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 Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Enumeration of equivalence classes 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 > < > : OY ′ , Oρ(e)

Y

∈ Acyc(Y ), OY ′ , Oρ(e)

Y

∈ Acyc(Y ) and OY (e) = (v, w) , OY ′′ , 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 Enumeration of equivalence classes 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 fα : SY − → Acyc(Y ) , fα(π) = Oπ

Y ,

where π is a linear extension of Oπ

Y .

12435 12453 14235 41235 41253 41523 14253 14523

1 2 3 4 5 fα

Figure: An element of Acyc(Circ5), 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 SY , and a bijection f ∗

Y : SY/∼α−

→ Acyc(Y ) , f ∗

Y ([π]α) = Oπ Y .

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

Equivalences on Acyc(Y )

The cyclic group Cn = σ acts on the set SY of orderings of v[Y ]: π1π2 · · · πn−1πn

σ

− → π2 · · · πn−1πnπ1 . Via the function fα : SY → Acyc(Y ), this corresponds to converting a source of OY 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 Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Enumeration of equivalence classes Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

The dihedral group Dn = σ, ρ acts on the set SY of orderings of v[Y ]: π1π2 · · · πn−1πn

σ

− → π2 · · · πn−1πnπ1 , π1π2 · · · πn−1πn

ρ

− → πnπn−1 · · · π2π1 . ◮ Via f ∗

α, the reflection ρ reverses all edge orientations of OY .

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 Cn,

Dn, and Aut(Y ) on SY : Cn Dn Acyc(Y ) Acyc(Y )/∼κ Acyc(Y )/∼δ Aut(Y ) Acyc(Y )/∼ ¯

α

Acyc(Y )/∼¯

κ

Acyc(Y )/∼¯

δ

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

Enumeration problems

α(Y ) := |Acyc(Y )| = TY (2, 0) satisfies α(Y ) = α(Y /e) + α(Y \ e) for any edge e ¯ α(Y ) := |Acyc(Y )/∼ ¯

α | =

1 Aut(Y ) X

γ∈Aut(Y )

α(γ \ Y ) κ(Y ) := |Acyc(Y )/∼κ | = TY (1, 0) satisfies κ(Y ) = κ(Y /e) + κ(Y \ e) for any cycle edge e δ(Y ) := |Acyc(Y )/∼δ | = ⌈κ(Y )/2⌉. ¯ κ(Y ) := |Acyc(Y )/∼¯

κ | =

1 Aut(Y ) X

γ∈Aut(Y )

|Fix(γ)| But what is |Fix(γ)| ???

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Enumeration of equivalence classes 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 Y = Circ4, 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). Each connected component corresponds with a unique element

• f Acyc(Circ4).

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Enumeration of equivalence classes 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 (the graph with n vertices and no edges). ◮ 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 Enumeration of equivalence classes 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 Enumeration of equivalence classes Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Let C(Y ) and D(Y ) be the graphs defined by v[C(Y )] = SY/∼α , e[C(Y )] = ˘ {[π]α, [shift(π)]α} | π ∈ SY ¯ , v[D(Y )] = SY/∼α , e[D(Y )] = ˘{[π]α, [refl(π)]α} | π ∈ SY ¯ ∪ e[C(Y )] . ◮ 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 = Circ4.

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 Enumeration of equivalence classes Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

Structure of C(Y ) and D(Y )

Proposition ([8])

Let Y be a connected graph on n vertices and let g, g ′ ∈ Cn 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 ′ ∈ Dn 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.

Acyclic Orientations Sequential Dynamical Systems Coxeter Groups Summary References Equivalences Enumeration of equivalence classes 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 ) = ∅.

Big idea: 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 Enumeration of equivalence classes 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

• Big idea: 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 Enumeration of equivalence classes 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. Big idea: “Put v and w as close together as possible, with v first. Then contract edge e = {v, w}, if doing so doesn’t create a cycle. Otherwise, delete it.”

