Factorizations of Coxeter Elements in Complex Reflection Groups

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 GroupsJuly 31, 2017 1 / 19 Table of Contents

1 Background Deﬁnitions

2 Previous Results

3 Question Framework

4 Results and Implications

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 2 / 19 Familiar Examples:

The dihedral group I2(n). The group B = G(2, 1, n) of signed n × n permutation matrices.

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.

dimC 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 GroupsJuly 31, 2017 3 / 19 Complex Reﬂection Groups

dimC 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 I2(n). The group B = G(2, 1, n) of signed n × n permutation matrices.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection GroupsJuly 31, 2017 3 / 19 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 N + N∗ h = 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 ζh-regular element.

Notation and Deﬁnitions

W will denote a well-generated, irreducible complex reﬂection group.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 4 / 19 The Coxeter number of W is the number N + N∗ h = 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 ζh-regular element.

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 GroupsJuly 31, 2017 4 / 19 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 ζh-regular element.

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 N + N∗ h = n

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 4 / 19 Notation and Deﬁnitions

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 ζh-regular element.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 4 / 19 Previous Results

Set n o fk = # (r1,..., rk ): c = r1 ... rk , ri ∈ R

Theorem (Chapuy-Stump, 2014, [5]) For any irreducible, well-generated complex reﬂection group, W of rank n,

k X t ∗ n FAC (t) = f = eNt/n − e−N t/n W k k! k≥0

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 5 / 19 Fact

Let Sm act on m-tuples C by permuting its entries. Then, for all ω ∈ Sm and all tuples C, g(C) = g(ω ·C)

Let

fm1,...,m` := g(R1,..., R1,..., R`,..., R`) | {z } | {z } m1 times m` times

Question Framework

For R = R1 ∪ · · · ∪ R` a partition of R with each Ri a union of conjugacy classes in R, and C = (C1,..., Cm) a tuple with Ci ∈ {R1,..., R`}. Set n o g(C) = # (r1,..., rm): c = r1 ... rm, ri ∈ Ci

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 6 / 19 Let

fm1,...,m` := g(R1,..., R1,..., R`,..., R`) | {z } | {z } m1 times m` times

Question Framework

Fact

Let Sm act on m-tuples C by permuting its entries. Then, for all ω ∈ Sm and all tuples C, g(C) = g(ω ·C)

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 6 / 19 Question Framework

Fact

Let Sm act on m-tuples C by permuting its entries. Then, for all ω ∈ Sm and all tuples C, g(C) = g(ω ·C)

Let

fm1,...,m` := g(R1,..., R1,..., R`,..., R`) | {z } | {z } m1 times m` times

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 6 / 19 Question

For what partitions R = R1 ∪ · · · ∪ R` does this function have a nice closed form expression?

Question Framework

Consider the generating function

` mi X Y ui FACW (u1,..., u`) = fm1,...,m` mi ! m1,...,m`≥0 i=1

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 7 / 19 Question Framework

Consider the generating function

` mi X Y ui FACW (u1,..., u`) = fm1,...,m` mi ! m1,...,m`≥0 i=1

Question

For what partitions R = R1 ∪ · · · ∪ R` does this function have a nice closed form expression?

Thomas Hameister Factorizations of Coxeter Elements in Complex Reflection GroupsJuly 31, 2017 7 / 19 For r ∈ R a reflection, let Hr be the hyperplane fixed by r. W acts on R∗ by right multiplication. Each conjugacy class C ⊂ R determines a unique W -orbit HC = Hr ⊂ V : r ∈ C

Hyperplane-Induced Partitions

Let CR(W ) be the set of conjugacy classes in R.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 8 / 19 W acts on R∗ by right multiplication. Each conjugacy class C ⊂ R determines a unique W -orbit HC = Hr ⊂ V : r ∈ C

Hyperplane-Induced Partitions

Let CR(W ) be the set of conjugacy classes in R.

For r ∈ R a reﬂection, let Hr be the hyperplane ﬁxed by r.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 8 / 19 Each conjugacy class C ⊂ R determines a unique W -orbit HC = Hr ⊂ V : r ∈ C

Hyperplane-Induced Partitions

Let CR(W ) be the set of conjugacy classes in R.

