Enumerating Projective Reflection Groups

Projective re ection groups have been recently dened by the second author. They include a special class of groups denoted G(r; p; s; n) which contains all classical Weyl groups and more generally all the complex re ection groups of type G(r; p; n). In this paper we dene some statistics analogous to descent number and major index over the projective re ection groups G(r; p; s; n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r; p; s; n), as distribution of one-dimensional characters and computation of Hilbert series of invariant algebras, are also treated.


Introduction
The study of permutation statistics, and in particular of those depending on up-down or descents patterns, is a very classical subject of study in algebraic combinatorics that goes back to the early 20th century to the work of MacMahon [15], and has found a new interest in the last decade after the fundamental work of Adin and Roichman [1]. In their paper they defined a new statistic, the flag major index that extend to the Weyl group of type B the concept of major index, which is a classical and well studied statistic over the symmetric group. This paper opened the way to several others concerning the study of statistics over classical Weyl groups, wreath products, and some types of complex reflection groups. Now we have a wide picture of several combinatorial and algebraic properties holding for all these families of groups.
Recently, the second author introduced a new class of groups called projective reflection groups [11]. Among them, there is an infinite family denoted G(r, p, s, n) where r, p, s, n are integers such that p and s divide r, and ps divides rn. They include the infinite family of irreducible complex reflection groups, in fact G(r, p, 1, n) = G(r, p, n). In this context is fundamental a notion of duality. Let G = G(r, p, s, n), then his dual group is G * = G(r, s, p, n), where the parameters p and s are interchanged. In [11] it is shown that much of the representation theory of reflection groups can be extended to projective reflection groups, and that the combinatorics of the group G = G(r, p, s, n) is strictly related to the invariant theory of the dual group G * .
In this paper we continue that study. We introduce two descent numbers des and fdes, and a color sum col over G(r, p, s, n), that together with the flag major index fmaj [11] allow the extension to the groups G(r, p, s, n) of classical results in permutation enumeration.
We show that the polynomial χ(g)q fmaj(g) , where χ denotes any one-dimensional character of G(r, p, s, n) and the sum is over g ∈ G(r, p, s, n), admits a nice product formula. This generalizes a classical result of Gessel and Simion [22] for the symmetric group, and the main results of Adin-Gessel-Roichman [3] for the Coxeter group of type B, and of the first author [6] for the type D case.
The enumeration of permutations by number of descents and major index yields a remarkable q-analogue of a well-known identity for the Eulerian polynomials, usually called Carlitz's identity. This identity has been generalized in several ways and to several groups, see e.g. [2,5,7,8,9,12,20]. All these extensions can be divided into two families, depending on the used descent statistic, either the number of geometric descents or the number of flag descents. We give a general method to compute the trivariate distribution of des (or fdes), fmaj and col over G(r, p, s, n). This unifies and generalizes all previous cited results.
We further exploit our method to compute the generating function of the six statistics des, ides, fmaj, ifmaj, col, icol over the group G(r, p, s, n), (ides(g) denotes des(g −1 ) an similarly for the others). A specialization of this identity leads us to the computation of the generating function of the Hilbert series of the diagonal invariant algebras of the groups G(r, p, s, n).
The definition of the above statistics depends on the particular order chosen on the colored integer numbers. By comparing our results with others in the literature, we deduce that a different choice of the order may lead to the same generating function. We conclude the paper by providing bijective explanations of this phenomenon.