Theorem ([9])

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 Enumeration of equivalence classes 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 Enumeration of equivalence classes Neutral networks for equivalence Poset structure of κ-equivalence classes Enumeration problems

κ(Y ) for some special graph classes

Proposition ([9])

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 Enumeration of equivalence classes 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 oriented (vi, vi+1), minus the number of edges oriented (vi+1, vi). Easy fact: If P is a cycle, then νP(OY ) is invariant under clicks. Let Y = Circn, and let P traverse Y once. The possible values for νP(Circn) are ±(n − 2), ±(n − 4), ±(n − 6), . . . . Therefore, κ(Circn) ≥ n − 1. By the recurrence κ(Y ) = κ(Y /e) = κ(Y \ e), and with base case κ(Tree) = 1, we get νP(Circn) = n − 1. ◮ Therefore, ν is a complete invariant of Acyc(Circn)/∼κ, i.e., if Y = Circn, νP(OY ) = νP(O′

Y )

⇐ ⇒ OY ∼κ O′

Y .

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

The ν-invariant (cont.)

+ ν = 4 ν = 2 ν = 0 ν = −2 ν = −4

Figure: The set Acycv(Circ6): A transversal for Acyc(Circ6)/∼κ.

In fact, taken over all cycles, ν is a complete invariant of Acyc(Y )/∼κ:

Theorem (M–, Mortveit [7])

If νC (OY ) = νC (O′

Y ) for every cycle C in Y , then OY ∼κ O′ Y .

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

Graph Dynamical Systems – GDSs:

◮ A Graph Dynamical System is a triple consisting of: A graph Y with vertex set v[Y ] = {1, 2, . . . , n}. For each vertex i a state xi ∈ K (e.g. K = {0, 1}) and a Y -local function Fi : K n − → K n Fi(x = (x1, x2, . . . , xn)) = (x1, . . . , xi−1, fi(x[i]) | {z }

vertex function

, xi+1, . . . , xn) . An update scheme that governs how the maps Fi are assembled to a map F: K n − → K n. ◮ Typical choices of update schemes: Parallel: Generalized Cellular Automata F(x1, . . . , xn)i = fi(x[i]) Sequential: Sequential Dynamical Systems [FY , w] = Fw(k) ◦ Fw(k−1) ◦ · · · ◦ Fw(1) (w = w(1) · · · w(k) – a word on v[Y ]) (Local dynamics)

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

An Example:

Graph Y = Circ4 State set K = {0, 1} System state x = (x1, x2, x3, x4) Restricted vertex state x[1] = (x1, x2, x4) Vertex functions: fi = nor3 : K 3 − → K by nor3(x, y, z) = (1 + x)(1 + y)(1 + z) Y -local maps: Nor1(x) = (nor3(x[1]), x2, x3, x4), etc. Update sequence π = (1, 2, 3, 4) SDS map: [NorY , π] = Nor4 ◦ Nor3 ◦ Nor2 ◦ Nor1 Sequential: [NorY , π](0, 0, 0, 0) = (1, 0, 1, 0) Parallel: Nor(0, 0, 0, 0) = (1, 1, 1, 1)

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 ∼α σ, then [FY , π] = [FY , σ].

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

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] . ◮ Thus, ¯ α(Y ) = |Acyc(Y )/∼ ¯

α | is an upper bound for the number of distinct SDS maps

[FY , π] for a fixed choice of FY , up to dynamical equivalence. 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) .

Theorem ([8])

For any π ∈ SY , the SDS maps [FY , π] and [FY , shift(π)] are cycle equivalent. Moreover, if K = F2, then these are cycle equivalent to [FY , refl(π)] as well. ◮ Thus, κ(Y ) = |Acyc(Y )/∼κ | is an upper bound for the number of distinct SDS maps [FY , π] for a fixed choice of FY , up to cycle equivalence. If K = F2, then δ(Y ) = |Acyc(Y )/∼δ | is an upper bound as well.

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 · [fα(π)] = [fα(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 (assuming FY is Aut(Y )-invariant). ◮ Thus, ¯ κ(Y ) = |Acyc(Y )/∼¯

κ | is an upper bound for the number of distinct SDS maps

[FY , π] for a fixed choice of FY , up to cycle equivalence. If K = F2, then ¯ δ(Y ) = |Acyc(Y )/∼¯

δ | is an upper bound as well.

• Question. Are these bounds sharp?

