Coxeter Theory and Discrete Dynamical Systems Matthew Macauley - - PowerPoint PPT Presentation

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 Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

A recursion for enumerating acyclic orientations

Let Y be an undirected graph. For e ∈ e[Y ], let Y ′

e and Y ′′ e

denote the graphs formed from Y by deleting and contracting e, respectively. ◮ For any e ∈ e[Y ], there is a bijection βe : Acyc(Y ) − → Acyc(Y ′

e ) ∪ Acyc(Y ′′ e )

defined by OY − → 8 > < > : O′

Y ,

Oρ(e)

Y

∈ Acyc(Y ), O′

Y ,

Oρ(e)

Y

∈ Acyc(Y ) and OY (e) = (v, w) , O′′

Y ,

Oρ(e)

Y

∈ Acyc(Y ) and OY (e) = (w, v) . ◮ Thus, the function α(Y ) := |Acyc(Y )| satisfies the recurrence α(Y ) = α(Y ′

e ) + α(Y ′′ e ) .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Acyclic orientations as posets

Let SY be the set of total orderings (or permutations) of v[Y ]. An element OY ∈ Acyc(Y ) defines a partial ordering on the vertex set v[Y ] by i ≤OY j if there is a directed path from i to j in OY . This induces a well-defined map fY : SY − → Acyc(Y ) , fY (π) = Oπ

Y ,

where π is a linear extension of Oπ

Y .

Explicitly, if π = π1π2 · · · πn then {i, j} ∈ e[Y ] is oriented (i, j) iff i appears before j in π. ◮ For any π, σ ∈ SY , define π ∼Y σ iff fY (π) = fY (σ). Denote the equivalence classes by, e.g., [π]Y . We have the bijection f ∗

Y : SY/∼Y −

→ Acyc(Y ) , f ∗

Y ([π]Y ) = Oπ Y .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Source-to-sink operations

◮ A cyclic 1-shift (left) of a linear extension of OY corresponds to converting a source

• f OY 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 OY corresponds to reversing all edge-

• rientations of OY .

◮ 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 Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Group actions on Acyc(Y )

Let σ, ρ ∈ Sn be the elements σ = (n, n − 1, . . . , 2, 1) , ρ = (1, n)(2, n − 1) · · · (⌈ n

2 ⌉, ⌊ n 2 ⌋ + 1) ,

and let Cn and Dn be the subgroups Cn = σ and Dn = σ, ρ . Cn and Dn act on SY via g · (π1, . . . , πn) = (πg−1(1), . . . , πg−1(n)) . Cn and Dn act on SY/∼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 Cn and Dn on Acyc(Y ).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Update graphs

Definition

The update graph U(Y ) has vertex set SY . The edge {π, σ} is present iff: π and σ differ by exactly an adjacent transposition (i, i + 1), {πi, πi+1} ∈ e[Y ].

• Example. Let Circ4 be the circular graph on 4 vertices.

1243 3241 2134 4132 1423 3421 2314 4312 1234 1432 3214 2341 4123 2143 4321 1324 3124 2413 4213 1342 3142 2431 4231 3412

Figure: The update graph U(Circ4).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Constructing U(Y ) from hyperplane arrangements

The n-permutahedron Πn is the convex hull of all permutations of the points (1, 2, . . . , n) ∈ Rn. It is an (n − 1)-dimensional polytope. The vertices and edges of Πn 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 (xi, xj) of the values of the two entries that are swapped. ◮ Πn is the update graph of En. ◮ Each transposition (i j) ∈ Sn corresponds with a complete set of parallel edges of Πn. ◮ The update graph U(Y ) can be constructed by “cutting” Πn with the normal central hyperplane Hn

i,j for every edge {i, j} ∈ e[Y ]. This is the graphic hyperplane arrangement

• f Y .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

An example

2 3 4 1 (a) Y

2143 1243 2314 1423 1432 2431

4231

4321 3241 4123 2134 2341 1234 3421 4132 4312 4213 2413 (1 3) (2 3) ( 1 2)

(b) Constructing U(Y )

Figure: Hyperplanes cuts corresponding with the edges {1, 2}, {2, 3}, and {1, 3} in Y < K4.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Neutral networks for κ- and δ-equivalence

