Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Coxeter Theory and Discrete Dynamical Systems Matthew Macauley Department of Mathematics University of California, Santa Barbara Network Dynamics and Simulation Science Laboratory Virginia Bioinformatics Institute Virginia Polytechnic Institute and State University University of California, Santa Barbara April 11, 2008
Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References 1 Acyclic Orientations Equivalences 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 Acknowledgments
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References 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 Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References 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 Y ( π ) = O π f Y : S Y − → Acyc( Y ) , Y , where π is a linear extension of O π Y . Explicitly, if π = π 1 π 2 · · · π n then { i , j } ∈ e [ Y ] is oriented ( i , j ) iff i appears before j in π . ◮ For any π, σ ∈ S Y , define π ∼ Y σ iff f Y ( π ) = f Y ( σ ). Denote the equivalence classes by, e.g., [ π ] Y . We have the bijection Y ([ π ] Y ) = O π f ∗ f ∗ Y : S Y / ∼ Y − → Acyc( Y ) , Y .
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References Source-to-sink operations ◮ A cyclic 1-shift (left) of a linear extension of O Y 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 . ◮ Reversing the order of a linear extension of O Y corresponds to reversing all edge- orientations of O Y . ◮ The source-to-sink operation with reflections puts a coarser equivalence relation on Acyc( Y ), denoted ∼ δ . ◮ Define the functions: κ ( Y ) = | Acyc( Y ) / ∼ κ | , δ ( Y ) = | Acyc( Y ) / ∼ δ | .
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References Group actions on Acyc( Y ) Let σ, ρ ∈ S n be the elements ρ = (1 , n )(2 , n − 1) · · · ( ⌈ n 2 ⌉ , ⌊ n σ = ( n , n − 1 , . . . , 2 , 1) , 2 ⌋ + 1) , and let C n and D n be the subgroups C n = � σ � and D n = � σ, ρ � . C n and D n act on S Y via g · ( π 1 , . . . , π n ) = ( π g − 1 (1) , . . . , π g − 1 ( n ) ) . C n and D n act on S Y / ∼ Y (and hence on Acyc( Y )) via g · [ π ] Y = [ g · π ] Y . ◮ Thus, we may interpret Acyc( Y ) / ∼ κ and Acyc( Y ) / ∼ δ as the set of orbits under the actions of C n and D n on Acyc( Y ).
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References 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 Circ 4 be 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 ).
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References 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 . ◮ 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 Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References 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 Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References Neutral networks for κ - and δ -equivalence Let C ( Y ) and D ( Y ) be the graphs defined by ¯ , e [ C ( Y )] = ˘ { [ π ] Y , [ σ 1 ( π )] Y } | π ∈ S Y v [ C ( Y )] = S Y / ∼ Y , ˘ ¯ v [ D ( Y )] = S Y / ∼ Y , e [ D ( Y )] = { [ π ] Y , [ ρ ( π )] Y } | π ∈ S Y ∪ e [ C ( Y )] . ◮ By construction, there is a bijection between the connected components of C ( Y ) (resp. D ( Y )) and Acyc( Y ) / ∼ κ (resp. Acyc( Y ) / ∼ δ ). Example . 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 Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References Structure of C ( Y ) and D ( Y ) Proposition ([7]) Let Y be a connected graph on n vertices and let g , g ′ ∈ C n with g � = g ′ . Then [ g · π ] Y � = [ g ′ · π ] Y . Proposition ([7]) Let Y be a connected graph on n vertices and let g , g ′ ∈ D n with g � = g ′ . If [ g · π ] Y = [ g ′ · π ] Y holds then Y is bipartite. Proposition ([7]) 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.
Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References Associating κ -classes with posets Throughout, let e = { v , w } be a fixed cycle-edge of the connected graph Y . Definition ( vw -interval) Let Acyc ≤ ( Y ) be the set of acyclic orientations of vertex-induced subgraphs of Y . Define the map I : Acyc( Y ) − → Acyc ≤ ( Y ) , by I ( O Y ) = [ v , w ] if v ≤ O Y w , and I ( O Y ) = ∅ otherwise. The map I can be extended to a map I ∗ : Acyc( Y ) / ∼ κ − I ∗ ([ O Y ]) = I ( O 1 → Acyc ≤ ( Y ) by Y ) , where O 1 Y ∈ [ O Y ] such that I ( O 1 Y ) � = ∅ . ◮ In other words: If O 1 Y ∼ κ O 2 Y w for i = 1 , 2 , then O 1 Y and O 2 Y and v ≤ O i Y have the same vw-interval.
� � � Acyclic Orientations Equivalences Sequential Dynamical Systems Neutral networks for equivalence Coxeter Groups Poset structure of κ -equivalence classes Summary Enumeration problems References The vw -interval under edge-deletion Proposition ([7]) There is a well-defined map I ∗ e that makes the following diagram commute: I ∗ Acyc( Y ) / ∼ κ Acyc ≤ ( Y ) � � � ε ∗ � I ∗ � � e � Acyc( Y ′ ) / ∼ κ ◮ In other words: Upon removing e = { v , w } , it is well-founded to define the vw-interval, AND it is a κ -invariant in Acyc( Y ′ e ) .
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