Geometry of a uniform minimal factorization Paul Thevenin CMAP, - PowerPoint PPT Presentation

Geometry of a uniform minimal factorization Paul Thevenin CMAP, Ecole Polytechnique June 14, 2019 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 1 / 32

Minimal factorizations 1 Definitions Geometrical coding of a factorization Connection with the Brownian excursion 2 Conclusion 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 2 / 32

Minimal factorizations Minimal factorizations 1 Definitions Geometrical coding of a factorization Connection with the Brownian excursion 2 Conclusion 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 3 / 32

Minimal factorizations Definitions Minimal factorizations 1 Definitions Geometrical coding of a factorization Connection with the Brownian excursion 2 Conclusion 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 4 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . The n-cycle is the permutation c n := (1 2 · · · n ). Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . The n-cycle is the permutation c n := (1 2 · · · n ). Minimal factorization of the n -cycle : an ordered ( n − 1)-tuple of transpositions whose product is c n . Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . The n-cycle is the permutation c n := (1 2 · · · n ). Minimal factorization of the n -cycle : an ordered ( n − 1)-tuple of transpositions whose product is c n . ( τ 1 , ..., τ n − 1 ) ∈ T n − 1 � , τ 1 · · · τ n − 1 = (1 2 · · · n ) � ⊂ S n M n := n is the set of minimal factorizations of c n . Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . The n-cycle is the permutation c n := (1 2 · · · n ). Minimal factorization of the n -cycle : an ordered ( n − 1)-tuple of transpositions whose product is c n . ( τ 1 , ..., τ n − 1 ) ∈ T n − 1 � , τ 1 · · · τ n − 1 = (1 2 · · · n ) � ⊂ S n M n := n is the set of minimal factorizations of c n . Convention We read transpositions in a factorization from left to right. Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Definitions Fix n ≥ 1 S n the set of permutations of � 1 , n � T n the set of transpositions of � 1 , n � . The n-cycle is the permutation c n := (1 2 · · · n ). Minimal factorization of the n -cycle : an ordered ( n − 1)-tuple of transpositions whose product is c n . ( τ 1 , ..., τ n − 1 ) ∈ T n − 1 � , τ 1 · · · τ n − 1 = (1 2 · · · n ) � ⊂ S n M n := n is the set of minimal factorizations of c n . Convention We read transpositions in a factorization from left to right. Example : (34)(89)(35)(13)(16)(18)(23)(78) ∈ M 9 . Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 5 / 32

Minimal factorizations Definitions Brief history and first properties Called minimal : one needs at least n − 1 transpositions to generate the n -cycle Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions Brief history and first properties Called minimal : one needs at least n − 1 transpositions to generate the n -cycle Study of minimal factorizations started in the 50’s. Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions Brief history and first properties Called minimal : one needs at least n − 1 transpositions to generate the n -cycle Study of minimal factorizations started in the 50’s. enes (’59) proves that # M n = n n − 2 Combinatorial approaches : D´ Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions Brief history and first properties Called minimal : one needs at least n − 1 transpositions to generate the n -cycle Study of minimal factorizations started in the 50’s. enes (’59) proves that # M n = n n − 2 Combinatorial approaches : D´ Moskowski (’89), Goulden & Pepper (’93) : bijective proofs of this result, using bijections between factorizations and trees. Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 6 / 32

Minimal factorizations Definitions A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ? Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ? typical : here, take a minimal factorization uniformly in M n . Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Definitions A new probabilistic point of view : F´ eray & Kortchemski (’17, ’18) Question : for n large, what does a typical minimal factorization look like ? typical : here, take a minimal factorization uniformly in M n . look like : find a way of representing a minimal factorization geometrically Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 7 / 32

Minimal factorizations Geometrical coding of a factorization Minimal factorizations 1 Definitions Geometrical coding of a factorization Connection with the Brownian excursion 2 Conclusion 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 8 / 32

Minimal factorizations Geometrical coding of a factorization Idea 1 Code a minimal factorization by a set of chords of the unit disk. Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization Idea 1 Code a minimal factorization by a set of chords of the unit disk. Each transposition is coded by a chord. Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization Idea 1 Code a minimal factorization by a set of chords of the unit disk. Each transposition is coded by a chord. � e − 2 i π a / n , e − 2 i π b / n � Transposition ( a b ) ↔ Chord Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 9 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32

Minimal factorizations Geometrical coding of a factorization (34)(89)(35)(13)(16)(18)(23)(78) 8 7 9 6 1 5 2 4 3 Paul Thevenin (CMAP, Ecole Polytechnique) Geometry of a uniform minimal factorization June 14, 2019 10 / 32