Growth series of braid monoid Resolution of Z Other types of monoids Growth function for a class of monoids Marie ALBENQUE and Philippe NADEAU Formal Power Series and Algebraic Combinatorics July, 24th 2009
Growth series of braid monoid Resolution of Z Other types of monoids First motivation = counting braids braid diagram = a sequence of strand crossings. σ t , s = σ s , t ( s < t ) = crossing of strands s and t , where strand s is above strand t braid diagram = word on the alphabet { σ s , t } 8 7 6 5 4 3 2 1 σ 1 , 3 σ 3 , 6 σ 2 , 7 Figure: A braid diagram and the corresponding word
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams 6 6 5 5 4 4 3 3 2 2 1 1 σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Equivalent diagrams 6 6 5 5 4 4 3 3 2 2 1 1 σ 1 , 4 σ 4 , 6 ≡ σ 4 , 6 σ 1 , 6 Braid = equivalence class of diagrams.
Growth series of braid monoid Resolution of Z Other types of monoids Presentation of the dual braid monoid. The set of generators of M is : S = { σ s , t = σ t , s pour 1 ≤ s < t ≤ n , } with the following equivalence relations : σ s , t σ u , v = σ u , v σ s , t si s < s t < s u < s v , σ s , t σ t , u = σ t , u σ u , s si s < s t < s u . where < s = cyclic order Z / n Z defined by : s < s s + 1 < s s + 2 < s . . . < s s − 1 . Length of a braid = | m | S
Growth series of braid monoid Resolution of Z Other types of monoids Presentation of the dual braid monoid. The set of generators of M is : S = { σ s , t = σ t , s pour 1 ≤ s < t ≤ n , } with the following equivalence relations : σ s , t σ u , v = σ u , v σ s , t si s < s t < s u < s v , σ s , t σ t , u = σ t , u σ u , s si s < s t < s u . where < s = cyclic order Z / n Z defined by : s < s s + 1 < s s + 2 < s . . . < s s − 1 . Length of a braid = | m | S
Growth series of braid monoid Resolution of Z Other types of monoids Presentation of the dual braid monoid. The set of generators of M is : S = { σ s , t = σ t , s pour 1 ≤ s < t ≤ n , } with the following equivalence relations : σ s , t σ u , v = σ u , v σ s , t si s < s t < s u < s v , σ s , t σ t , u = σ t , u σ u , s si s < s t < s u . where < s = cyclic order Z / n Z defined by : s < s s + 1 < s s + 2 < s . . . < s s − 1 . Length of a braid = | m | S
Growth series of braid monoid Resolution of Z Other types of monoids How many braids ? a k = number of braids of length k � a k t k = a 0 + a 1 t + a 2 t 2 · · · F n ( t ) = k ≥ 0 Theorem (A., Nadeau ‘08) The growth function of the dual braid monoid on n strands is : � n − 1 � − 1 ( n − 1 + k )!( − t ) k � F n ( t ) = . ( n − 1 − k )! k !( k + 1 )! k = 0
Growth series of braid monoid Resolution of Z Other types of monoids Steps of the proofs Computation of the growth function of the monoid Involution Ψ Alternating generating series of lcm
Growth series of braid monoid Resolution of Z Other types of monoids A few definition about lcm σ ≺ m = there exists a diagram of m whose first letter is σ Definition J ⊂ S is a clique if it admits a common multiple. The set of cliques is denoted J If J ∈ J , then a least common multiple (lcm) exists, is unique and is denoted M J . We fix arbitrarily a linear ordering on S , and denote a clique as J = σ 1 . . . σ n , with σ i < σ i + 1
Growth series of braid monoid Resolution of Z Other types of monoids A few definition about lcm σ ≺ m = there exists a diagram of m whose first letter is σ Definition J ⊂ S is a clique if it admits a common multiple. The set of cliques is denoted J If J ∈ J , then a least common multiple (lcm) exists, is unique and is denoted M J . We fix arbitrarily a linear ordering on S , and denote a clique as J = σ 1 . . . σ n , with σ i < σ i + 1
Growth series of braid monoid Resolution of Z Other types of monoids A few definition about lcm σ ≺ m = there exists a diagram of m whose first letter is σ Definition J ⊂ S is a clique if it admits a common multiple. The set of cliques is denoted J If J ∈ J , then a least common multiple (lcm) exists, is unique and is denoted M J . We fix arbitrarily a linear ordering on S , and denote a clique as J = σ 1 . . . σ n , with σ i < σ i + 1
Growth series of braid monoid Resolution of Z Other types of monoids Theorem �� � � J ∈J ( − 1 ) | J | M J · ( m ∈ M m ) = 1 Corollary (Bronfman ’05, Kraamer ’05) The growth series of the monoid verifies then: �� J ∈J ( − 1 ) | J | t | M J | � F ( t ) = 1
Growth series of braid monoid Resolution of Z Other types of monoids A large class of monoids Our approach works for every monoid M which admits a presentation with generators and relations and which is: • atomic, • left-cancellable : a , u , v ∈ M , au = av ⇒ u = v , • if a subset of generators has a right common multiple then it has a least common multiple. [Bronfman, 00], [Krammer, 04] Trace monoids, Garside monoids, Artin-Tits monoids, . . . To get the growth series from the involution, the relations must besides preserve the length.