For r ∈ R a reﬂection, let Hr be the hyperplane ﬁxed by r. W acts on R∗ by right multiplication.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 8 / 19 Hyperplane-Induced Partitions

Let CR(W ) be the set of conjugacy classes in R.

For r ∈ R a reﬂection, let Hr be the hyperplane ﬁxed by r. W acts on R∗ by right multiplication. Each conjugacy class C ⊂ R determines a unique W -orbit HC = Hr ⊂ V : r ∈ C

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 8 / 19 R = R1 ∪ · · · ∪ R` as above will be called a hyperplane-induced partition of R.

Hyperplane-Induced Partitions

Deﬁne the equivalence relation on CR(W ) by

C1 ∼ C2 ⇐⇒ HC1 = HC2

Let Θ1,..., Θ` be the equivalence classes of CR(W ) under ∼ and set Ri = # r ∈ R: r ∈ C for some C ∈ Θi .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 9 / 19 Hyperplane-Induced Partitions

Deﬁne the equivalence relation on CR(W ) by

C1 ∼ C2 ⇐⇒ HC1 = HC2

Let Θ1,..., Θ` be the equivalence classes of CR(W ) under ∼ and set Ri = # r ∈ R: r ∈ C for some C ∈ Θi .

R = R1 ∪ · · · ∪ R` as above will be called a hyperplane-induced partition of R.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 9 / 19 The Hurwitz Action and Numbers ni

Deﬁnition

Say rank(W ) = n. The Hurwitz action of the braid group of type An−1 on factorizations (t1,..., tn) of c is given by generators

−1 ei · (t1,..., tn) = (t1,..., ti ti+1ti , ti ,..., tn)

Theorem (Bessis, 2003 [1]) The Hurwitz action is transitive on the set of minimal-length factorizations (t1,..., tn) of any ﬁxed Coxeter element c.

The Hurwitz action preserves the multiset of conjugacy classes {Ci : ti ∈ Ci }.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 10 / 19 Let R = R1 ∪ · · · ∪ R` be a hyperplane-induced partition of R, and ∗ let Hi be the W -orbit of R corresponding to Ri . Set ∗ Ni := #Ri and Ni = #Hi .

Constants associated to Hyperplane-Induced Partitions

Deﬁnition

For any factorization c = t1 ··· tn of a Coxeter element c, set ni = # j : tj ∈ Ri

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 GroupsJuly 31, 2017 11 / 19 Constants associated to Hyperplane-Induced Partitions

Deﬁnition

For any factorization c = t1 ··· tn of a Coxeter element c, set ni = # j : tj ∈ Ri

This deﬁnition is independent of the choice of factorization by the transitivity of the Hurwitz action.

Let R = R1 ∪ · · · ∪ R` be a hyperplane-induced partition of R, and ∗ let Hi be the W -orbit of R corresponding to Ri . Set ∗ Ni := #Ri and Ni = #Hi .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 11 / 19 Schematic Interpretation of ni

The data of the theorem can be read oﬀ a Coxeter-Shephard diagram.

Example: W = G26 has diagram 3 4 3 3 2 3 4 3 3 2 ⇓ Remove edges with even label. 3 3 3 2

⇓ ni = size of connected component. n1 = 2 and n2 = 1

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 12 / 19 Compare with Chapuy-Stump:

∗ n 1 Nt − N t FAC (t) = e n − e n W |W |

Main Result

Theorem Let W be an irreducible, well-generated complex reﬂection group with hyperplane-induced partition of reﬂections R = R1 ∪ · · · ∪ R`. Let ∗ ni , Ni , Ni be as before. Then,

∗ ` N u N u ni 1 Y i i − i i FAC (u ,..., u ) = e ni − e ni W 1 ` |W | i=1

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 13 / 19 Main Result

Theorem Let W be an irreducible, well-generated complex reﬂection group with hyperplane-induced partition of reﬂections R = R1 ∪ · · · ∪ R`. Let ∗ ni , Ni , Ni be as before. Then,

∗ ` N u N u ni 1 Y i i − i i FAC (u ,..., u ) = e ni − e ni W 1 ` |W | i=1

Compare with Chapuy-Stump:

∗ n 1 Nt − N t FAC (t) = e n − e n W |W |

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 13 / 19 Main Result

Theorem Let W be an irreducible, well-generated complex reﬂection group with hyperplane-induced partition of reﬂections R = R1 ∪ · · · ∪ R`. Let ∗ ni , Ni , Ni be as before. Then,

∗ ` N u N u ni 1 Y i i − i i FAC (u ,..., u ) = e ni − e ni W 1 ` |W | i=1

Example: W = G26,

∗ n1 = 2, N1 = 24, N1 = 12, ∗ n2 = 1, N2 = 12, N2 = 9

2 1 12u −6u 9t −9t FACG26 (u, t) = e − e e − e |G26|

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 14 / 19 Remarks on the Main Theorem

The multivariate generating function in our work specializes to that of Chapuy-Stump:

FACW (u1,..., u`) = FACW (t) u1=···=u`=t

For any R = R1 ∪ · · · ∪ R` a hyperplane-induced partition of reﬂections, ` is at most 2.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 15 / 19 In the real case, this is explained by a proposition of Bourbaki: Proposition

([3], Ch VI, Section 11, Prop 33) If si are reﬂections corresponding to a basis of an irreducible root system R, the cyclic subgroup Γ = hci of order h generated by c = s1s2 ... sl acts freely on R and there exist representatives θ1, . . . , θm of the Γ-orbits such that each θi is in the W -orbit of a simple root.

A Corollary to the Main Theorem

Corollary Let W be an irreducible, well-generated Coxeter group with Coxeter ∗ Ni +Ni number h. Set hi = . Then hi = h for every i. ni

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 16 / 19 A Corollary to the Main Theorem

Corollary Let W be an irreducible, well-generated Coxeter group with Coxeter ∗ Ni +Ni number h. Set hi = . Then hi = h for every i. ni

In the real case, this is explained by a proposition of Bourbaki: Proposition

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 16 / 19 ⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 ∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 ⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 ∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 Sketch of Case-Free Proof of Corollary in Real Case

∗ Let O1,..., O` be W -orbits of R .

⇒ ni counts number of θj with hyperplane in Oi .

∗ Because real, Ni = Ni and number of roots with hyperplane in Oi is 2Ni .

Because Γ acts freely, get 2Ni /h orbits in Oi .

⇒ ni = 2Ni /h

∗ ⇒ h = 2Ni /ni = (Ni + Ni )/ni = hi Remark In the complex case, Jean-Michel has kindly provided an argument using results of Bessis and Brou´e-Malle-Rouquier.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 17 / 19 Acknowledgements

This research was carried out as part of the 2017 summer REU program at the School of Mathematics, University of Minnesota, Twin Cities, and was supported by NSF RTG grant DMS-1148634 and by NSF grant DMS-1351590.

Thanks goes out to Victor Reiner, Elise delMas, and Craig Corsi for their mentorship and support throughout this project.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 18 / 19 References

David Bessis. The dual braid monoid. In Annales scientiﬁques de l’Ecole normale sup´erieure, volume 36, pages 647–683. Elsevier, 2003.

David Bessis. Finite complex reﬂection arrangements are k (pi, 1). arXiv preprint math/0610777, 2006.

Nicolas Bourbaki. Lie groups and lie algebras. chapters 4–6. translated from the 1968 french original by andrew pressley, 2002.

Michel Broué,Gunter Malle, and RaphaëlRouquier. Complex reflection groups, braid groups, hecke algebras. 1997. Guillaume Chapuy and Christian Stump. Journal of the London Mathematical Society, 90 (3):919-939 (2014), 11 2012.

Gustav I Lehrer and Donald E Taylor. Unitary reﬂection groups, volume 20 of australian mathematical society lecture series. Cambridge University Press, 26:27–40, 2009.

Jean Michel. Deligne-lusztig theoretic derivation for weyl groups of the number of reﬂection factorizations of a coxeter element. Proceedings of the American Mathematical Society, 144(3):937–941, 2016.

Thomas Hameister Factorizations of Coxeter Elements in Complex Reﬂection GroupsJuly 31, 2017 19 / 19