Let C(Y ) and D(Y ) be the graphs defined by v[C(Y )] = SY/∼Y , e[C(Y )] = ˘{[π]Y , [σ1(π)]Y } | π ∈ SY ¯ , v[D(Y )] = SY/∼Y , e[D(Y )] = ˘ {[π]Y , [ρ(π)]Y } | π ∈ SY ¯ ∪ 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 2341 1234 4321 1432 3412 2143 3214 4123 2413 1324 2314 4132 3241

Figure: The graphs C(Circ4) and D(Circ4). The dashed lines are edges in D(Circ4) but not in C(Circ4). Clearly, κ(Circ4) = 3 and δ(Circ4) = 2.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Structure of C(Y ) and D(Y )

Proposition ([7])

Let Y be a connected graph on n vertices and let g, g ′ ∈ Cn 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 ′ ∈ Dn 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 Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

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(OY ) = [v, w] if v ≤OY w, and I(OY ) = ∅ otherwise. The map I can be extended to a map I∗ : Acyc(Y )/∼κ− → Acyc≤(Y ) by I∗([OY ]) = I(O1

Y ) ,

where O1

Y ∈ [OY ] such that I(O1 Y ) = ∅.

◮ In other words: If O1

Y ∼κ O2 Y and v ≤Oi

Y w for i = 1, 2, then O1

Y and O2 Y have the

same vw-interval.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

The vw-interval under edge-deletion

Proposition ([7])

There is a well-defined map I∗

e that makes the following diagram commute:

Acyc(Y )/∼κ

I∗

• ε∗
• Acyc≤(Y )

Acyc(Y ′)/∼κ

I∗

e

• ◮ 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 ).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

A recursion for κ(Y )

◮ The map I∗

e is crucial in proving that for any cycle-edge e ∈ e[Y ], the map

Θ: Acyc(Y )/∼κ − → ` Acyc(Y ′

e )/∼κ

´ [ ` Acyc(Y ′′

e )/∼κ

´ defined by [OY ]

Θ

− → ( [Oπ

Y ′′

e ],

∃Oπ

Y ∈ [OY ] with π = vwπ3 · · · πn

[Oπ

Y ′

e ],

• therwise.

(for any π = vπ2 · · · πn with w = vi minimal), is a bijection.

Theorem ([7])

For any cycle-edge e ∈ e[Y ], κ(Y ) = κ(Y ′

e ) + κ(Y ′ e ) .

Moreover, κ(Y ) is unchanged upon removal of bridge edges.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

The Tutte polynomial

The Tutte polynomial [12] of a graph Y is defined as follows: If Y has no cycle-edges, b bridges, and ℓ loops, then TY (x, y) = xbyℓ. If e is a cycle-edge, then TY (x, y) = TY ′

e (x, y) + TY ′′ e (x, y).

◮ Any quantity that satisfies a recursion via edge-deletion and contraction is an evaluation of the Tutte polynomial at some (x0, y0). Examples. TY (1, 1) is the number of spanning trees of Y TY (2, 1) is the number of acyclic subgraphs of Y . TY (1, 2) is the number of spanning edge-induced subgraphs of Y . TY (2, 0) = α(Y ), the number of acyclic orientations of Y . TY (1, 0) = κ(Y ). ◮ The chromatic polynomial of Y is (−1)n−1kTY (1 − k, 0). ◮ The HOMFLY (and hence Jones and Alexander) polynomial can be computed from the Tutte polynomial of a related graph.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

κ(Y ) for some special graph classes

Proposition ([8])

For v ∈ v[Y ], let |Acycv(Y )| be the number of acyclic orientations of Y where v is the unique source. There is a bijection φv : Acycv(Y ) − → Acyc(Y )/∼κ .

Corollary

For any vertex v of Y the set Acycv(Y ) is a transversal of Acyc(Y )/∼κ. ◮ If Y is a tree, then κ(Y ) = 1. ◮ If Y is an n-cycle, then κ(Y ) = n − 1. ◮ If Y ⊕ v is the vertex join of Y , then κ(Y ⊕ v) = α(Y ). ◮ κ(Kn) = (n − 1)!.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

The ν-invariant

◮ Let P = v1v2 · · · vk be a path in Y . Define νP(OY ) to be the number of edges

• riented (vi, vi+1), minus the number of edges oriented (vi+1, vi).

