Factorizations of Coxeter Elements in Complex Reflection Groups University of Minnesota-Twin Cities 2017 REU Thomas Hameister July 31, 2017 Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 1 / 19
Table of Contents Background 1 Definitions Previous Results 2 Question Framework 3 Results and Implications 4 Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 2 / 19
Complex Reflection Groups Definition Let V be a finite dimensional complex vector space of dimension n . A complex reflection is an element r ∈ GL ( V ) such that r has finite order, The fixed space of r is a hyperplane in V , i.e. dim C ker( r − 1 ) = n − 1. Definition A complex reflection group is a finite subgroup of GL ( V ) generated by reflections. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 3 / 19
Complex Reflection Groups Definition Let V be a finite dimensional complex vector space of dimension n . A complex reflection is an element r ∈ GL ( V ) such that r has finite order, The fixed space of r is a hyperplane in V , i.e. dim C ker( r − 1 ) = n − 1. Definition A complex reflection group is a finite subgroup of GL ( V ) generated by reflections. Familiar Examples: The dihedral group I 2 ( n ). The group B = G (2 , 1 , n ) of signed n × n permutation matrices. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 3 / 19
Notation and Definitions W will denote a well-generated, irreducible complex reflection group. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 4 / 19
Notation and Definitions W will denote a well-generated, irreducible complex reflection group. R = set of reflections in W , N = |R| . R ∗ = set of hyperplanes in V fixed by some element of R , N ∗ = |R ∗ | . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 4 / 19
Notation and Definitions W will denote a well-generated, irreducible complex reflection group. R = set of reflections in W , N = |R| . R ∗ = set of hyperplanes in V fixed by some element of R , N ∗ = |R ∗ | . The Coxeter number of W is the number h = N + N ∗ n Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 4 / 19
Notation and Definitions W will denote a well-generated, irreducible complex reflection group. R = set of reflections in W , N = |R| . R ∗ = set of hyperplanes in V fixed by some element of R , N ∗ = |R ∗ | . The Coxeter number of W is the number h = N + N ∗ n Definition A ζ -regular element is a c ∈ W with eigenvalue ζ and corresponding eigenvector not contained in any H ∈ R ∗ . A Coxeter element is a ζ h -regular element. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 4 / 19
Previous Results Set � � f k = # ( r 1 , . . . , r k ): c = r 1 . . . r k , r i ∈ R Theorem (Chapuy-Stump, 2014, [5]) For any irreducible, well-generated complex reflection group, W of rank n, t k � e Nt / n − e − N ∗ t / n � n � FAC W ( t ) = f k k ! = k ≥ 0 Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 5 / 19
Question Framework For R = R 1 ∪ · · · ∪ R ℓ a partition of R with each R i a union of conjugacy classes in R , and C = ( C 1 , . . . , C m ) a tuple with C i ∈ {R 1 , . . . , R ℓ } . Set � � g ( C ) = # ( r 1 , . . . , r m ): c = r 1 . . . r m , r i ∈ C i Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 6 / 19
Question Framework For R = R 1 ∪ · · · ∪ R ℓ a partition of R with each R i a union of conjugacy classes in R , and C = ( C 1 , . . . , C m ) a tuple with C i ∈ {R 1 , . . . , R ℓ } . Set � � g ( C ) = # ( r 1 , . . . , r m ): c = r 1 . . . r m , r i ∈ C i Fact Let S m act on m-tuples C by permuting its entries. Then, for all ω ∈ S m and all tuples C , g ( C ) = g ( ω · C ) Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 6 / 19
Question Framework For R = R 1 ∪ · · · ∪ R ℓ a partition of R with each R i a union of conjugacy classes in R , and C = ( C 1 , . . . , C m ) a tuple with C i ∈ {R 1 , . . . , R ℓ } . Set � � g ( C ) = # ( r 1 , . . . , r m ): c = r 1 . . . r m , r i ∈ C i Fact Let S m act on m-tuples C by permuting its entries. Then, for all ω ∈ S m and all tuples C , g ( C ) = g ( ω · C ) Let f m 1 ,..., m ℓ := g ( R 1 , . . . , R 1 , . . . , R ℓ , . . . , R ℓ ) � �� � � �� � m 1 times m ℓ times Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 6 / 19
