Factorisations of a group element, Hurwitz action and shellability Vivien Ripoll University of Vienna, Austria Combinatorial Algebra meets Algebraic Combinatorics London, Ontario 2016, January 24th uhle (´ joint work with Henri M¨ Ecole Polytechnique, France)
Outline Framework and example: 1 generated group, Hurwitz action, factorisations, shellability Motivations: 2 noncrossing partition lattices of reflection groups Some results and a conjecture: 3 compatible order on the generators, Hurwitz-transitivity, shellability Vivien Ripoll Factorisations, Hurwitz action and shellability
Outline Framework and example: 1 generated group, Hurwitz action, factorisations, shellability Motivations: 2 noncrossing partition lattices of reflection groups Some results and a conjecture: 3 compatible order on the generators, Hurwitz-transitivity, shellability Vivien Ripoll Factorisations, Hurwitz action and shellability
Generated group and reduced decompositions ( G , A ) generated group A ⊆ G generates G as a monoid Let g ∈ G . Write g = a 1 a 2 . . . a n , with a i ∈ A . Length of g : ℓ A ( g ) := minimal such n . Reduced decompositions of g Red A ( g ) := { ( a 1 , . . . , a n ) | a i ∈ A , g = a 1 . . . a n } , where n = ℓ A ( g ) . Example. G = S 4 A = T := { all transpositions ( i j ) } . g = (1 2 3 4) ℓ T ( g ) = 3 Reduced decompositions of g : g = (12)(23)(34) = (23)(13)(34) = (13)(12)(34) = (13)(34)(12) = (14)(13)(12) = (34)(14)(12) = (34)(12)(24) = (34)(24)(14) = (24)(23)(14) = (23)(34)(14) = (23)(14)(13) = (12)(34)(24) = (12)(24)(23) = (24)(14)(23) = (14)(12)(23) = (14)(23)(13) Vivien Ripoll Factorisations, Hurwitz action and shellability
Generated group and reduced decompositions ( G , A ) generated group A ⊆ G generates G as a monoid Let g ∈ G . Write g = a 1 a 2 . . . a n , with a i ∈ A . Length of g : ℓ A ( g ) := minimal such n . Reduced decompositions of g Red A ( g ) := { ( a 1 , . . . , a n ) | a i ∈ A , g = a 1 . . . a n } , where n = ℓ A ( g ) . Example. G = S 4 A = T := { all transpositions ( i j ) } . g = (1 2 3 4) ℓ T ( g ) = 3 Reduced decompositions of g : g = (12)(23)(34) = (23)(13)(34) = (13)(12)(34) = (13)(34)(12) = (14)(13)(12) = (34)(14)(12) = (34)(12)(24) = (34)(24)(14) = (24)(23)(14) = (23)(34)(14) = (23)(14)(13) = (12)(34)(24) = (12)(24)(23) = (24)(14)(23) = (14)(12)(23) = (14)(23)(13) Vivien Ripoll Factorisations, Hurwitz action and shellability
Generated group and reduced decompositions ( G , A ) generated group A ⊆ G generates G as a monoid Let g ∈ G . Write g = a 1 a 2 . . . a n , with a i ∈ A . Length of g : ℓ A ( g ) := minimal such n . Reduced decompositions of g Red A ( g ) := { ( a 1 , . . . , a n ) | a i ∈ A , g = a 1 . . . a n } , where n = ℓ A ( g ) . Example. G = S 4 A = T := { all transpositions ( i j ) } . g = (1 2 3 4) ℓ T ( g ) = 3 Reduced decompositions of g : g = (12)(23)(34) = (23)(13)(34) = (13)(12)(34) = (13)(34)(12) = (14)(13)(12) = (34)(14)(12) = (34)(12)(24) = (34)(24)(14) = (24)(23)(14) = (23)(34)(14) = (23)(14)(13) = (12)(34)(24) = (12)(24)(23) = (24)(14)(23) = (14)(12)(23) = (14)(23)(13) Vivien Ripoll Factorisations, Hurwitz action and shellability
Hurwitz action Hurwitz moves Fix g ∈ G . Take ( a 1 , . . . , a n ) ∈ Red A ( g ). For 1 ≤ i ≤ n − 1 define: σ i · ( a 1 , . . . , a i − 1 , a i , a i +1 , a i +2 , . . . , a n ) a i a i +1 a − 1 = ( a 1 , . . . , a i − 1 , , a i , a i +2 , . . . , a n ) i Assumption: For any ( a 1 , . . . , a n ) ∈ Red A ( g ) and any 1 ≤ i ≤ n − 1, a i a i +1 a − 1 and a − 1 i +1 a i a i +1 ∈ A . (e.g., A stable by conjugacy) i This defines an action on Red A ( g ) by the braid group B n [Hurwitz action]. B n = � σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i if | i − j | > 1 � grp � General Question 1: Is the Hurwitz action transitive on Red A ( g )? Vivien Ripoll Factorisations, Hurwitz action and shellability
Hurwitz action Hurwitz moves Fix g ∈ G . Take ( a 1 , . . . , a n ) ∈ Red A ( g ). For 1 ≤ i ≤ n − 1 define: σ i · ( a 1 , . . . , a i − 1 , a i , a i +1 , a i +2 , . . . , a n ) a i a i +1 a − 1 = ( a 1 , . . . , a i − 1 , , a i , a i +2 , . . . , a n ) i Assumption: For any ( a 1 , . . . , a n ) ∈ Red A ( g ) and any 1 ≤ i ≤ n − 1, a i a i +1 a − 1 and a − 1 i +1 a i a i +1 ∈ A . (e.g., A stable by conjugacy) i This defines an action on Red A ( g ) by the braid group B n [Hurwitz action]. B n = � σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i if | i − j | > 1 � grp � General Question 1: Is the Hurwitz action transitive on Red A ( g )? Vivien Ripoll Factorisations, Hurwitz action and shellability