Notation and preliminaries
In this section we collect the notations that are used in this paper as well as the preliminary results that are needed.
We let Z be the set of integer numbers and N be the set of nonnegative integer numbers. For a, b ∈ Z, with a ≤ b we let [a, b] = {a, a + 1, . . . , b} and, for n ∈ N we let [n] we denote by ζ r the primitive r-th root of unity ζ r def = e 2πi r . For x ∈ Q, we denote ⌊x⌋ the biggest integer smaller than or equal to x. As usual for n ∈ N, we set [n] q def = 1 + q + q 2 + . . . + q n−1 , and P n the set of nondecreasing sequences of nonnegative integers (λ 1 , λ 2 , . . . , λ n ), that is partitions of length less than or equal to n. Let F ∈ C[[q 1 , q 2 , . . .]] be a formal power series with complex coefficients, and M a monomial. We denote by {F } M the power series obtained from F discarding all its homogeneous components not divisible by M .
The main subject of this work are the complex reflection groups [16], or simply reflection groups. The most important example of a complex reflection group is the group of permutations of [n], known as the symmetric group, that we denote by S n . We know by the work of Shephard-Todd [17] that all but a finite number of irreducible reflection groups are the groups G(r, p, n), where r, p, n are positive integers with p|r, that we are going to describe. If A is a matrix with complex entries we denote by |A| the real matrix whose entries are the absolute values of the entries of A. The wreath product G(r, n) = G(r, 1, n) is given by all n × n matrices satisfying the following conditions: the nonzero entries are r-th roots of unity; there is exactly one nonzero entry in every row and every column (i.e. |A| is a permutation matrix). If p divides r then the reflection group G(r, p, n) is the subgroup of G(r, n) given by all matrices A ∈ G(r, n) such that det A det |A| is a r p -th root of unity. A projective reflection group is a quotient of a reflection group by a scalar subgroup (see [11]). Observe that a scalar subgroup of G(r, n) is necessarily a cyclic group C s of order s, generated by ζ s I, for some s|r, where I denotes the identity matrix. It is also easy to characterize all possible scalar subgroups of the groups G(r, p, n): in fact the scalar matrix ζ s I belongs to G(r, p, n) if and only if s|r and ps|rn. In this case we let If G = G(r, p, s, n) then the projective reflection group G * def = G(r, s, p, n), where the roles of the parameters p and s are interchanged, is always well-defined. We say that G * is the dual of G and we refer the reader to [11] for the main properties of this duality.
We sometimes think of an element g ∈ G(r, n) as a colored permutation. If the nonzero entry in the i-th row of g ∈ G(r, n) is ζ ci r we let c i (g) def = R r (c i ) and say that c 1 (g), . . . , c n (g) are the colors of g. Now it is easy to see that an element g ∈ G(r, n) is uniquely determined by the permutation |g| ∈ S n defined by |g|(i) = j if g i,j = 0, and by its colors c i (g) for all i ∈ [n]. More precisely, when we consider an element of G(r, n) as a colored permutation we represent its elements either as couples (c 1 , . . . , c n ; σ), where σ = σ(1) · · · σ(n) is a permutation in S n and (c 1 , . . . , c n ) is the sequence of its colors, or in window notation as We call σ(i) the absolute value of g(i), denoted |g(i)|. When it is not clear from the context, we will denote c i by c i (g). Moreover, if c i = 0, it will be omitted in the window notation of g. In this notation the multiplication in G(r, n) = {(c 1 , . . . , c n ; σ) | c i ∈ [0, r − 1], σ ∈ S n } is given by (d 1 , . . . , d n ; τ )(c 1 , . . . , c n ; σ) = (R r (c 1 + d σ(1) ), . . . , R r (c n + d σ(n) ); τ σ).

Descent-type statistics
In this section we recall the notions of descents, flag descents, and flag major index for the wreath products. Then we introduce analogous definitions for descents and flag descents for projective reflection groups.
Definition. In all the paper we use the following order, called color order For g ∈ G(r, n) we define the descent set as where g(0) := 0, and denote its cardinality by des G (g). A geometric interpretation of this set in terms of Coxeter-like generators and length can be given as in [19, §7]. If we consider only positive descents we obtain and we denote des A (g) its cardinality. The flag major index [1] and the flag descent number [2], [5] are defined by fmaj(g) def = r maj(g) + col(g) (5) where as usual the major index maj(g) = i∈DesA(g) i is the sum of all positive descents of g.
For g ∈ G(r, p, s, n), we let The sequence λ(g) def = (λ 1 (g), . . . , λ n (g)) is a partition. The flag-major index for g ∈ G(r, p, s, n) is defined by [11, §5] fmaj(g) We define the descent number and the flag descent number of g ∈ G(r, p, s, n) respectively by des(g) def = sλ 1 (g) + r − s r and (10) Finally, for g = (c 1 , . . . , c n ; σ) ∈ G(r, p, s, n) we define the color of g by Note that the previous definitions do not depend on the particular representative of g ∈ G(r, p, s, n) chosen among its s lifts in G(r, n).
Remark. From [11,Lemma 5.1], it follows that for p = s = 1 the flag major index on G(r, p, s, n) defined in (9) coincides with the flag major index of Adin and Roichman for wreath products G(r, n) defined in (5). Moreover the definition of fdes is (11) is consistent with that in (6), and so where the last equality holds since These equalities give motivations to the definitions of the previous statistics.