Question Framework Consider the generating function ℓ u m i � � i FAC W ( u 1 , . . . , u ℓ ) = f m 1 ,..., m ℓ m i ! m 1 ,..., m ℓ ≥ 0 i =1 Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 7 / 19
Question Framework Consider the generating function ℓ u m i � � i FAC W ( u 1 , . . . , u ℓ ) = f m 1 ,..., m ℓ m i ! m 1 ,..., m ℓ ≥ 0 i =1 Question For what partitions R = R 1 ∪ · · · ∪ R ℓ does this function have a nice closed form expression? Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 7 / 19
Hyperplane-Induced Partitions Let C R ( W ) be the set of conjugacy classes in R . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 8 / 19
Hyperplane-Induced Partitions Let C R ( W ) be the set of conjugacy classes in R . For r ∈ R a reflection, let H r be the hyperplane fixed by r . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 8 / 19
Hyperplane-Induced Partitions Let C R ( W ) be the set of conjugacy classes in R . For r ∈ R a reflection, let H r be the hyperplane fixed by r . W acts on R ∗ by right multiplication. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 8 / 19
Hyperplane-Induced Partitions Let C R ( W ) be the set of conjugacy classes in R . For r ∈ R a reflection, let H r be the hyperplane fixed by r . W acts on R ∗ by right multiplication. Each conjugacy class C ⊂ R determines a unique W -orbit � � H C = H r ⊂ V : r ∈ C Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 8 / 19
Hyperplane-Induced Partitions Define the equivalence relation on C R ( W ) by C 1 ∼ C 2 ⇐ ⇒ H C 1 = H C 2 Let Θ 1 , . . . , Θ ℓ be the equivalence classes of C R ( W ) under ∼ and set � � R i = # r ∈ R : r ∈ C for some C ∈ Θ i . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 9 / 19
Hyperplane-Induced Partitions Define the equivalence relation on C R ( W ) by C 1 ∼ C 2 ⇐ ⇒ H C 1 = H C 2 Let Θ 1 , . . . , Θ ℓ be the equivalence classes of C R ( W ) under ∼ and set � � R i = # r ∈ R : r ∈ C for some C ∈ Θ i . R = R 1 ∪ · · · ∪ R ℓ as above will be called a hyperplane-induced partition of R . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 9 / 19
The Hurwitz Action and Numbers n i Definition Say rank ( W ) = n . The Hurwitz action of the braid group of type A n − 1 on factorizations ( t 1 , . . . , t n ) of c is given by generators e i · ( t 1 , . . . , t n ) = ( t 1 , . . . , t i t i +1 t − 1 , t i , . . . , t n ) i Theorem (Bessis, 2003 [1]) The Hurwitz action is transitive on the set of minimal-length factorizations ( t 1 , . . . , t n ) of any fixed Coxeter element c. The Hurwitz action preserves the multiset of conjugacy classes { C i : t i ∈ C i } . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 10 / 19
Constants associated to Hyperplane-Induced Partitions Definition For any factorization c = t 1 · · · t n of a Coxeter element c , set � � n i = # j : t j ∈ R i This definition is independent of the choice of factorization by the transitivity of the Hurwitz action. Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 11 / 19
Constants associated to Hyperplane-Induced Partitions Definition For any factorization c = t 1 · · · t n of a Coxeter element c , set � � n i = # j : t j ∈ R i This definition is independent of the choice of factorization by the transitivity of the Hurwitz action. Let R = R 1 ∪ · · · ∪ R ℓ be a hyperplane-induced partition of R , and let H i be the W -orbit of R ∗ corresponding to R i . Set N ∗ N i := # R i and i = # H i . Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 11 / 19
Schematic Interpretation of n i The data of the theorem can be read off a Coxeter-Shephard diagram. Example: W = G 26 has diagram 3 4 3 3 2 3 4 3 3 2 ⇓ Remove edges with even label. 3 3 3 2 ⇓ n i = size of connected component. n 1 = 2 and n 2 = 1 Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection Groups July 31, 2017 12 / 19
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