Coxeter Groups and Root Systems via Automatic Structuresedgartj/automatic.pdf · Coxeter Groups are...

Introduction Automata Groups Coxeter Groups Regularity Questions

Coxeter Groups and Root Systems viaAutomatic Structures

Tom Edgar

Department of MathematicsPacific Lutheran University

Tacoma, WA

Colloquium, February 2, 2012Western Washington University


Introduction

Coxeter Groups - Important class of finitely generated groups

Finite Group Theory: Coxeter groups↔ reflection groups- dihedral groups- symmetric group

Representation Theory: Weyl groups- semi-simple Lie algebras- Lie group theory- E8

CombinatoricsAutomatic Group Theory


Introduction

Finite State Automata and Regular Languages

Automatic Groups - Definitions and Examples

Coxeter Groups, Root Systems, and some Automata

Deeper Regularity Properties and Consequences

Further Questions and Open Problems


Finite State Automata

Let S be a finite set (called an alphabet).

DefinitionA finite state automaton for the alphabet S is a tuple(A,a0, µ,Af ) where

A is a finite set (called the states of the automaton)

a0 ∈ A (called the initial state)

µ : S × A→ A (called the transition function)

Af ⊆ A (called the final states or accept states)


Examples

ExampleS = {s} A = {•, •} a0 = • Af = {•}

µ is best described by the edges of a graph (shown below):

• •s // see

Remarks:1. Vertices←→ A. Each vertex should have one originating,

directed, labeled edge for each element of S.2. We often write s · a instead of µ(s,a).


Examples

ExampleS = {r , s} A = {•, •, •, •, •} a0 = • Af = {•, •}

Again, we describe µ by a graph:

• • • •

•

s // s // r //

r ,s

ZZ

r

��??????????????

r

��''''''''''

s

��

r

||yyyyyyyyyyyyyyyy

r

||


Regular Languages

Let S be a finite alphabet and A = (A,a0, µ,Af ).

1. S∗ is the set of all words on S (including the empty word).2. L ⊆ S∗ is called a language.3. We can easily extend µ so that µ : S∗ × A→ A.4. Let w = s1...sn ∈ S∗. We say A accepts w if

w · a0 := µ(w ,a0) ∈ Af .

5. Let L(A) ⊆ S∗ to be the collection of words accepted by A.6. A language L ⊆ S∗ is regular over S if L = L(A) for some

finite state automaton A.


Examples, Revisited

ExampleS = {s} A = {•, •} a0 = • Af = {•}

µ from graph:

• •s // see

L(A) = {sm | m ≥ 1} = S∗ \ {ε}


Examples

ExampleS = {r , s} A = {•, •, •, •, •} a0 = • Af = {•, •}

µ from graph:

• • • •

•

s // s // r //

r ,s

ZZ

r

��??????????????

r

��''''''''''

s

��

r

||yyyyyyyyyyyyyyyy

s

||

L(A) = {(ssr)m | m ≥ 1} ∪ {(ssr)mss | m ≥ 0}


More Examples

ExampleLet S = {r , s}. Define L = {(sr)n | n ∈ N}. Then L is regular.

• ••

•

s //

r��??????????

r88

r ,s

ZZ

sxx

s

��r

��

ExampleS = {r , s}. Define L = {snrn | n ∈ N}. Then L is not regular.


Groundhog Day

Let S = {s, n} and A = {F2,EW ,B}.

Then if a0 = F2 and Af = {EW} and µ is given by

F2B EW

•

soo n //

n,s

zzttttttttttttttttt

n,s

n,s

$$JJJJJJJJJJJJJJJJJJ

Then L(A) = {n}.

Point: Any finite language is regular.


Closure

TheoremIf L and K are both regular languages over S, then L ∩ K ,L ∪ K , and Lc are all regular.

Proof Idea.Let A = (A,a0, µ,Af ) and B = (B,b0, ν,Bf ) be the FSAs for Land K respectively.

For Lc use (A,a0, µ,A \ Af )

For L ∪ K use (A× B, (a0,b0), µ× ν, (A× Bf ) ∪ (Af × B))

For L ∩ K use (A× B, (a0,b0), µ× ν,Af × Bf )

Note: L ∩ K = (Lc ∪ K c)c by DeMorgan’s Laws.


Finitely Generated Groups

DefinitionA group is a set G with a operation such that

1. Operation is associative: (ab)c = a(bc) for all triples

2. There is an identity element: 1g = g1 = g for all g

3. Every element has an inverse: g−1g = gg−1 = 1.

DefinitionWe say G is generated by S ⊆ G if every element of G can bewritten as a product of elements in S ∪ S−1. We write G = 〈S〉.

DefinitionWe say G = 〈S〉 is finitely generated if S is finite. (G = 〈S | R〉).


