A refined count of Coxeter element reflection factorizations

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the number of reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.


Introduction
A complex reflection group is a finite subgroup W of GL(V ), where V = C n , generated by the set of all reflections t in W , that is, the elements t whose fixed space V t := ker(t − 1) is a hyperplane H, meaning dim H = n − 1.Let R denote the set of all reflections in W , and R * the collection of all reflecting hyperplanes.An important numerological role is played by the cardinalities of R, R * , denoted N, N * , respectively.This paper focusses on the complex reflection groups W which act irreducibly on V = C n , and which are well-generated in the sense that they can be generated by n reflections.For such a group W , one can define the Coxeter number as h := N +N * n , and then the Coxeter elements c in W are the elements c which have at least one eigenvector v in V reg := V \ ∪ H∈R * H with eigenvalue ζ h := e 2πi h .It is known that there is only one conjugacy class of Coxeter elements c; see, for example, [2,5].
Having fixed one Coxeter element c, one can ask for the number f counting reflection factorizations of c having length , that is, sequences (t 1 , . . ., t ) ∈ R with c = t 1 t 2 • • • t .The main result of Chapuy and Stump [5] is the following amazingly simple product formula for the exponential generating function of the sequence (f ) =0,1,2,... : In particular, this power series starts at x n , as shortest factorizations of c have length n.
Our main result refines equation ( 1), accounting for how many reflections t j appearing in c = t 1 t 2 • • • t have their reflecting hyperplane ker(t j − 1) lying in the various W -orbits R * 1 , . . ., R * p that decompose R * = p i=1 R * i .Stating it requires some numerology associated to each orbit R * i for i = 1, 2, . . ., p. Let R i denote the subset of reflections whose reflecting hyperplane lies in It is not obvious that these numbers n i are well-defined, independent of the choice of a length n factorization for c, but this follows from work of Bessis [2, Prop.7.6], who showed that any two such shortest factorizations can be connected by a sequence of Hurwitz moves Let f 1 , 2 ,..., p be the number of tuples (t 1 , . . ., t ) factoring c = t 1 t 2 • • • t having the first 1 reflections t 1 , t 2 , . . ., t 1 in R 1 , the next 2 reflections in R 2 , etc. (so = p i=1 i ).One can show using the Hurwitz moves above (or see Proposition 3 below), that f 1 , 2 ,..., p also counts factorizations in which the elements of R i occur in any prescribed set of the i positions, rather than all t j in R 1 first, then R 2 second, etc. Theorem 1.For any irreducible, well-generated complex reflection group, and notation as above, one has .
the electronic journal of combinatorics 25(1) (2018), #P1.28 Our proof is the same as Chapuy and Stump's proof of equation ( 1), via the classification1 of irreducible, well-generated reflection groups, and Frobenius's character-theoretic technique for counting factorizations, reviewed in Section 2. Since there is little novelty in the methods, the proof in Section 4 is abbreviated as much as possible.
One caveat: The phrasing of Theorem 1, while convenient, may seem deceptively general, since the classification of irreducible complex reflection groups shows that p = 1 or 2 in every case.When p = 1, Theorem 1 is the same as equation ( 1), giving no further information.The remaining cases where p = 2 are listed in the table below, with the factorization in the theorem shown, using variables (x, y) instead of (x 1 , x 2 ): The second column of the table gives the Coxeter-Shephard diagram for these groups, reflecting the case-by-case observation that irreducible, well-generated groups W with p = 2 are all Shephard groups, that is, symmetry groups of regular complex (or real) polytopes.This implies (see [6]) that they have a Shephard presentation where here the integer p i 2 labels the node for s i , and the integer m ij 2 labels the edge from s i to s j , with m ij = 2 whenever |i − j| 2 (and no edge from s i to s j is shown).It is known for Coxeter groups and Shephard groups that one can choose s 1 , . . ., s n in such a way that their product factors a Coxeter element c = s 1 s 2 • • • s n .The hyperplane orbits R * 1 , R * 2 correspond to the connected components obtained when one erases the edges with even labels m ij in the Coxeter-Shephard diagram, and in this case, n 1 , n 2 may be re-interpreted as the number of nodes in the corresponding connected component.
We also explain (Proposition 5) how Theorem 1 necessarily specializes to recover the Chapuy-Stump formula (1).Comparing the two results then gives our first proof of the following seemingly new fact about the Coxeter number h.
Corollary 2. For irreducible, well-generated complex reflection groups, and notation as above, each hyperplane orbit R * i for i = 1, 2, . . ., p satisfies Because it uses both Theorem 1 and equation ( 1), this first proof of Corollary 2 relies on case-by-case checks.We also give a second proof which is case-free, but applies only to real reflection groups, and a third proof for the general case supplied by J. Michel, proving a more general assertion about regular elements (Theorem 6), which he has kindly allowed us to reproduce here.