◮ If P is a cycle, then νP(OY ) is invariant under clicks. If Y = Circn, then κ(Y ) = n − 1. Let P traverse Y once. The possible values for νP(Circn) are ±(n − 1), ±(n − 3), ±(n − 5), . . . . Thus for Y = Circn, νP(OY ) = νP(O′

Y ) iff OY ∼κ O′ Y .

Problem 1: Does this hold in general, i.e., if νP(OY ) = νP(O′

Y ) for every cycle P,

then does OY ∼κ O′

Y ? If not, then what graphs does it hold for? (Maybe appeal to

Robertson-Seymour?)

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Sequential dynamical systems

◮ A sequential dynamical system (SDS) is a triple consisting of: A graph Y with vertex set v[Y ] = {1, 2, . . . , n}. For each vertex i a state yi ∈ K (e.g. F2 = {0, 1}) and a local function Fi : K n − → K n Fi(y = (y1, y2, . . . , yn)) = (y1, . . . , yi−1, fi(y[i]) | {z }

vertex function

, yi+1, . . . , yn) . A ordering π = π1π2 · · · πn ∈ SY of the vertex set. ◮ The SDS map generated by the triple (Y , (Fi )n

1, π) is

[FY , π] = Fπn ◦ Fπn−1 ◦ · · · ◦ Fπ1 .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Functional equivalence

Definition

Two SDS maps [FY , π] and [FY , σ] are functionally equivalent if they are equivalent as functions K n − → K n.

• Question. For a fixed FY , how many different SDS maps are there of the form [FY , π],

for π ∈ SY ?

• Proposition. If π ∼Y σ, then [FY , π] = [FY , σ].

◮ Thus, α(Y ) = |Acyc(Y )| is an upper bound for the number of functionally distinct SDS maps [FY , π] for a fixed choice of FY . Let NorY be the logical NOR functions. Then [NorY , π] = [NorY , σ] if and only if π ∼Y σ. ◮ Therefore, α(Y ) is a sharp upper bound for functional equivalence of SDSs.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Dynamical equivalence

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 . The group Aut(Y ) acts on SY/∼Y by γ · [π]Y = [γπ]Y .

Proposition

If FY is a sequence of Aut(Y )-invariant local functions, then for any γ ∈ Aut(Y ), γ ◦ [FY , π] ◦ γ−1 = [FY , γπ] .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Dynamical equivalence (cont.)

◮ For a fixed FY , the number of orbits of this action is an upper bound for the number

• f SDS maps [FY , π] for π ∈ SY , up to dynamical equivalence.

◮ This bound can be computed explicitly as follows: ¯ α(Y ) = 1 |Aut(Y )| X

γ∈Aut(Y )

α(γ \ Y ) , where γ \ Y is orbit graph of γ and Y , (see [1, 2]). ◮ This bound is known to be sharp for certain graph classes, but in the general case this is still an open problem.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Cycle equivalence

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) . Let σ, ρ ∈ Sn be σ = (n, n − 1, . . . , 2, 1) , ρ = (1, n)(2, n − 1) · · · (⌈ n

2 ⌉, ⌊ n 2 ⌋ + 1) ,

and let Cn and Dn be the groups Cn = σ Dn = σ, ρ . ◮ These groups act on update orders π = π1π2 · · · πn by shift and reflection as follows: σ(π) := σ · π = π2π3 · · · πnπ1 , ρ(π) := ρ · π = πnπn−1 · · · π2π1 .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Cycle equivalence of SDSs

Theorem ([6])

For any π ∈ SY , the SDS maps [FY , π] and [FY , σ(π)] are cycle equivalent. Moreover, if K = F2, then these are cycle equivalent to [FY , ρ(π)] as well. ◮ Recall that shifting the update order corresponds to a source-to-sink operation on the acyclic orientation.

Corollary

For a fixed choice of functions FY , the function κ(Y ) is an upper bound for the number

• f SDS maps [FY , π] with π ∈ SY , up to cycle equivalence. If K = F2, then δ(Y ) is an

upper bound as well.

