Coxeter numbers: From fake degree palindromicity to the enumeration - PowerPoint PPT Presentation

Coxeter numbers: From fake degree palindromicity to the enumeration of reflection factorizations. Theo Douvropoulos � � Paris VII, IRIF ERC CombiTop November 21, 2018 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 1 / 23

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 November 21, 2018 2 / 23

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 November 21, 2018 2 / 23

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 , with Coxeter number h, there are h n n ! | W | (minimal length) reflection factorizations t 1 · · · t n = c of the Coxeter element c. Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 2 / 23

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 November 21, 2018 3 / 23

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 November 21, 2018 3 / 23

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 November 21, 2018 3 / 23

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 November 21, 2018 4 / 23

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 November 21, 2018 4 / 23

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 November 21, 2018 4 / 23

A brief history of the W -Hurwitz number h n n ! | W | Some proofs and some mathematical shadows: Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 5 / 23

A brief history of the W -Hurwitz number h n n ! | W | Some proofs and some mathematical shadows: Deligne-Tits-Zagier, rediscovered by Reading. 1 Enumerate factorizations t 1 · · · t n = c with respect to the c -orbit of t n : � Hur( W ) = h Hur( W � s � ) 2 s ∈ S Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 5 / 23

A brief history of the W -Hurwitz number h n n ! | W | Some proofs and some mathematical shadows: Deligne-Tits-Zagier, rediscovered by Reading. 1 Enumerate factorizations t 1 · · · t n = c with respect to the c -orbit of t n : � Hur( W ) = h Hur( W � s � ) 2 s ∈ S Chapoton. Interpretation as the number of maximal chains of NC ( W ): 2 � � � n X n hX + d i Hur( W ) = n ! d i i =1 Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 5 / 23

A brief history of the W -Hurwitz number h n n ! | W | Some proofs and some mathematical shadows: Deligne-Tits-Zagier, rediscovered by Reading. 1 Enumerate factorizations t 1 · · · t n = c with respect to the c -orbit of t n : � Hur( W ) = h Hur( W � s � ) 2 s ∈ S Chapoton. Interpretation as the number of maximal chains of NC ( W ): 2 � � � n X n hX + d i Hur( W ) = n ! d i i =1 Lyashko-Looijenga and Bessis. 3 There exist two subgroups G 1 ≤ G 2 ≤ B n of the braid group B n on n strands, with finite indexes ν 1 and ν 2 such that: ν 1 = h n n ! ν 2 = # { reduced reflection factorizations of c } | W | Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 5 / 23

How to count, the Frobenius way 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 November 21, 2018 6 / 23

How to count, the Frobenius way 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 November 21, 2018 6 / 23

How to count, the Frobenius way 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 November 21, 2018 6 / 23

How to count, the Frobenius way 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 November 21, 2018 6 / 23

How to count, the Frobenius way 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 � � � R N · c − 1 � · t N 1 = | W | · dim( χ ) · χ N ! N ≥ 0 χ ∈ � W Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 6 / 23

How to count, the Frobenius way Consider the central element R := � t ∈R t of the group algebra C [ W ]. � � � R N · c − 1 � t N 1 = | W | · dim( χ ) · χ · N ! N ≥ 0 χ ∈ � W Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 7 / 23

How to count, the Frobenius way Consider the central element R := � t ∈R t of the group algebra C [ W ]. � � � R N · c − 1 � t N 1 = | W | · dim( χ ) · χ · N ! N ≥ 0 χ ∈ � W � χ ( R ) � N � � t N 1 · χ ( c − 1 ) · = | W | · χ (1) · χ (1) N ! N ≥ 0 χ ∈ � W Theo Douvropoulos (Paris VII, IRIF) How to count reflection factorizations November 21, 2018 7 / 23