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

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 |SY | = 8! = 40320, but: Cn Dn α(Q3

2) = 1862

κ(Q3

2) = 133

δ(Q3

2 ) = 67

Aut(Y ) ¯ α(Q3

2) = 54

¯ κ(Q3

2) = 8

¯ δ(Q3

2) = 8

◮ If Y = Q3

2, then for a fixed choice of (Aut(Y )-invariant) 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 mij = mji. For i = j, si and sj commute iff mij = 2. 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 systems and Coxeter graphs

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, Γ: v[Γ] = S, e[Γ] = {{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 system is simply-laced if mij ≤ 3. Example: 4 5 ∞ s1 s2 s3 s4

Figure: Coxeter graph for W = s1, s2, s3, s4 | s2

1, s2 2, s2 3, s2 4 , (s1s3)2, (s2s4)2, (s1s2)3, (s2s3)4, (s3s4)5.

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

Coxeter elements and source-to-sink operations

◮ 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 ) and the acyclic

• rientations of Γ.

◮ 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 . Example:

1 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 5 5 5 5 4 4 4 5 6 6 6 6 3 6

s1s2s4s5s3s6 s4s2s5s3s6s1 s2s5s3s6s1s4 =s5s2s3s6s1s4 s2s3s6s1s4s5 =s6s2s3s1s4s5 s2s3s1s4s5s6

Figure: Conjugating a Coxeter element

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

The conjugacy problem for Coxeter elements

Therefore, the equivalence relation ∼κ on Acyc(Γ) carries over to C(W ). ◮ Clearly, if c ∼κ c′, then c and c′ are conjugate in W . It follows that κ(Γ) is an upper bound on | Conj(C(W ))|, the number of conjugacy classes of Coxeter elements. ◮ Open question. Is this bound sharp, i.e., does the converse of the statement above hold? In other words, we have a surjection fκ making the following diagram commute: C(W )

fα

• πc
• Acyc(Γ)

πκ

• Conj(C(W ))

Acyc(Γ)/∼κ

fκ

Is it a bijection?

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

Conjugacy in simply-laced Coxeter groups

Theorem (H. Eriksson, 1994 [2])

Let (W , S) be a simply-laced Coxeter system where Γ = Circn (i.e., W = ˜ An−1 is the affine Weyl group). Then two Coxeter elements c, c′ ∈ C(W ) are conjugate if and only if c ∼κ c′.

Theorem (J.-Y. Shi, 2001 [13])

Let (W , S) be a simply-laced Coxeter system where Γ is unicyclic. Then two Coxeter elements c, c′ ∈ C(W ) are conjugate if and only if c ∼κ c′.

Theorem (M–, Mortveit, 2008 [7])

Let (W , S) be any simply-laced Coxeter system. Then two Coxeter elements c, c′ ∈ C(W ) are conjugate if and only if c ∼κ c′. Open question: Can we extend this to non simply-laced Coxeter systems?

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

Standard geometric representation

Let (W , S) be a Coxeter system of rank n. Let V n be a vector space, with basis {ei}. Equip V n with a symmetric bilinear form · , · , defined by ei, ej = − cos π mij . There is an injection σ: W − → GL(V n) , si − → In − 2Ei,i + X

j : mij≥3

ai,jEi,j , called the standard geometric representation (where ai,j = cos π

mij ).

In other words, the Coxeter group W is the subgroup of GL(V n) generated by the reflections across the coordinate hyperplanes.

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

Standard geometric representation (cont.)

Explicity, the standard geometric representation sends each generator si ∈ S to a matrix, as follows: 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 where ai,j = cos

π mij .

Henceforth, we will identity w ∈ W with the corresponding matrix in the standard geometric representation.

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. If w and w′ are conjugate in W , then they have the same spectral class. Question [13]. Given a Coxeter system (W , S) with Coxeter graph Γ, how many spectral classes do the Coxeter elements fall into? Two κ-classes that have respective acyclic

• rientations

OΓ and O′

Γ

such that ϕ: OΓ − → O′

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

◮ Therefore, ¯ κ(Γ) is an upper bound for the number of spectal classes in C(W ).

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

A complete invariant of spectral classes for unicyclic graphs

Let (W , S) be a Coxeter system where Γ is unicyclic. The possible values for ν(OΓ) are ±(n − 2), ±(n − 4), etc. + ν = 4 ν = 2 ν = 0 ν = −2 ν = −4

Figure: The five distinct κ-classes of Acycv(Circ6).

Note: If c ∈ C(W ), then ν(c) = −ν(c−1).