Corollary

◮ If Y is a tree, then any two permutation SDS maps over FY are cycle equivalent. ◮ If Y is a tree, then all SDS maps [ParY , π], where π ∈ SY , are dynamically equivalent.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

• Example. Define the function nork : Fk

2 −

→ F2 by nork(x) =

k

Y

i=1

(1 + xi). ◮ [NorCirc4, π] for given update orders:

1000 0010 0100 0001 1010 0000 0101 0011 1011 0111 1111 1101 (1234) 0110 1110 1001 1100 (1423) 0000 1100 0110 0010 1000 0101 1010 1010 1101 1110 1111 1011 0111 1001 0100 0001

0111 1111 1101 1010 0000 0101 0010 1000 1110 1100 0110 0011 1001 1011 0001 0100 (1324)

There are ¯ α(Circ4) = 3 dynamically inequivalent phase spaces and δ(Circ4) = 2 cycle inequivalent phase spaces.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

Aut(Y )-actions

The group Aut(Y ) acts on Acyc(Y )/∼κ (and Acyc(Y )/∼δ) by g · [Oπ

Y ] = g · [fY (π)] = [fY (g · π)] .

◮ If [Oπ

Y ] and [Oσ Y ] are in the same orbit under this action, then [FY , π] and [FY , σ] are

cycle equivalent. ◮ Define the number of orbits of this action on Acyc(Y )/∼κ (resp. Acyc(Y )/∼δ) to be ¯ κ(Y ) (resp. ¯ δ(Y )). ◮ For a fixed choice of functions FY , the function ¯ κ(Y ) is an upper bound for the number of SDS maps [FY , π] with π ∈ SY , up to cycle equivalence. If K = F2, then ¯ δ(Y ) is an upper bound as well. Problem 2. Are these bounds sharp? Problem 3. What’s a good way to compute these bounds?

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Functional equivalence Dynamical equivalence Cycle equivalence Aut(Y )-actions

An example

Let Q3

2 be the binary 3-cube. A tedious calculation gives α(Y ) = 1862.

κ ! = κ ! + κ ! = κ ! + 2κ ! + κ ! = κ ! + 2κ ! + 2κ ! + κ ! + κ ! = κ ! + 4κ ! + 2κ ! + κ ! + κ ! = 27 + 64 + 16 + 12 + 14 = 133 ◮ In summary, we have: α(Q3

2) = 1862 ,

¯ α(Q3

2) = 54 ,

κ(Q3

2) = 133 ,

δ(Q3

2) = 67 ,

¯ κ(Q3

2) = ¯

δ(Q3

2) = 8 .

◮ If Y = Q3

2, then for a fixed choice of functions FY , there are at most 8 possible cycle

structures of the SDS map [FY , π], up to isomorphism.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Coxeter groups

Definition

A Coxeter group is a group with presentation s1, . . . , sn | sis

mij j

• where mij > 1 iff i = j.

It follows easily that |si| = 2, and |sisj| = |sjsi|. Think of a Coxeter group as a generalized reflection group (more on this later). Recall, for any non-zero vectors v, w ∈ Rn, the reflection of v across the hyperplane

• rthogonal to w is

v − 2 v, w w, w w .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Coxeter graphs and acyclic orientations

A Coxeter system is a pair (W , S) where W is a Coxeter group, and S is the set of reflections that generate W . ◮ A concise way to describe a Coxeter system (W , S) is with a Coxeter graph, Y = Y (W , S): v[Y ] = S, e[Y ] = {{si , sj} | mij ≥ 3} . Additionally, each edge {si, sj} is labeled with mij (usually the label is omitted for mij = 3 because these are the most common). Note: Edges correspond to non-commuting pairs of reflections. ◮ A Coxeter element is the product of the generators in any order. ◮ There is a bijection between the set of Coxeter elements C := C(W , S) and the acyclic

• rientations of Y .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Source-to-sink operations

◮ Conjugating a Coxeter element by a simple reflection cyclically shifts the word, and corresponds to a source-to-sink operation (or “click”): sπ(1)(sπ(1)sπ(2) · · · sπ(n))sπ(1) = sπ(2)sπ(3) · · · sπ(n)sπ1 . Therefore, the equivalence relation ∼κ carries over to C(W , S). ◮ Clearly, if c ∼κ c′, then c and c′ are conjugate in W . ◮ Therefore, κ(Y ) is an upper bound on the number of conjugacy classes of Coxeter elements. Problem 4. Is this bound sharp, i.e., does the converse of the statement above hold?

