What do Coxeter Groups and Boolean Networks have in Common? Matthew - PowerPoint PPT Presentation

Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References What do Coxeter Groups and Boolean Networks have in Common? Matthew Macauley Department of Mathematical Sciences CLEMSON UNIVERSITY Clemson, South Carolina, USA 29634 AMS Sectional Meeting Baylor University Waco, Texas October 16th, 2009

Graph dynamical systems Coxeter groups Interplay between Coxeter theory and SDSs Summary References Outline 1 Graph dynamical systems Definitions Examples Applications Basic Questions 2 Coxeter groups Definitions The word problem & Matsumoto’s theorem 3 Interplay between Coxeter theory and SDSs Acyclic orientations and source-to-sink moves Dynamics groups of SDSs The Root automaton of a Coxeter group 4 Summary

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References Graph Dynamical Systems – GDSs: ◮ A Graph Dynamical System (see [15]) is a triple consisting of: A graph Y with vertex set v [ Y ] = { 1 , 2 , . . . , n } . For each vertex i a state x i ∈ K (e.g. K = { 0 , 1 } ) and a Y -local function F i : K n − → K n F i ( x = ( x 1 , x 2 , . . . , x n )) = ( x 1 , . . . , x i − 1 , f i ( x [ i ]) , x i +1 , . . . , x n ) . | {z } vertex function An update scheme that governs how the maps F i are assembled to a map F : K n − → K n . ◮ Typical choices of update schemes: Parallel: Generalized Cellular Automata F ( x 1 , . . . , x n ) i = f i ( x [ i ]) Sequential: Sequential Dynamical Systems [ F Y , w ] = F w ( k ) ◦ F w ( k − 1) ◦ · · · ◦ F w (1) ( w = w (1) · · · w ( k ) – a word on v [ Y ]) (Local dynamics)

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References An Example: Graph Y = Circ 4 State set K = { 0 , 1 } System state x = ( x 1 , x 2 , x 3 , x 4 ) Restricted vertex state x [1] = ( x 1 , x 2 , x 4 ) Vertex functions: f i = nor 3 : K 3 − → K by nor 3 ( x , y , z ) = (1 + x )(1 + y )(1 + z ) Y -local maps: Nor 1 ( x ) = ( nor 3 ( x [1]) , x 2 , x 3 , x 4 ), etc. Update sequence π = (1 , 2 , 3 , 4) SDS map: [ Nor Y , π ] = Nor 4 ◦ Nor 3 ◦ Nor 2 ◦ Nor 1 Sequential: [ Nor Y , π ](0 , 0 , 0 , 0) = (1 , 0 , 1 , 0) Parallel: Nor (0 , 0 , 0 , 0) = (1 , 1 , 1 , 1)

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References Remark . The global dynamics can be sensitive to changes in the update order. ◮ [ Nor Circ 4 , π ] for given update sequences: (1234) 1000 1100 (1423) 0101 0010 1001 0001 1100 0100 0011 0111 0110 0101 0100 1001 0000 1011 0111 1101 1010 1111 0000 1010 1000 1010 0001 1011 0010 1110 0110 1110 1111 1101 0101 0011 0110 0111 (1324) 1100 0001 0100 1001 0010 1000 1101 0000 1110 1111 1011 1010

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References Applications of SDSs Large complex networks [15]. Epidemiology. Disease propogation over social contact graphs. Agent-based transportation simulations. Packet flow in wireless networks. Gene annotation (Functional Linkage Networks) [8] Transport computation on irregular grids (e.g., heat, radiation). Image processing and pattern recognition [3]. Discrete event simulations (e.g., chemical reaction networks) [6].

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References Fundamental Questions Let S ∗ be the free monoid generated by v [ Y ] (i.e., words in the vertex set of Y ). There are infinitely many words w = w 1 w 2 w 3 · · · w n in S ∗ . . . . . . but only finitely many functions F n → F n 2 − 2 . Question 1 . When are two SDS maps [ F Y , w ] and [ F Y , w ′ ] “the same”? Question 2 . What do we really mean by “the same”?

Graph dynamical systems Definitions Coxeter groups Examples Interplay between Coxeter theory and SDSs Applications Summary Basic Questions References ◮ Question 2 : What does it mean for two SDS s to be “the same”? (see [12]) Definition Two SDSs are functionally equivalent if their SDS maps are identical as functions K n − → K n . Definition Two finite dynamical systems φ, ψ : K n − → K n are dynamically equivalent if there is a bijection h : K n − → K n such that ψ ◦ h = h ◦ φ . (i.e., phase spaces are isomorphic). Definition Two finite dynamical systems φ, ψ : K n → K n are cycle equivalent if there exists a bijection h : Per( φ ) − → Per( ψ ) such that ψ | Per( ψ ) ◦ h = h ◦ φ | Per( φ ) . (i.e., phase spaces are isomorphic when restricted to the periodic points).