Frobenius's method
Frobenius gave a method, using character theory, for counting factorizations of an element in any finite group W as a product of elements from specified conjugacy-closed subsets.
Recall that (finite-dimensional, complex) representations W → V ).For subsets A ⊆ W , define χ(A) := w∈A χ(w).Proposition 3. (Frobenius; see, e.g., [7, Thm A.1.9])For A 1 , . . ., A subsets of a finite group W , with each A i closed under conjugation, and c in W , the number of factorizations where the sum is over all the characters χ of the inequivalent irreducible representations of G.
To apply this, recall that for Coxeter elements c in W a well-generated complex reflection group, we defined f 1 ,..., p as the number of sequences (t 1 , . . ., t ) factoring c = t 1 • • • t in which exactly i of the factors t j lie in R i , with the factors from R 1 all coming first in the sequence, those from R 2 coming next, etc. Corollary 4. With the above notations, 3 Proofs of Corollary 2.
Before proving Theorem 1, we explain how it specializes to equation ( 1), and why this implies Corollary 2.
Note that in each summand on the right in Corollary 4, the order of the factors χ(R 1 ) 1 . . .χ(R p ) p does not matter.This explains an assertion from the Introduction: f 1 ,..., p also counts sequences (t 1 , . . ., t ) factoring c = t 1 • • • t in which exactly i of the factors t j lie in R i , but where one fixes any of the 1 ,..., p choices of the positions in which the factors from R 1 , . . ., R p should occur.This then has the following implication.
Proposition 5.The exponential generating functions in Theorem 1 and equation (1) are related by specialization: One way to see this claim is to first write where the last equality holds since e z −1 ] has constant term 1.Thus at least the product a 1 • • • a p is determined by P (x).Naming the coefficients c k in the unique expansion log e z −1 ] lets one read off from P (x) all of the power sums {a k 1 + • • • + a k p } k=1,2,... , via this calculation: But then these power sums uniquely determine the multiset (a 1 , . . ., a p ).
As mentioned in the Introduction, the above first proof of Corollary 2 relies on Theorem 1 and equation (1), both proven via case-by-case arguments.We therefore seek case-free proofs.The second proof will apply only when W is a real reflection group.
Second proof of Corollary 2, for real W , but case-free.Let W be an irreducible real reflection group, with simple reflections S = {s 1 , . . ., s n }, root system Φ, and corresponding simple roots {α 1 , . . ., α n }.Then it is known that the Coxeter element c = s 1 s 2 • • • s n generates a cyclic subgroup c of order h acting freely on the root system, decomposing Φ = n i=1 Φ i into n orbits Φ i .Furthermore, one has c -orbit representatives θ j := s n s n−1 • • • s j+1 (α j ), so that θ j lies in the W -orbit of α j ; see 2 while the third equality comes from the fact that the θ j represent the orbits for the free c -action on Φ.
The promised third proof of Corollary due to J. Michel, is case-free and even proves a more general assertion.Recall that a positive integer d is called a regular number for W if there is a regular element w in W (one with an eigenvector v in V reg ) having order d.Recall also that it is a consequence of a characterization of regular numbers (originally proven case-by-case by Lehrer and Springer [9], and later in a case-free fashion by Lehrer and Michel [8]) that the Coxeter number h is a regular number for every well-generated group.Theorem 6. (J.Michel) A complex reflection group W has every regular number d dividing N i + N * i for each i = 1, 2, . . ., p.In particular, when W is irreducible, well-generated and d = h, one has The proof uses the theory of the braid group B := π 1 (V reg /W ) associated to a complex reflection group W ; see Broué, Malle, and Rouquier [4], further developments by Bessis [2], and the exposition in Broué [3].
This theory emphasizes a certain generating set {s H } H∈R * for W , where s H is the distinguished reflection fixing H, the one having det(s H ) = ζ #W H , where W H is the cyclic subgroup pointwise fixing H.
Two surjections out of B play an important role here.First is the surjection B W sending b → w, which arises because the quotient map V reg → V reg /W is a Galois covering with Galois group W ; say that b lifts w in this situation.For each hyperplane H, there is an important family of lifts of s H to elements s H,γ in B, called braid reflections; all of these braid reflection lifts s H,γ of s H lie in the same B-conjugacy class.
Second is the abelianization map The composite map B → Z p can be defined by the following property [4, Thm.2.17]: if H lies in the W -orbit R * i inside R * , then each braid reflection s H,γ lifting s H maps to the i th standard basis vector of Z p .