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Natural reflection representation

Define ai,j = cos

π mij .

The natural reflection representation of W is defined on the generators s ∈ S by si − → In − 2Ei,i + X

j : mij≥3

ai,jEi,j . Example. si

• 2

6 6 6 6 6 6 6 6 6 6 6 4 1 ... 1 · · · ai−1,i −1 ai,i+i · · · 1 ... 1 3 7 7 7 7 7 7 7 7 7 7 7 5

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Spectral classes

The spectral class of a linear operator is the multiset of eigenvalues. As a convention, we identity w ∈ W with the corresponding linear transformation in the natural reflection representation. If w and w′ are conjugate in W , then they have the same spectral class. Problem 5 [13]. Given a Coxeter system (W , S) with Coxeter graph Y , how many spectral classes do the Coxeter elements in C(W , S) fall into? Two κ-classes that have respective acyclic orientations OY and O′

Y

such that ϕ: OY − → O′

Y for some ϕ ∈ Aut(Y ) also have the same spectral class.

◮ Therefore, ¯ κ(Y ) is an upper bound for the number of spectal classes.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

An example

Let Y = K2,3, with vertex set {1, 3, 5} ⊔ {2, 4}. α(Y ) = 46, κ(Y ) = 7, and ¯ κ(Y ) = 2. There are 2 spectral classes (See Shi, 2001 [13]): {12345, 23451, 52341, 51234, 45123, 34512} {12543, 25431, 32541, 31254, 43125, 54312} {32145, 35214, 52143, 21435, 14352, 43521} {14523, 45231, 34512, 31452, 23145, 52314} {14325, 43251, 54321, 51432, 25143, 32514} {34124, 41235, 54123, 35412, 23541, 12354} {24351, 21354, 13524, 41352, 52431, 15243, 12435, 31245, 32451, 35241} Elements in the first six classes have characteristic polynomial f (x) = x5 − 3x4 − 6x3 − 6x2 − 3x + 1. Elements in the last class have characteristic polynomial f (x) = x 5−x4−8x3−8x2−x+1.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

An example (cont.)

Figure: The update graph U(K2,3).

Consider the mapping (sπi )i

φ

− → (πi mod 2)i . Non-adjacency in Y coincides with parity, that is, if c = c′, then φ(c) = φ(c′). 12 size-1 components: 10101 24 size-2 components: 01011, 11010, 01101, 10110. 6 size-4 components: 10011, 11001. 2 size-6 components: 01110 2 size-12 components: 11100, 00111.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

An example (cont.)

15243 13245 35241 12435 32415 52413 41352 13524 24135 21354 45123 34512 51234 15432 21543 32154 14325 54321 52341 12345 43215 23451

×1 ×3

Figure: The graph C(K2,3) contains the component on the left, and three isomorphic copies of the structure on the right (but with different vertex labels).

Component at left: φ(π) ∈ {01101, 11010, 10101, 01011, 10110}. Component at right: φ(π) ∈ {11100, 11001, 10011, 00111, 01110}.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Word-independence of SDSs

◮ A sequence FY is π-independent (w-independent) if Per[FY , w] = Per[FY , w′] for all w and w′ in SY (fair words in WY ). In fact, these conditions are equivalent, so we call them both word-independence.

Proposition

If FY is w-independent, then each Fi is bijective on P := Per(FY ). Let [FY , ω]∗ denote the restriction of [FY , ω] to the set of periodic points. ◮ If W ′ ⊆ WY then the group H(FY , W ′) = [FY , ω]∗ : ω ∈ W ′ is called the dynamics group of FY and W ′. Full dynamics group: G(FY ) := H(FY , WY ) = F ∗

i : Fi ∈ FY ,

Permutation dynamics group: H(FY ) := H(FY , SY ) = [FY , π]∗ : π ∈ SY .

