Reduced words in Coxeter Groups Philippe Nadeau (CNRS & Univ. - - PowerPoint PPT Presentation

Reduced words in Coxeter Groups

Philippe Nadeau (CNRS & Univ. Lyon 1)

MADACA conference, June 21 2016

Introduction

Coxeter groups are ubiquitous structures in mathematics, which can be thought of as generalized reflection groups. They are presented with generators and relations, and we will be interested in minimal words used to represent elements. In this talk I will present some interplay (some of it conjectural) between these words and the roots coming from the geometric representation. This is a joint work with Christophe Hohlweg and Nathan Williams.

Coxeter group

S a finite set; M = (mst)s,t∈S a symmetric matrix. M must satisfy mss = 1 and mst ∈ {2, 3, . . .} ∪ {∞} Definition The Coxeter group W associated to M has generators S and relations (st)mst = 1 for all s, t ∈ S.

Coxeter group

S a finite set; M = (mst)s,t∈S a symmetric matrix. M must satisfy mss = 1 and mst ∈ {2, 3, . . .} ∪ {∞} Equivalently:    s2 = 1 sts · · ·

mst

= tst · · ·

mst

Braid relations Definition The Coxeter group W associated to M has generators S and relations (st)mst = 1 for all s, t ∈ S.

Coxeter group

S a finite set; M = (mst)s,t∈S a symmetric matrix. M must satisfy mss = 1 and mst ∈ {2, 3, . . .} ∪ {∞} Equivalently:    s2 = 1 sts · · ·

mst

= tst · · ·

mst

Braid relations Definition The Coxeter group W associated to M has generators S and relations (st)mst = 1 for all s, t ∈ S. Coxeter graph Labeled graph Γ encoding M, with vertices S, edge if mst ≥ 3, and label mst when mst ≥ 4. 4 5

s1 s2 s0

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

Families of Coxeter groups

• 1. Finite groups

These correspond exactly to finite reflection groups.

Families of Coxeter groups

• 1. Finite groups
• 2. Affine groups

These correspond exactly to finite reflection groups. These are essentially the groups of isometries generated by

• rthogonal affine reflections.

Families of Coxeter groups

• 1. Finite groups
• 2. Affine groups

These correspond exactly to finite reflection groups. These are essentially the groups of isometries generated by

• rthogonal affine reflections.

A complete classification exists for both families. Finite: An−1, Bn, Dn and I2(m), F4, H3, H4, E6, E7, E8. Affine: An−1, Bn, Cn, Dn and G2, F4, E6, E7, E8.

4 An Bn Dn I2(m) m H3 H4 E7 E6 E8 F4 4 5 5

Families of Coxeter groups

• 1. Finite groups
• 2. Affine groups
• 3. All other Coxeter groups

These correspond exactly to finite reflection groups. These are essentially the groups of isometries generated by

• rthogonal affine reflections.

These correspond to certain groups of linear transformations of Rn generated by reflections which are not orthogonal.

Families of Coxeter groups

• 1. Finite groups
• 2. Affine groups
• 3. All other Coxeter groups

These correspond exactly to finite reflection groups. These are essentially the groups of isometries generated by

• rthogonal affine reflections.

→ Study of sub families: right-angled groups, simply-laced groups, crystallographic groups, hyperbolic groups, . . . These correspond to certain groups of linear transformations of Rn generated by reflections which are not orthogonal.

Families of Coxeter groups

• 1. Finite groups
• 2. Affine groups
• 3. All other Coxeter groups

These correspond exactly to finite reflection groups. These are essentially the groups of isometries generated by

• rthogonal affine reflections.

→ Study of sub families: right-angled groups, simply-laced groups, crystallographic groups, hyperbolic groups, . . . These correspond to certain groups of linear transformations of Rn generated by reflections which are not orthogonal. Geometry: Every Coxeter group has a faithful geometric representation in σ : W → GL(⊕s∈SRαs) where:

• Each s ∈ S is a reflection through a hyperplane (s2 = 1);
• if s = t, st is a rotation of order mst ((st)mst = 1).

Triangle group T(2, 4, 5)

4 5

s1 s2 s0

s2

0 = s2 1 = s2 2 = 1

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

Triangle group T(2, 4, 5)

4 5

s1 s2 s0 s1 s2s0

s2

0 = s2 1 = s2 2 = 1

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

Triangle group T(2, 4, 5)

4 5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 1 1 2 2 1 2

s1 s2 s0

1 2

s1 s2s0

s2

0 = s2 1 = s2 2 = 1

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

Triangle group T(2, 4, 5)

4 5

s1 s2 s0 s1 s2s0

s2

0 = s2 1 = s2 2 = 1

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

Triangle group T(2, 4, 5)

4 5

s1 s2 s0 s1 s2s0

s2

0 = s2 1 = s2 2 = 1

s0s2 = s2s0 s0s1s0s1 = s1s0s1s0 s1s2s1s2s1 = s2s1s2s1s2

W ↔ Chambers Word ↔ Path

Length function

The geometric representation is thus a realization of the Cayley graph of (W, S) with vertices W and edges (w, ws).

