter Groups Neutral networks for equivalence Summary Poset structure of κ -equivalence classes References Enumeration problems Structure of C ( Y ) and D ( Y ) Proposition ([8]) Let Y be a connected graph on n vertices and let g , g ′ ∈ C n with g � = g ′ . Then [ g · π ] κ � = [ g ′ · π ] κ . Proposition ([8]) Let Y be a connected graph on n vertices and let g , g ′ ∈ D n with g � = g ′ . If [ g · π ] κ = [ g ′ · π ] κ holds then Y is bipartite. Proposition ([8]) Let Y be a connected undirected graph. If Y is not bipartite then δ ( Y ) = 1 2 κ ( Y ) . If Y is bipartite then δ ( Y ) = 1 2 ( κ ( Y ) + 1) . Corollary A connected graph Y is bipartite if and only if κ ( Y ) is odd.