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Example: Asynchronous Cellular Automata

Let Y = Circn, the circular graph on n vertices. If k = a7a6a5a4a3a2a1a0 in binary, then Wolfram rule k is defined by wolf(k) : (yi−1, yi, yi+1) → zi by the following table. yi−1yiyi+1 111 110 101 100 011 010 001 000 zi a7 a6 a5 a4 a3 a2 a1 a0 Let Wolf(k) : Fn

2 → Fn 2 be the corresponding local function, and Wolf (k) n

= (Wolf(k)) the sequence of local functions of Circn. ◮ The SDS map [Wolf (k)

n , π], where π ∈ SY , is an asynchronous cellular automata

(ACA).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Word-independent ACAs

Theorem (Hansson, Mortveit, Reidys [4])

Of the 16 symmetric Wolfram rules, exactly 11 are w-independent for all n > 3.

Theorem (M–, McCammond, Mortveit [5])

Of the 256 Wolfram rules, exactly 104 are w-independent. More precisely, Wolf (k)

n

is w-independent for all n > 3 iff k ∈ {0, 1, 4, 5, 8, 9, 12, 13, 28, 29, 32, 40, 51, 54, 57, 60, 64, 65, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 92, 93, 94, 95, 96, 99, 102, 105, 108, 109, 110, 111, 124, 125, 126, 127, 128, 129, 132, 133, 136, 137, 140, 141, 147, 150, 152, 153, 156, 157, 160, 164, 168, 172, 184, 188, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 204, 205, 206, 207, 216, 218, 220, 221, 222, 223, 224, 226, 228, 230, 232, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255}. ◮ These 104 rules constitute 41 distinct classes up to dynamical equivalence (inversion and reflection).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Table of the 104 rules [5]

p3

• 1

1

• x

x p2

• 1
• 1

x

• x

p4p1 72 64 8 74 88 90 66 24 18

• 132

204 196 140 132 206 220 222 198 156 150 0- 4 76 68 12 4 78 92 94 70 28

128 200 192 136 128 202 216 218 194 152 1- 164 236 228 172 164 238 252 254 230 188

• 1

133 205 197 141 133 207 221 223 199 157 10 160 232 224 168 160 234 248 250 226 184 01 5 77 69 13 5 79 93 95 71 29 00 72 64 8 x0 32 96 40 32 0x 1 73 65 9 1

• x

129 201 193 137 129 195 153 147 x- 36 108 110 124 126 102 60 54 x1 37 109 111 125 127 1x 161 235 249 251 11 165 237 239 253 255 xx 33 105 99 57 51

Table: The 104 w-independent rules arranged by symmetric and asymmetric parts of their tags.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Computation of dynamics groups

◮ If K = F2, the dynamics group is the homomorphic image of a Coxeter group, because |Fi| ≤ 2 and |FiFj| = mij ◮ If K = F2, the dynamics group is sometimes the homomorphic image of an Artin group. Problem 6. Can we determine this homomorphism, i.e., the “extra relations”? ◮ Of the 41 non-equivalent word-independent ACAs, 26 of them have a trivial dynamics

• group. For the remaining 15, mij ∈ {1, 2, 3, 4, 6, 12}, and the groups are:

SL(n) or AL(n): Rules 1, 9, 110, 126 Zn

2: Rules 28, 29, 51

A2n or A2n−1: Rules 54, 57 GL(n, 2): Rule 60 Not sure: Rules 73, 105, 108, 150, 156

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Conjugacy of Coxeter elements Spectral classes Word-independence and dynamics groups

Table of the 104 rules, arranged by mij [9]

p3 x

• x
• 1

1 p2 x x

• 1
• 1

p4p1 18 66 24 72 64 8 74 88 90 0x 1 6 6 6 6 x1 37 6 6 6 6

• x

129 12 4 4 6 6 6 6 x- 36 12 4 4 6 6 6 6 xx 33 2 12 12 3

• 132

3 6 6 1 1 1 1 1 1 1

128 1 1 1 1 1 1 1 1 1 1- 164 1 1 1 1 1 1 1 1 1 10 160 1 1 1 1 1 1 1 1 1

• 1

133 2 2 1 1 1 1 1 1 1 0- 4 2 2 1 1 1 1 1 1 1 01 5 2 2 1 1 1 1 1 1 1 00 1 1 1 1 11 165 1 1 1 1 1x 161 1 1 1 x0 32 1 1 1 Table: The 104 w-independent rules arranged by mij