Finitely Generated Groups

ExampleZ = 〈1〉Zn = 〈1 | 1 + 1 + · · ·+ 1︸︷︷︸

n

= 0〉

ExampleDn = 〈r , s | rn = s2 = 1; rs = sr−1〉

= 〈a,b | a2 = b2 = 1; (ab)n = 1〉

Example

Sn =

⟨{s1, ..., sn−1}

∣∣∣∣ s2i =1;

si si+1si =si+1si si+1;si sj =sj si (|i − j| 6= 1)

⟩


Automatic Groups

Let G be a finitely generated group G = 〈S〉.

DefinitionAn automatic structure on G consists of a finite stateautomaton A over S and a finite state automataMx over (S,S)for each x ∈ S ∪ {ε}, satisfying:

1. There is a surjective map π : L(A)→ G.2. (w1,w2) ∈ L(Mx ) if and only if w1x = w2 in G.

Remarks:A is called the word acceptor.Mx are called the multiplier (and equality) automata.Automata over (S,S) can be made precise (more time).Above, we abuse notation by letting wi ∈ S∗ and wi ∈ G.


Examples

Suppose G is finite. Then G = 〈G〉 we define µ(g,h) = gh.

A = (G,1, µ,G) is the word acceptor for G.Multipliers are obtained by modifying A.

ExampleLet G = Z3

0 1

2

1

332

ss

2&&

0

1ff

1

�� 2

FF

0

0

Note: In general, for S ⊆ G, you can use the Cayley graph for (G,S).


Examples

Example (Integers)Z = 〈{1,−1}〉. Let A = {•, •, •, •}, a0 = •, Af = {•, •, •} andsuppose that µ is described by the graph

•• •

•

1 //−1oo 1−1

1 ''OOOOOOOOOOO

−1wwooooooooooo

1,−1

This recognizes reduced expressions of integers.

Multiplier automata can be built from this (length).


ExamplesExample (Free Group)Let F2 = 〈{x , y , x−1, y−1}〉.

• ••

•

•

x//

x−1oo

y

��

y−1

OO

x

y��

y−1__?????????

x−1

y��?????????

y−1??��

y

x

TT

x−1

JJ

y−1

x

x−1

��

Missing edges go to a fail state.

Word acceptor shown, can be modified to multipliers.


Coxeter GroupsLet S = {s1, ..., sn}.

DefinitionAn n × n Coxeter matrix is a matrix m satisfying

mi,j = mj,i

mi,i = 1mi,j ∈ Z≥2 ∪ {∞} for i 6= j .

Given m, we let W = 〈S | (sisj)mi,j = 1}

We call (W ,S) a Coxeter system

Notes:mi,i = 1 means s2

i = 1mi,j = 2 means sisj = sjsi

mi,j = n means sisj ...︸︷︷︸n times

= sjsi ...︸︷︷︸n times


Examples

Example (Dihedral Groups)

Let S = {a,b} and m =

(1 nn 1

). Then

Dn = 〈a,b | a2 = b2 = 1; (ab)n = 1〉

Remark: a and b are both reflections of n-gon.


Examples

Example (Symmetric Groups)

Let S = {s1, ..., sn−1}. Define m by letting mi,i = 1 and for i 6= j

mi,j =

{2 |i − j | 6= 13 otherwise

.

Then

Sn =

⟨s1, ..., sn−1

∣∣∣∣ s2i =1;

si si+1si =si+1si si+1;si sj =sj si (|i − j| 6= 1)

⟩Remark: si = (i , i + 1) is a simple transpositions.


Examples

Example (Infinite Case)

Let S = {r , s,u} and define the Coxeter matrix by

m =

1 3 43 1 ∞4 ∞ 1

Then

W = 〈r , s,u | r2 = s2 = u2 = 1; rsr = srs; ruru = urur〉

Remark: ∞ means there is no relation.


Facts and Terminology

Every element of S∗ gives element in W .

There is a length function ` : W → N.

w = si1 · · · sir is a reduced expression for w if `(w) = r .i.e. D4, w = ababbaabab = abaaabab = abababso `(w) = 6 and ababab is a reduced expression.

There is a classification of the finite Coxeter groups by matrixFour infinite families (A,B,D, I2)Six sporadic


Coxeter Groups are Automatic

How can we show that Coxeter groups are automatic?i.e. infinite Coxeter groups

Develop some combinatorial/geometric data

Use the data to build a word acceptorRecognize reduced expressionsFind one that recognizes some unique reduced expressionModify to build the multiplier automata

What good does this do us?


Root Systems

Fix a Coxeter system (W ,S) with |S| = n.

Let V be an n-dimensional R-vector spaceChoose an S-indexed basis Π := {αs | s ∈ S} for VDefine bilinear form, (− | −), on V by

(αsi | αsj ) =

{− cos

(π

mi,j

)mi,j 6=∞

bi,j ≤ −1 o.w.

