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

Coxeter group S a finite set; M = ( m st ) s,t ∈ S a symmetric matrix. M must satisfy m ss = 1 and m st ∈ { 2 , 3 , . . . } ∪ {∞} Definition The Coxeter group W associated to M has generators S and relations ( st ) m st = 1 for all s, t ∈ S .  s 2 = 1  Equivalently: sts · · · = tst · · · Braid relations  � �� � � �� � m st m st

Coxeter group S a finite set; M = ( m st ) s,t ∈ S a symmetric matrix. M must satisfy m ss = 1 and m st ∈ { 2 , 3 , . . . } ∪ {∞} Definition The Coxeter group W associated to M has generators S and relations ( st ) m st = 1 for all s, t ∈ S .  s 2 = 1  Equivalently: sts · · · = tst · · · Braid relations  � �� � � �� � m st m st Coxeter graph Labeled graph Γ encoding M , with vertices S , edge if m st ≥ 3 , and label m st when m st ≥ 4 . s 0 s 2 = s 2 s 0 4 5 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 s 0 s 1 s 2 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups.

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups. 2. Affine groups These are essentially the groups of isometries generated by orthogonal affine reflections.

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups. 2. Affine groups These are essentially the groups of isometries generated by orthogonal affine reflections. A complete classification exists for both families. Finite : A n − 1 , B n , D n and I 2 ( m ) , F 4 , H 3 , H 4 , E 6 , E 7 , E 8 . Affine : � A n − 1 , � B n , � C n , � D n and � G 2 , � F 4 , � E 6 , � E 7 , � E 8 . 4 A n D n B n m 4 5 5 I 2 ( m ) F 4 H 3 H 4 E 6 E 7 E 8

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups. 2. Affine groups These are essentially the groups of isometries generated by orthogonal affine reflections. 3. All other Coxeter groups These correspond to certain groups of linear transformations of R n generated by reflections which are not orthogonal.

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups. 2. Affine groups These are essentially the groups of isometries generated by orthogonal affine reflections. 3. All other Coxeter groups These correspond to certain groups of linear transformations of R n generated by reflections which are not orthogonal. → Study of sub families: right-angled groups, simply-laced groups, crystallographic groups, hyperbolic groups, . . .

Families of Coxeter groups 1. Finite groups These correspond exactly to finite reflection groups. 2. Affine groups These are essentially the groups of isometries generated by orthogonal affine reflections. 3. All other Coxeter groups These correspond to certain groups of linear transformations of R n generated by reflections which are not orthogonal. → Study of sub families: right-angled groups, simply-laced groups, crystallographic groups, hyperbolic groups, . . . Geometry: Every Coxeter group has a faithful geometric representation in σ : W → GL ( ⊕ s ∈ S R α s ) where: • Each s ∈ S is a reflection through a hyperplane ( s 2 = 1 ); • if s � = t , st is a rotation of order m st ( ( st ) m st = 1 ).

Triangle group T (2 , 4 , 5) 4 5 s 0 s 1 s 2 s 2 0 = s 2 1 = s 2 2 = 1 s 0 s 2 = s 2 s 0 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2

Triangle group T (2 , 4 , 5) 4 5 s 0 s 1 s 2 s 2 0 = s 2 1 = s 2 2 = 1 s 0 s 2 = s 2 s 0 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 s 1 s 2 s 0 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2

Triangle group T (2 , 4 , 5) 4 5 s 0 s 1 s 2 0 2 1 2 0 0 1 s 2 0 = s 2 1 = s 2 0 2 = 1 0 1 0 0 1 0 0 1 0 2 2 s 0 s 2 = s 2 s 0 0 1 1 2 1 0 1 1 2 1 0 1 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 1 2 2 2 s 1 2 1 2 2 0 2 s 2 s 0 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2 0 1 1 0 1 2 1 1 0 2 0 0 0 0 0 0 0 0 1 1 2 2 1 0 1 2 0 2 1 1 2 1 2 2 2 2 2 2 1 2 0 1 0 2 1 1 0 1 0 2 0 1 2 1 0 0 0 2 1 0 0 0 0

Triangle group T (2 , 4 , 5) 4 5 s 0 s 1 s 2 s 2 0 = s 2 1 = s 2 2 = 1 s 0 s 2 = s 2 s 0 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 s 1 s 2 s 0 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2

Triangle group T (2 , 4 , 5) 4 5 s 0 s 1 s 2 s 2 0 = s 2 1 = s 2 2 = 1 s 0 s 2 = s 2 s 0 s 0 s 1 s 0 s 1 = s 1 s 0 s 1 s 0 s 1 s 2 s 0 s 1 s 2 s 1 s 2 s 1 = s 2 s 1 s 2 s 1 s 2 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 The geometric representation is thus a realization of the Cayley graph of ( W, S ) with vertices W and edges ( w, ws ) . Definition Length ℓ ( w )= minimal l such that w = s 1 s 2 . . . s l . The minimal words are the reduced decompositions of w . Example In type A n − 1 ≃ S n , ℓ ( w ) is the number of inversions of the permutation w .

Length function The geometric representation is thus a realization of the Cayley graph of ( W, S ) with vertices W and edges ( w, ws ) . Definition Length ℓ ( w )= minimal l such that w = s 1 s 2 . . . s l . The minimal words are the reduced decompositions of w . Example In type A n − 1 ≃ S n , ℓ ( w ) is the number of inversions of the permutation w . The length measures the distance to the identity in the Cayley graph, and reduced words are in bijection with geodesics from the identity. 1 0 1 2 2 0 s 2 s 1 s 0 s 1 s 2 s 0 s 1 s 2 1 2

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

Reduced words Let Red ( w ) be the set of reduced words for the element w , and Red W the union of Red ( w ) for all w ∈ W . → Two natural combinatorial problems: (1) Structure of Red ( w ) for w ∈ W . (2) Structure of Red W . Seminal work by Brink and Howlett in 1993: they prove that Red W is a regular language on the alphabet S , i.e. a subset of S ∗ “recognized by a finite automaton”. ⇒ � w | Red ( w ) | q ℓ ( w ) is a rational function.

Reduced words Let Red ( w ) be the set of reduced words for the element w , and Red W the union of Red ( w ) for all w ∈ W . → Two natural combinatorial problems: (1) Structure of Red ( w ) for w ∈ W . (2) Structure of Red W . Seminal work by Brink and Howlett in 1993: they prove that Red W is a regular language on the alphabet S , i.e. a subset of S ∗ “recognized by a finite automaton”. ⇒ � w | Red ( w ) | q ℓ ( w ) is a rational function. I will describe this result, and some recent work together with C. Hohlweg and N. Williams around various automata recognizing Red W .

Reduced words Theorem [Brink, Howlett ’93] Red W is a regular language.

Reduced words Theorem [Brink, Howlett ’93] Red W is a regular language. Proof : Red W is recognized by a finite automaton A BH ( W ) . a b c � A 2 c � Bj¨ orner-Brenti

The automaton of Brink and Howlett The construction of A BH ( W ) is based on roots of W . 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 automaton of Brink and Howlett The construction of A BH ( W ) is based on roots of W . 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.