Hurwitz action Hurwitz moves Fix g ∈ G . Take ( a 1 , . . . , a n ) ∈ Red A ( g ). For 1 ≤ i ≤ n − 1 define: σ i · ( a 1 , . . . , a i − 1 , a i , a i +1 , a i +2 , . . . , a n ) a i a i +1 a − 1 = ( a 1 , . . . , a i − 1 , , a i , a i +2 , . . . , a n ) i Assumption: For any ( a 1 , . . . , a n ) ∈ Red A ( g ) and any 1 ≤ i ≤ n − 1, a i a i +1 a − 1 and a − 1 i +1 a i a i +1 ∈ A . (e.g., A stable by conjugacy) i This defines an action on Red A ( g ) by the braid group B n [Hurwitz action]. B n = � σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i if | i − j | > 1 � grp � General Question 1: Is the Hurwitz action transitive on Red A ( g )? Vivien Ripoll Factorisations, Hurwitz action and shellability
Hurwitz action Hurwitz moves Fix g ∈ G . Take ( a 1 , . . . , a n ) ∈ Red A ( g ). For 1 ≤ i ≤ n − 1 define: σ i · ( a 1 , . . . , a i − 1 , a i , a i +1 , a i +2 , . . . , a n ) a i a i +1 a − 1 = ( a 1 , . . . , a i − 1 , , a i , a i +2 , . . . , a n ) i Assumption: For any ( a 1 , . . . , a n ) ∈ Red A ( g ) and any 1 ≤ i ≤ n − 1, a i a i +1 a − 1 and a − 1 i +1 a i a i +1 ∈ A . (e.g., A stable by conjugacy) i This defines an action on Red A ( g ) by the braid group B n [Hurwitz action]. B n = � σ 1 , . . . , σ n − 1 | σ i σ i +1 σ i = σ i +1 σ i σ i +1 , σ i σ j = σ j σ i if | i − j | > 1 � grp � General Question 1: Is the Hurwitz action transitive on Red A ( g )? Vivien Ripoll Factorisations, Hurwitz action and shellability
� � Example: Hurwitz graph of Red T (1 2 3 4) 12 | 24 | 23 12 | 23 | 34 12 | 34 | 24 14 | 12 | 23 24 | 14 | 23 13 | 12 | 34 23 | 13 | 34 14 | 23 | 13 23 | 14 | 13 14 | 13 | 12 13 | 34 | 12 24 | 23 | 14 23 | 34 | 14 34 | 12 | 24 34 | 14 | 12 34 | 24 | 14
Factorisation poset Prefix order Equip G with a partial order ≤ A : x ≤ A y ⇔ x is a prefix of a reduced decomposition of y ℓ A ( x ) + ℓ A ( x − 1 y ) = ℓ A ( y ) ⇔ Factorisation poset of g [ e , g ] A := { x ∈ G | x ≤ A g } [ e , g ] A is a graded poset (by ℓ A ); Hasse diagram of the poset [ e , g ] A corresponds to geodesics from e to g in the Cayley graph of ( G , A ); for x , y ∈ [ e , g ] A : x ≤ A y if and only if a reduced decomposition of x is a subword of a reduced decomposition of y . [by assumption on conjugacy-stability] Vivien Ripoll Factorisations, Hurwitz action and shellability
Factorisation poset Prefix order Equip G with a partial order ≤ A : x ≤ A y ⇔ x is a prefix of a reduced decomposition of y ℓ A ( x ) + ℓ A ( x − 1 y ) = ℓ A ( y ) ⇔ Factorisation poset of g [ e , g ] A := { x ∈ G | x ≤ A g } [ e , g ] A is a graded poset (by ℓ A ); Hasse diagram of the poset [ e , g ] A corresponds to geodesics from e to g in the Cayley graph of ( G , A ); for x , y ∈ [ e , g ] A : x ≤ A y if and only if a reduced decomposition of x is a subword of a reduced decomposition of y . [by assumption on conjugacy-stability] Vivien Ripoll Factorisations, Hurwitz action and shellability
Factorisation poset Prefix order Equip G with a partial order ≤ A : x ≤ A y ⇔ x is a prefix of a reduced decomposition of y ℓ A ( x ) + ℓ A ( x − 1 y ) = ℓ A ( y ) ⇔ Factorisation poset of g [ e , g ] A := { x ∈ G | x ≤ A g } [ e , g ] A is a graded poset (by ℓ A ); Hasse diagram of the poset [ e , g ] A corresponds to geodesics from e to g in the Cayley graph of ( G , A ); for x , y ∈ [ e , g ] A : x ≤ A y if and only if a reduced decomposition of x is a subword of a reduced decomposition of y . [by assumption on conjugacy-stability] Vivien Ripoll Factorisations, Hurwitz action and shellability
Factorisation poset Prefix order Equip G with a partial order ≤ A : x ≤ A y ⇔ x is a prefix of a reduced decomposition of y ℓ A ( x ) + ℓ A ( x − 1 y ) = ℓ A ( y ) ⇔ Factorisation poset of g [ e , g ] A := { x ∈ G | x ≤ A g } [ e , g ] A is a graded poset (by ℓ A ); Hasse diagram of the poset [ e , g ] A corresponds to geodesics from e to g in the Cayley graph of ( G , A ); for x , y ∈ [ e , g ] A : x ≤ A y if and only if a reduced decomposition of x is a subword of a reduced decomposition of y . [by assumption on conjugacy-stability] Vivien Ripoll Factorisations, Hurwitz action and shellability
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