Length function

Definition Length ℓ(w)= minimal l such that w = s1s2 . . . sl. The minimal words are the reduced decompositions of w. Example In type An−1 ≃ Sn, ℓ(w) is the number of inversions of the permutation w. The geometric representation is thus a realization of the Cayley graph of (W, S) with vertices W and edges (w, ws).

Length function

Definition Length ℓ(w)= minimal l such that w = s1s2 . . . sl. The minimal words are the reduced decompositions of w. The length measures the distance to the identity in the Cayley graph, and reduced words are in bijection with geodesics from the identity. Example In type An−1 ≃ Sn, ℓ(w) is the number of inversions of the permutation w. 2 2 1 1 1 s2s1s0s1s2s0s1s2 2 The geometric representation is thus a realization of the Cayley graph of (W, S) with vertices W and edges (w, ws).

Reduced words

→ Two natural combinatorial problems: (1) Structure of Red(w) for w ∈ W. (2) Structure of RedW. Let Red(w) be the set of reduced words for the element w, and RedW the union of Red(w) for all w ∈ W.

Reduced words

→ Two natural combinatorial problems: (1) Structure of Red(w) for w ∈ W. (2) Structure of RedW. Seminal work by Brink and Howlett in 1993: they prove that RedW is a regular language on the alphabet S, i.e. a subset

• f S∗ “recognized by a finite automaton”.

⇒

w |Red(w)|qℓ(w) is a rational function.

Let Red(w) be the set of reduced words for the element w, and RedW the union of Red(w) for all w ∈ W.

Reduced words

→ Two natural combinatorial problems: (1) Structure of Red(w) for w ∈ W. (2) Structure of RedW. Seminal work by Brink and Howlett in 1993: they prove that RedW is a regular language on the alphabet S, i.e. a subset

• f S∗ “recognized by a finite automaton”.

I will describe this result, and some recent work together with

• C. Hohlweg and N. Williams around various automata

recognizing RedW. ⇒

w |Red(w)|qℓ(w) is a rational function.

Let Red(w) be the set of reduced words for the element w, and RedW the union of Red(w) for all w ∈ W.

Reduced words

Theorem [Brink, Howlett ’93] RedW is a regular language.

Reduced words

Theorem [Brink, Howlett ’93] RedW is a regular language. Proof: RedW is recognized by a finite automaton ABH(W).

c Bj¨

• rner-Brenti

a b c

• A2

The automaton of Brink and Howlett

Small roots To a Coxeter group W is attached a set of vectors, the roots, on which W acts. The set Φ of roots is partitioned into negative and positive roots, Φ = Φ+ ⊔ Φ−. Fact The number |N(w)| of positive roots which are sent to a negative root by w ∈ W is equal to the length ℓ(w). The construction of ABH(W) is based on roots of W.

The automaton of Brink and Howlett

Small roots To a Coxeter group W is attached a set of vectors, the roots, on which W acts. The set Φ of roots is partitioned into negative and positive roots, Φ = Φ+ ⊔ Φ−. Fact The number |N(w)| of positive roots which are sent to a negative root by w ∈ W is equal to the length ℓ(w). We say that α dominates β in Φ+ if whenever wα ∈ Φ− then wβ ∈ Φ−. Define α to be small if it dominates no other positive root. The construction of ABH(W) is based on roots of W.

The automaton of Brink and Howlett

Small roots To a Coxeter group W is attached a set of vectors, the roots, on which W acts. The set Φ of roots is partitioned into negative and positive roots, Φ = Φ+ ⊔ Φ−. Fact The number |N(w)| of positive roots which are sent to a negative root by w ∈ W is equal to the length ℓ(w). We say that α dominates β in Φ+ if whenever wα ∈ Φ− then wβ ∈ Φ−. Define α to be small if it dominates no other positive root. Theorem [BH] The set Σ of small roots is finite. It is then easy to define ABH(W) recognizing RedW with states of the form N(w) ∩ Σ. The construction of ABH(W) is based on roots of W.

New automata for RedW

Definition (Weak order) We write x ≤ y if the equality ℓ(x) + ℓ(x−1y) = ℓ(y) holds. This means that x is on a geodesic from 1 to y.

New automata for RedW

Definition A ⊆ W is a Garside shadow if S ⊆ A, A is closed under joins (for ≤) and under taking suffixes. Define a function πA : W → A by πA(w) = {a ∈ A; a ≤ w} Definition (Weak order) We write x ≤ y if the equality ℓ(x) + ℓ(x−1y) = ℓ(y) holds. This means that x is on a geodesic from 1 to y.