Graph dynamical systems Coxeter groups Definitions Interplay between Coxeter theory and SDSs The word problem & Matsumoto’s theorem Summary References Coxeter groups Definition A Coxeter group is a group with presentation m ( s i , s j ) � s 1 , . . . , s n | s 2 i = 1 , s i s = 1 � j where m ij := m ( s i , s j ) ≥ 2 if i � = j . m ( s , t ) = 2 = ⇒ st = ts (short braid relation) m ( s , t ) = 3 = ⇒ sts = tst (braid relation) m ( s , t ) = 4 = ⇒ stst = tsts (braid relation) . . . ◮ A Coxeter group is a generalized reflection group .

Graph dynamical systems Coxeter groups Definitions Interplay between Coxeter theory and SDSs The word problem & Matsumoto’s theorem Summary References The word problem for Coxeter groups There are infinitely many words w = w 1 w 2 w 3 · · · w n in S ∗ . . . . . . and sometimes (usually), infinitely many group elements in W . Question . Given two words w = s 1 s 2 · · · s n and w ′ = s ′ 1 s ′ 2 · · · in S ∗ , when do they give rise to the same group element? This question has an simple answer (see [4]): Matsumoto’s Theorem : Any two reduced expressions for the same group element differ by braid relations.

Graph dynamical systems Coxeter groups Acyclic orientations and source-to-sink moves Interplay between Coxeter theory and SDSs Dynamics groups of SDSs Summary The Root automaton of a Coxeter group References Role of acyclic orientations in Coxeter groups and SDSs Coxeter groups Sequential dynamical systems Base graph ← → Coxeter graph Γ Dependency graph Y Acyc(Γ) ← → Coxeter elements SDS maps c = s π (1) s π (2) · · · s π ( n ) [ F Y , π ] = F π ( n ) ◦ · · · ◦ F π (2) ◦ F π (1) . Source-to- ← → Conjugacy classes Cycle-equivalence classes sink moves of Coxeter elements of SDS maps Aut(Γ) ← → Spectral classes Cycle-equivalence classes orbits of Coxeter elements of SDS maps (finer) This can be extended beyond Coxeter elements using labeled heaps instead of acyclic orientations.

Graph dynamical systems Coxeter groups Acyclic orientations and source-to-sink moves Interplay between Coxeter theory and SDSs Dynamics groups of SDSs Summary The Root automaton of a Coxeter group References SDS dynamics revealed as quotients of Coxeter groups ◮ A sequence F Y is w-independent if Per[ F Y , w ] = Per[ F Y , w ′ ] for all w and w ′ in S ∗ . Proposition If F Y is w-independent, then each F i is bijective on P := Per( F Y ) . ◮ Let G ( F Y ) be the group of permutations of P generated by { F 1 , . . . , F n } . This is called the dynamics group of F Y . ◮ If K = F 2 , the dynamics group is the homomorphic image of a Coxeter group, because | F i | ≤ 2 and | F i F j | = m ij . (see [10]) ◮ If K � = F 2 , the dynamics group is the homomorphic image of an Artin group. Question . Can we determine this homomorphism, i.e., the “extra relations”?

Graph dynamical systems Coxeter groups Acyclic orientations and source-to-sink moves Interplay between Coxeter theory and SDSs Dynamics groups of SDSs Summary The Root automaton of a Coxeter group References The root automaton of a Coxeter group (an infinite SDS) Let Φ = Φ + ⊔ Φ − be the set of all roots of a Coxeter group W . We can represent roots as vectors in R n and partially order them by ≤ componentwise, to get the root poset. Each generator s i ∈ S acts on Φ by reflection: „ π n « X s i �− → z + 2 cos z j e i . z m ij j =1 ◮ In summary, multiplication by s i flips the sign of the i th entry and adds each neighboring state z j weighted by 2 cos( π/ m ij ) ≥ 1. We can represent this as a local function F i : R n − → R n : „ π n « X F i : ( z 1 , . . . z i − 1 , z i , z i +1 , z n ) �− → ( z 1 , . . . , z i − 1 , z i + 2 cos z j , z i +1 , . . . , z n ) . m ij j =1

Graph dynamical systems Coxeter groups Acyclic orientations and source-to-sink moves Interplay between Coxeter theory and SDSs Dynamics groups of SDSs Summary The Root automaton of a Coxeter group References The root automaton detecting reduced words Consider a word w = s x 1 s x 2 s x 3 · · · s x k − 1 s x k ∈ S ∗ . Start at the vector e x 1 ∈ Φ + (a positive root), and follow the paths labeled s x 2 , s x 3 , s x 4 , . . . . If s x k is the first instance of crossing over to the negative roots Φ − , then w is not a reduced expression, and moreover, s x 1 s x 2 s x 3 · · · s x k − 1 s x k = s x 2 s x 3 · · · s x k − 1 . ◮ Thus, the root automaton detects reduced words.