Proof of Theorem 1
As explained in the caveat following Theorem 1, the number p of W -orbits of hyperplanes is either 1 or 2. When p = 1, the theorem is equivalent to equation (1), and so there is nothing further to prove.The irreducible, well-generated groups W having p = 2 appear in the table following the caveat, with only two infinite families G(m, m, 2), G(r, 1, n), and several exceptional groups.Just as in [5], one can use Frobenius's Proposition 3 to verify the table entries-we give here the general calculations for the two infinite families in the next two subsections.The exceptional cases were handled via computer, accessing in SAGE (via the Gap3 package Chevie, see [11]) the irreducible complex reflection groups and their character tables; we discuss the exceptional cases no further here.

4.1
The dihedral group G(m, m, 2) for even m.
The group G(m, m, 2) turns out to be the complexification of a real reflection group, the dihedral group of type I 2 (m) with Coxeter presentation Here the sets R * , R of reflecting hyperplanes (lines) and reflections both have size m.Both R * , R have a single W -orbit when m is odd, but when m is even, they decompose two orbits of size , indexed here so that s i lies in R i for i = 1, 2. Furthermore, c = s 1 s 2 , and n 1 = n 2 = 1.
Using the p = 2 case of Corollary 4 then gives the following: the electronic journal of combinatorics 25(1) (2018), #P1.28 and hence, in agreement with Theorem one calculates #W The accompanying decomposition of the reflections R = R 1 R 2 has R 1 consisting of the N 1 = (r − 1)n reflections that scale one of the n coordinates by ζ r for some 1 r − 1, and fix all other coordinates, while R 2 is the collection of N 2 = N * 2 = r n 2 order two reflections in each of the hyperplanes of R * 2 .To finish the computation, we use the character-theoretic analysis already detailed in [5, §5.3].There the authors show that the only W -irreducible characters χ which do not vanish on c −1 form a two-parameter family denoted {χ q h n k } where 0 q r − 1 and 0 k n − 1, with these values: Using the p = 2 case of Corollary 4, one has Therefore one can check agreement with Theorem 1 as follows: #W x 1

1 x − 1 .
for i=1,2,...,p .Proof.The discussion of the preceding paragraph shows that f = ( 1 ,..., p)∈N p : i i = 1 , . . ., p f 1 ,..., p and the rest is simple manipulation of summations and factorials.From this one can now see why Theorem 1 and equation (1) imply Corollary 2. First proof of Corollary 2. Plugging equation (1) into the left of Proposition 5 and plugging Theorem 1 into the right, gives this equality: e − N * n x = e − N * n x (e N +N * n x − 1) = e − N * n x (e hx − 1) on the left, and similarly on the right, gives e −N * x (e hx − 1) n = e −x p the electronic journal of combinatorics 25(1) (2018), #P1.28 the other hand, by definition, p i=1 N * i = N * , and hence (e hx − 1) n = Then the desired equality N i +N * i n i = h for i = 1, 2, . . ., p follows from this claim: Claim: Any power series of the form P (x) = p i=1 (e a i x − 1) in R[[x]] uniquely determines the multiset (a 1 , . . ., a p ).

2 x e m 2 y − e − m 2 y . 4 . 2
The monomial groups G(r, 1, n) for r 2The group W = G(r, 1, n) is the set of n × n matrices with one nonzero entry in each row and column, and that nonzero entry is an r th root-of-unity in C, a power of the primitive root ζ r = e 2πi r .The reflecting hyperplane W -orbit decomposition is R