Growth series of braid monoid Resolution of Z Other types of monoids A large class of monoids Our approach works for every monoid M which admits a presentation with generators and relations and which is: • atomic, • left-cancellable : a , u , v ∈ M , au = av ⇒ u = v , • if a subset of generators has a right common multiple then it has a least common multiple. [Bronfman, 00], [Krammer, 04] Trace monoids, Garside monoids, Artin-Tits monoids, . . . To get the growth series from the involution, the relations must besides preserve the length.
Growth series of braid monoid Resolution of Z Other types of monoids A large class of monoids Our approach works for every monoid M which admits a presentation with generators and relations and which is: • atomic, • left-cancellable : a , u , v ∈ M , au = av ⇒ u = v , • if a subset of generators has a right common multiple then it has a least common multiple. [Bronfman, 00], [Krammer, 04] Trace monoids, Garside monoids, Artin-Tits monoids, . . . To get the growth series from the involution, the relations must besides preserve the length.
Growth series of braid monoid Resolution of Z Other types of monoids A large class of monoids Our approach works for every monoid M which admits a presentation with generators and relations and which is: • atomic, • left-cancellable : a , u , v ∈ M , au = av ⇒ u = v , • if a subset of generators has a right common multiple then it has a least common multiple. [Bronfman, 00], [Krammer, 04] Trace monoids, Garside monoids, Artin-Tits monoids, . . . To get the growth series from the involution, the relations must besides preserve the length.
Growth series of braid monoid Resolution of Z Other types of monoids Proof of the inversion formula �� � � � J ∈J ( − 1 ) | J | M J ( − 1 ) | J | M J m = 1 ( ∈ M m ) = ( J , m ) Ψ is an involution with only ( 1 , 1 ) as fixed point : Ψ : J × M → J × M ( J , m ) �→ ( J ′ , m ′ ) with M J m = M J ′ m ′ and | J ∆ J ′ | = 1 � � σ m = max σ such that σ ≺ M J m �� � J ∪ { σ m } , ( M J ∪{ σ m } ) − 1 · m if σ m / ∈ J Ψ( J , m ) = � J \{ σ m } , ( M J \{ σ m } ) − 1 M J · m � otherwise
Growth series of braid monoid Resolution of Z Other types of monoids Proof of the inversion formula �� � � � J ∈J ( − 1 ) | J | M J ( − 1 ) | J | M J m = 1 ( ∈ M m ) = ( J , m ) Ψ is an involution with only ( 1 , 1 ) as fixed point : Ψ : J × M → J × M ( J , m ) �→ ( J ′ , m ′ ) with M J m = M J ′ m ′ and | J ∆ J ′ | = 1 � � σ m = max σ such that σ ≺ M J m �� � J ∪ { σ m } , ( M J ∪{ σ m } ) − 1 · m if σ m / ∈ J Ψ( J , m ) = � J \{ σ m } , ( M J \{ σ m } ) − 1 M J · m � otherwise
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