On enumerating factorizations in reflection groups. Theo - PowerPoint PPT Presentation

On enumerating factorizations in reflection groups. Theo Douvropoulos � � Paris VII, IRIF ERC CombiTop FPSAC Ljubljana, July 5, 2019 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 1 / 19

The number of reduced reflection factorizations of c Theorem (Hurwitz, 1892) There are n n − 2 (minimal length) factorizations t 1 · · · t n − 1 = (12 · · · n ) ∈ S n where the t i ’s are transpositions. Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 2 / 19

The number of reduced reflection factorizations of c Theorem (Hurwitz, 1892) There are n n − 2 (minimal length) factorizations t 1 · · · t n − 1 = (12 · · · n ) ∈ S n where the t i ’s are transpositions. For example, the 3 1 factorizations (12)(23) = (123) (13)(12) = (123) (23)(13) = (123) . Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 2 / 19

The number of reduced reflection factorizations of c Theorem (Hurwitz, 1892) There are n n − 2 (minimal length) factorizations t 1 · · · t n − 1 = (12 · · · n ) ∈ S n where the t i ’s are transpositions. For example, the 3 1 factorizations (12)(23) = (123) (13)(12) = (123) (23)(13) = (123) . Theorem (Deligne-Arnol’d-Bessis) For a well-generated, complex reflection group W and a Coxeter element c, there are h n n ! | W | (minimal length) reflection factorizations t 1 · · · t n = c where h = | c | . Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 2 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact S n , c ( N ) t N FAC S n , c ( t ) = N ! . N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 3 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact S n , c ( N ) t N FAC S n , c ( t ) = N ! . N ≥ 0 Theorem (Jackson, ’88) If c = (12 · · · n ) ∈ S n , then FAC S n , c ( t ) = e t ( n 2 ) � 1 − e − tn � n − 1 . n ! Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 3 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact S n , c ( N ) t N FAC S n , c ( t ) = N ! . N ≥ 0 Theorem (Jackson, ’88) If c = (12 · · · n ) ∈ S n , then FAC S n , c ( t ) = e t ( n 2 ) � 1 − e − tn � n − 1 . n ! Notice that � � t n − 1 FAC S n , c ( t ) = 1 n ! · ( n ) n − 1 · ( n − 1)! = n n − 2 . ( n − 1)! Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 3 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact W , c ( N ) t N FAC W , c ( t ) = N ! . N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 4 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact W , c ( N ) t N FAC W , c ( t ) = N ! . N ≥ 0 Theorem (Chapuy-Stump, ’12) If W is well-generated, of rank n, and h is the order of the Coxeter element c, then FAC W , c ( t ) = e t |R| � 1 − e − th � n . | W | Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 4 / 19

Arbitrary length reflection factorizations of c If R denotes the set of reflections of W , we write Fact W , c ( N ) := # { ( t 1 , · · · , t N ) ∈ R n | t 1 · · · t N = c } . Now, consider the exponential generating function: � Fact W , c ( N ) t N FAC W , c ( t ) = N ! . N ≥ 0 Theorem (Chapuy-Stump, ’12) If W is well-generated, of rank n, and h is the order of the Coxeter element c, then FAC W , c ( t ) = e t |R| � 1 − e − th � n . | W | Notice that � t n � | W | · h n · n ! = h n n ! 1 FAC W , c ( t ) = | W | . n ! Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 4 / 19

There must be an example we can do by hand? Let C 2 := { Id , c } be the group of order 2 and R = { c } . Then, t + t 3 3! + t 5 FAC C 2 , c ( t ) = 5! + · · · Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 5 / 19

There must be an example we can do by hand? Let C 2 := { Id , c } be the group of order 2 and R = { c } . Then, t + t 3 3! + t 5 FAC C 2 , c ( t ) = 5! + · · · e t − e − t = e t � 1 − e − 2 t � = 2 · 2 And a non-example? Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 5 / 19

There must be an example we can do by hand? Let C 2 := { Id , c } be the group of order 2 and R = { c } . Then, t + t 3 3! + t 5 FAC C 2 , c ( t ) = 5! + · · · e t − e − t = e t � 1 − e − 2 t � = 2 · 2 And a non-example? For C n := { Id , c , · · · , c n − 1 } if we pick factors only from U := { c } , we again have t n +1 t 2 n +1 FAC C n , c ( t ) = t + ( n + 1)! + (2 n + 1)! + · · · Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 5 / 19

There must be an example we can do by hand? Let C 2 := { Id , c } be the group of order 2 and R = { c } . Then, t + t 3 3! + t 5 FAC C 2 , c ( t ) = 5! + · · · e t − e − t = e t � 1 − e − 2 t � = 2 · 2 And a non-example? For C n := { Id , c , · · · , c n − 1 } if we pick factors only from U := { c } , we again have t n +1 t 2 n +1 FAC C n , c ( t ) = t + ( n + 1)! + (2 n + 1)! + · · · � e t + ξ − 1 · e ξ · t + ξ − 2 · e ξ 2 · t + · · · + ξ − n +1 · e ξ n − 1 · t � 1 = n · with ξ a n-th root of unity . Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 5 / 19

How to count, the Frobenius way � � � � (12) + (13) + (23) · (12) + (13) + (23) = 3 · Id +3 · (123) + 3 · (132) Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 6 / 19

How to count, the Frobenius way � � � � (12) + (13) + (23) · (12) + (13) + (23) = 3 · Id +3 · (123) + 3 · (132) Consider the central element R := � t ∈R t of the group algebra C [ W ]. � · t N # { ( t 1 , · · · , t N ) ∈ R N | t 1 · · · t N = c } N ! N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 6 / 19

How to count, the Frobenius way � � � � (12) + (13) + (23) · (12) + (13) + (23) = 3 · Id +3 · (123) + 3 · (132) Consider the central element R := � t ∈R t of the group algebra C [ W ]. � · t N # { ( t 1 , · · · , t N ) ∈ R N | t 1 · · · t N = c } N ! N ≥ 0 � � � · t N R N = c N ! N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 6 / 19

How to count, the Frobenius way � � � � (12) + (13) + (23) · (12) + (13) + (23) = 3 · Id +3 · (123) + 3 · (132) Consider the central element R := � t ∈R t of the group algebra C [ W ]. � · t N # { ( t 1 , · · · , t N ) ∈ R N | t 1 · · · t N = c } N ! N ≥ 0 � � � · t N R N = c N ! N ≥ 0 � � � � R N · c − 1 � · t N = id N ! N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 6 / 19

How to count, the Frobenius way � � � � (12) + (13) + (23) · (12) + (13) + (23) = 3 · Id +3 · (123) + 3 · (132) Consider the central element R := � t ∈R t of the group algebra C [ W ]. � · t N # { ( t 1 , · · · , t N ) ∈ R N | t 1 · · · t N = c } N ! N ≥ 0 � � � · t N R N = c N ! N ≥ 0 � � � � R N · c − 1 � · t N = id N ! N ≥ 0 ! = ! � � R N · c − 1 � · t N 1 | W | Tr C [ W ] N ! N ≥ 0 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations FPSAC Ljubljana, July 5, 2019 6 / 19