For each s ∈ S, σs : V → V given by

σs(β) = β − 2(αs | β)αs

Extend to action of W on V : w 7→ σw ∈ End(V )

We call this the standard reflection representation.


Root SystemsAbuse notation, for β ∈ V , w(β) := σw (β)

DefinitionThe root system of (W ,S) is Φ = {w(αs) | w ∈W ; αs ∈ Π}.

Φ+ := {β ∈ Φ | β =∑si∈S

ciαsi ; ci ≥ 0}

Φ− := {β ∈ Φ | β =∑si∈S

ciαsi ; ci ≤ 0}

Fact: −Φ+ = Φ−

Fact: Φ = Φ+ t Φ−

DefinitionFor any w ∈W , define Φw = {β ∈ Φ+ | w(β) ∈ Φ−}.

Fact: Let s ∈ S. Then `(ws) > `(w) if and only if αs 6∈ Φw


A Finite State Automaton for a Coxeter Group

DefinitionLet α, β ∈ Φ+. We say α dominates β if, for all w ∈W ,w(α) ∈ Φ− implies that w(β) ∈ Φ− (write β � α).

TheoremThe relation � is a partial order on Φ+.

TheoremLet Σ be the minimal elements of Φ+ with respect to �. Then Σis finite.

DefinitionFor any w ∈W , we define DΣ(w) = Φw ∩ Σ.Note: A = {DΣ(w) | w ∈W} is finite.


A Finite State Automaton for a Coxeter Group

TheoremLet w ∈W and s ∈ S with αs 6∈ DΣ(w). Then

DΣ(ws) = {αs} ∪(s(DΣ(w)) ∩ Σ

)TheoremCoxeter groups are automatic.

Proof.Let Af = {DΣ(w) | w ∈W}, A = Af ∪ {F}, a0 = DΣ(1) and

µ(s,B) =

{αs} ∪ (s(B) ∩ Σ) B ∈ Af ; αs 6∈ BF B ∈ Af ;αs ∈ BF B = F


Regular Subsets

We can generalize the set Σ to

Σn = {β ∈ Φ+ | β dominates n positive roots}

TheoremFor every n, Σn is finite.

This creates a large family of automata for a Coxeter group

DΣn (w) = Φw ∩ Σn.Fix k , can recognize words with k -reductions.Important subsets of Coxeter groups.


Regular Subsets

Examples of Regular Subsets

1. For J ⊆ S we get WJ (standard parabolic subgroup)

2. W ′ reflection subgroup, W ′\W/WJ (minimal coset reps)

3. For α, β ∈ Φ, {w ∈W | w(α) = β}

4. For K ⊆ S, {w ∈W | `(wr) < `(w) for all r ∈ K}5. For t = utu−1 be a reflection

{w ∈W | `(tw) < `(w)} (half space)CW (t) (centralizer in W )

6. Many others including finite unions, intersections, andcomplements.


Poincare Series

DefinitionFor a subset U ⊆W we define the Poincare series of U to be

P(U; W ) =∑w∈U

q`(w).

TheoremIf U is a regular subset of W, then P(U; W ) has a rationalexpression.

Remark: Allows us to effectively count elements of U of a fixedlength.


Open Questions in Coxeter groups

1. How big is Σ (or Σn)?

2. Can you effectively construct and use these automata?

3. Which subgroups regular (in particular are reflectionsubgroups regular)?

4. What are the representation-theoretic implications?(Lusztig’s a-function, etc.)


Biautomatic Groups and Associated Problems

DefinitionA group is biautomatic if it is automatic and has a family of leftmultiplier automata as well.

Automatic groups have solvable word problem.

Biautomatic groups have solvable conjugacy problem.

Questions:1. Are Coxeter groups biautomatic?2. Is there an automatic group that is not biautomatic?3. Does solvable conjugacy problem imply biautomaticity?

(word problem 6⇒ automatic)4. Do automatic groups have solvable conjugacy problem?


Artin Groups

DefinitionLet S be a finite set and m be a Coxeter matrix. Define

W = 〈S | sisj ...︸︷︷︸mi,j

= sjsi ...︸︷︷︸mi,j

when i 6= j}.

W is called an Artin group.Questions:

1. Are all Artin groups automatic?2. Are all Artin groups biautomatic?


Questions

Thanks!

REFERENCES

Epstein, D, et al. Word Processing in Groups.

Humphreys, J. Reflection Groups and Coxeter Groups.

Bjorner and Brenti. Combinatorics of Coxeter Groups.

Edgar, T. Dominance and Regularity in Coxeter Groups.


Regular Subset ExampleLet W = B2, given by m =

1 4 24 1 42 4 1

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