Theorem ([1])

Let (W , S) be a Coxeter system where Γ is unicyclic, and let c, c′ ∈ C(W ). Then c and c′ have the same spectral class if and only if |ν(c)| = |ν(c′)|. ◮ Therefore, to solve the conjugacy problem for general (non simply-laced) Coxeter groups, it suffices to prove that if νC(c) = 0, then c and c−1 are not conjugate.

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

Extending the unicyclic case to the general case

Classic result: For unicyclic graphs, the ν-function is a complete invariant of the κ-equivalence (and hence conjugacy) classes. In [1], it is shown that for unicyclic graphs, the ν-function (up to sign) is a complete invariant of spectral classes for Coxeter elements. Recently, we’ve shown that taken over all cycles in a graph, the generalized ν-function is a complete invariant of κ-equivalence (and for conjugacy classes if (W , S) is simply-laced). Goal: Use the generalized ν-function to prove the following:

Conjecture

Let (W , S) be a Coxeter system, and let c, c′ ∈ C(W ). Then c and c′ have the same spectral class if and only if |νC(c)| = |νC (c′)| for every cycle C in Γ.

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

An example

Let Γ = K2,3, with vertex set {1, 3, 5} ⊔ {2, 4}. α(Γ) = 46, κ(Γ) = 7, and ¯ κ(Γ) = 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 Γ 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 the homomorphic image of an Artin group.

• Question. 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 More complicated: 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 [6]

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

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

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

Summary of SDS / Coxeter theory connections

Coxeter groups Sequential dynamical systems Base graph ← → Coxeter graph Γ 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

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

Summary of future research

Combinatorics Is there a nice closed-form or easily computable solution to ¯ κ(Y ) and ¯ δ(Y )? (Burnside’s Theorem?) Sequential dynamical systems Are ¯ κ(Y ) and ¯ δ(Y ) sharp upper bounds for the number of SDS maps up to cycle equivalence? Coxeter groups Prove that two Coxeter elements are conjugate iff they are κ-equivalent. 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? Prove that two Coxeter elements have the same spectral class iff |νC (c)| = |νC(c′)| for every cycle C in Γ.

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

Summary of future research (cont.)

◮ Dynamics groups Can one characterize the “extra relations” of a dynamics groups from the graph and the local functions? Finish characterizing the dynamics groups of the 104 word-independent ACAs. Analyze the 152 non word-independent CA rules, and compare the dynamics to classical (synchronous) CAs. 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

References

[1]

• A. Boldt. The spectral classes of unicyclic graphs. Journal for Pure and Applied Algebra 133:39–49, 1998.

[2]

• H. Eriksson. Computational and combinatorial aspects of Coxeter groups. Ph.D. thesis, KTH, Stockholm, 1994.

[3]

• K. Eriksson. Node firing games on graphs. Contemp. Math. 178, 1994:117–127.

[4]

• A. ˚
• A. Hansson, H. S. Mortveit, and C. M. Reidys. On asynchronous cellular automata. Adv. Comp. Sys. 8 (4), 2005:521–538.

[5]

• M. Macauley, J. McCammond, and H. S. Mortveit. Order independence in asynchronous cellular automata. J. Cell. Autom. 3 (1),

2008:37–56. [6]

• M. Macauley, J. McCammond, and H. S. Mortveit. Dynamics groups of asynchronous cellular automata. Submitted to Foundat.
• Comput. Math., 2008.

[7]

• M. Macauley, H. S. Mortveit. A solution to the conjugacy problem for Coxeter elements in simply laced Coxeter groups. Submitted

to Adv. Math., 2008. [8]

• M. Macauley, H. S. Mortveit. Cycle equivalence of graph dynamical systems. Nonlinearity 22, 2009:421–436.

[9]

• M. Macauley, H. S. Mortveit. On enumeration of conjugacy classes of Coxeter elements. Proc. Amer. Math. Soc. 136, 2008:4157–

4165. [10]

• R. Marsh, M. Reineke and A. Zelevinsky. Generalized associahedra via quiver representations. Trans. AMS 355, 2003:4171–4186.

[11]

• H. S. Mortveit, C. M. Reidys. An introduction to sequential dynamical systems. Springer Verlag, 2007.

[12]

• W. T. Tutte. A contribution to the theory of chromatic polynomials. Canad. J. Math. 6:80–91, 1954.

[13] J.-Y. Shi. Conjugacy relation on Coxeter elements. Adv. Math., 161, 2001:1–19.