New automata for RedW

Definition A ⊆ W is a Garside shadow if S ⊆ A, A is closed under joins (for ≤) and under taking suffixes. Define a function πA : W → A by πA(w) = {a ∈ A; a ≤ w} Automaton Let A(A) be the deterministic automaton on the alphabet S with states a for a ∈ A, initial state 1, all states final, and partial transition function δ defined by δ(a, s) = πA(sa) if ℓ(sa) > ℓ(a). Theorem[Hohlweg, N.,Williams] A(A) recognizes RedW . Definition (Weak order) We write x ≤ y if the equality ℓ(x) + ℓ(x−1y) = ℓ(y) holds. This means that x is on a geodesic from 1 to y.

New automata for RedW

Definition A ⊆ W is a Garside shadow if S ⊆ A, A is closed under joins (for ≤) and under taking suffixes. Define a function πA : W → A by πA(w) = {a ∈ A; a ≤ w} Automaton Let A(A) be the deterministic automaton on the alphabet S with states a for a ∈ A, initial state 1, all states final, and partial transition function δ defined by δ(a, s) = πA(sa) if ℓ(sa) > ℓ(a). Theorem[Hohlweg, N.,Williams] A(A) recognizes RedW . Definition (Weak order) We write x ≤ y if the equality ℓ(x) + ℓ(x−1y) = ℓ(y) holds. By a result of [Dehornoy,Dyer, Hohlweg ’14] there always exists a Garside shadow A = Alow(W) of finite cardinality. This means that x is on a geodesic from 1 to y.

New automata for RedW

ABH ։ A(Alow) ։ A( S) ։ Amin Given W, we can define four automata recognizing RedW. Here S is the smallest Garside shadow, i.e. the intersection of all Garside shadows, and Amin is the minimal automaton recognizing RedW.

New automata for RedW

ABH ։ A(Alow) ։ A( S) ։ Amin Given W, we can define four automata recognizing RedW. Here S is the smallest Garside shadow, i.e. the intersection of all Garside shadows, and Amin is the minimal automaton recognizing RedW. Conjecture 1 [Hohlweg,N.,Williams] The automata A( S) and Amin are isomorphic for any W.

New automata for RedW

ABH ։ A(Alow) ։ A( S) ։ Amin Given W, we can define four automata recognizing RedW. Here S is the smallest Garside shadow, i.e. the intersection of all Garside shadows, and Amin is the minimal automaton recognizing RedW. Conjecture 1 [Hohlweg,N.,Williams] The automata A( S) and Amin are isomorphic for any W. Let us say that a positive root β is spherical if the parabolic subgroup Wsupp(β) is finite. Let Φ+

sph be the set of these roots (one has always Φ+ sph ⊆ Σ).

New automata for RedW

ABH ։ A(Alow) ։ A( S) ։ Amin Given W, we can define four automata recognizing RedW. Here S is the smallest Garside shadow, i.e. the intersection of all Garside shadows, and Amin is the minimal automaton recognizing RedW. Conjecture 1 [Hohlweg,N.,Williams] The automata A( S) and Amin are isomorphic for any W. Let us say that a positive root β is spherical if the parabolic subgroup Wsupp(β) is finite. Let Φ+

sph be the set of these roots (one has always Φ+ sph ⊆ Σ).

Conjecture 2 [Hohlweg,N.,Williams] If W is irreducible, then ABH(W) is minimal if and only if Σ = Φ+

sph.

New automata for RedW

Conjecture 2 [Hohlweg,N.,Williams] If W is irreducible, then ABH(W) is minimal if and only if Σ = Φ+

sph.

We proved that the condition is sufficient.

New automata for RedW

The conjecture holds for all the following cases:

• 1. W is finite.
• 2. W is right-angled, i.e. mst = 2 or ∞ for all s = t
• 3. ΓW is a complete graph, i.e. mst > 2 for all s = t.
• 4. W is of type

An−1.

• 5. W has rank 3.
• 6. W has rank 4 and mst ≤ 10 for all s, t.

Conjecture 2 [Hohlweg,N.,Williams] If W is irreducible, then ABH(W) is minimal if and only if Σ = Φ+

sph.

We proved that the condition is sufficient.

New automata for RedW

The conjecture holds for all the following cases:

• 1. W is finite.
• 2. W is right-angled, i.e. mst = 2 or ∞ for all s = t
• 3. ΓW is a complete graph, i.e. mst > 2 for all s = t.
• 4. W is of type

An−1.

• 5. W has rank 3.
• 6. W has rank 4 and mst ≤ 10 for all s, t.

Conjecture 2 [Hohlweg,N.,Williams] If W is irreducible, then ABH(W) is minimal if and only if Σ = Φ+

sph.

We proved that the condition is sufficient. Point 6. was checked by computer, which we used to also check Conjecture 1 in such cases.