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Quiver representations [10]

A quiver is a finite directed graph (loops and multiple edges allowed). A quiver Q with a field K gives rise to a path algebra KQ. There is a natural correspondence (categorial equivalence) between KQ-modules, and K-representations of Q. ◮ A path algebra is finite-dimensional if and only if the quiver is acyclic. Modules over finite-dimensional path algebras form a reflective subcategory. ◮ A reflection functor maps representations of a quiver Q to representations of a quiver Q′, where Q′ differs from Q by a source-to-sink operation. ◮ A composition of n = |v[Q]| distinct reflection functors is not the identity, but a Coxeter functor.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Node-firing games [3]

◮ In the chip-firing game, each vertex of a graph is given some number (possibly zero)

• f chips.

If vertex i has degree di, and at least di chips, then a legal move (or a “click”) is a transfer of one chip to each neighbor. A legal move is in a sense a generalization of a source-to-sink operation. ◮ In the node-firing game, each vertex of a graph is assigned an integer value, and the edges are weighted according to the mij relations of the Coxeter group. The legal sequences of moves in the numbers game are in 1–1 correspondence with the reduced words of the Coxeter group with that Coxeter graph.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Summary of SDS / Coxeter theory connections

Coxeter groups Sequential dynamical systems Base graph ← → Coxeter graph Y (W , S) Dependency graph Y Acyc(Y ) ← → Coxeter elements SDS maps c = sπ(1)sπ(2) · · · sπ(n) [FY , π] = Fπ(n) ◦ · · · ◦ Fπ(2) ◦ Fπ(1). Clicks ← → Conjugacy classes Cycle-equivalence classes

• f Coxeter elements
• f SDS maps

Aut(Y ) ← → Spectral classes Cycle-equivalence classes

• rbits
• f Coxeter elements
• f SDS maps (finer)

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Connections to quiver representations and chip firing

Quiver representations Chip-firing game Base graph ← → Undirected quiver ¯ Q Underlying graph Y Acyc(Y ) ← → Quiver Q of a Configurations, or states finite-dimensional path-algebra KQ

• f the game

Clicks ← → Reflection functors Legal moves Aut(Y ) ← → Vector space isomorphisms Equivalent states

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Summary of future research

◮ Combinatorics Problem 1: Prove that ν is a complete invariant of Acyc(Y )/∼κ, or characterize all such counterexamples. (Robertson-Seymour?) Problem 3. Is there a nice closed-form or easily computable solution to ¯ κ(Y ) and ¯ δ(Y )? (Burnside’s Theorem?) ◮ Sequential dynamical systems Problem 2. Are ¯ κ(Y ) and ¯ δ(Y ) sharp upper bounds for the number of SDS maps up to cycle equivalence? ◮ Coxeter groups Problem 4. Prove (or disprove) that two Coxeter elements are conjugate iff they are κ-equivalent. Problem 5. Is ¯ κ(Y ) a sharp upper bound for the number of spectral classes of Coxeter elements of (W , S)? If not, for which graphs does it fail, and by how much?

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Summary of future research (cont.)

◮ Dynamics groups Problem 6. Can one characterize the “extra relations” of a dynamics groups from the graph and the local functions? Problem 7. Finish characterizing the dynamics groups of the 104 word-independent ACAs. Problem 8. Analyze the other 152 rules, and compare the dynamics to classical (synchronous) CAs. Problem 9. Can the dynamics group be generalized to non word-independent systems? (Gold standard: Seifert-van Kampenish)

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Connections to other areas of mathematics Future research Acknowledgments

Acknowledgments

Special thanks: University of California, Santa Barbara, Network Dynamics and Simulation Science Laboratory (NDSSL), at the Virginia Bioinformatics Institute, Los Alamos National Laboratory, The Fields Institute, Clemson University, Bjorn Birnir, Bud Brown, Elena Dimitrova, Ed Green, Ning Jia, Reinhard Laubenbacher, Nick Loehr, Jon McCammond, Ken Millett, Henning Mortveit, Landon Rabern, Anil Vullikanti, et al. NDSSL: Web: http://ndssl.vbi.vt.edu

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References

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[2]

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[7]

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[9]

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[10]

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[11]

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[12]

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