One-dimensional characters and flag major index
The generating function of the major index with the unique nontrivial one-dimensional character admits a nice factorization formula over the symmetric group, as shown in Theorem 4.2 by Gessel-Simion (see [22]). The same happens for the other classical Weyl groups, as proved by Adin-Gessel-Roichman [3] for the type B case, and by the first author [6] for the type D case. In this section we generalize these results to all projective groups of type G(r, p, s, n).
The irreducible representations of G(r, p, s, n) are classified in [11, §6]. In particular, for n > 2, the one-dimensional characters of G(r, p, s, n) are all of the form where ǫ = ±1, and k ∈ [0, r p − 1] with the further condition that s divides kn, and c(g) is the sum of the colors of any element in G(r, p, n) representing the class of g (in particular c(g) = col(g) if s = 1). As usual for σ ∈ S n , we denote by inv(σ) def = |{(i, j) ∈ [n] × [n] | i < j and σ(i) > σ(j)}|, the number of its inversions, and by sign(σ) def = (−1) inv(σ) its sign. Our main results is the following one.
We recall that {F } q p is the polynomial obtained from F discarding all the homogeneous components in the variable q of degree not divisible by p. Hence for r = 2 and p = s = 1 in Theorem 4.1, we obtain [3, Theorems 5.1, 6.1, 6.2]; for r = s = 2 and p = 1 [6,Theorem 4.8].
In order to prove Theorem 4.1 we proceed in several steps passing through G(r, n) and G(r, 1, s, n) until the general case of G(r, p, s, n). The basic stone is given by the following results for the symmetric group.
The particular case of the Weyl groups B n def = G(2, n) is treated in [3]. In this paper the authors focus on the sum where ℓ(g) is the Coxeter length of g with respect to a given set of simple reflections, making use of a combinatorial interpretation of ℓ(g) due to Brenti [10]. Theorem 4.1 is then achieved in this case since Although one can define an analogue of the Coxeter length for the wreath products G(r, n) (see e.g., [4], [19]), this does not lead to a one-dimensional character of the group, and the corresponding sum does not factorize nicely. This is why we focus on one-dimensional characters, obtaining in particular also a new proof in the case of Weyl groups of type B that does not make use of the combinatorial interpretation of the length function.
The following is a technical lemma which is the multivariate extension of a result appearing in [3, §5]. For r > 0 and k r−1 , k r−2 , . . . , k 0 ∈ N such that k r−1 + k r−2 + · · · + k 0 = n (i.e. (k r−1 , . . . , k 0 ) is a composition of n) we let Proof. The window notation of σ ∈ S(k r−1 , . . . , k 0 ) can be split into r blocks of length k r−1 , . . . , k 0 , the entries in each block being increasing. Consider the following involution on S(k r−1 , . . . , k 0 ) We recall that the window notation of (2i − 1, 2i)σ is obtained from that of σ by exchanging 2i − 1 and 2i, and so this map is actually an involution on S(k r−1 , . . . , k 0 ). Since σ and (2i − 1, 2i)σ have opposite signs we can restrict our attention to those σ ∈ S(k r−1 , . . . , k 0 ) such that 2i − 1 and 2i belong to the same block, for all i ∈ {1, . . . , ⌊ n 2 ⌋}. It is clear that if two among the k i 's are odd such σ does not exist and the result follows in this case. We can thus assume that at most one of the k i 's is odd, and in this case all entries of the window notation of σ are uniquely determined by the even entries. This is because an odd entry j different from n belongs to the same block of j + 1 and, if n is odd, n necessarily belongs to the unique block of odd length. The number of even entries is ⌊ n 2 ⌋ and these should be combined into r parts of cardinality ⌊ kr−1 2 ⌋, . . . , ⌊ k0 2 ⌋. It remains to show that all these elements have even sign. We first note that there are no inversions of the form (σ −1 (2j − 1), σ −1 (2j)). Then we observe that, for all i < 2j − 1 < n, This completes the proof if n is even. If n is odd we only have to make the final remark that there are an even number of inversions of the form (σ −1 (n), σ −1 (i)), since n appears in an odd position.
One can show that G(r, n) = U S n , with U ∩ S n = 1, i.e. for each g ∈ G(r, n) there exist unique τ ∈ U and σ ∈ S n such that g = τ σ (see [9,Proposition 4.1]). Moreover we have col(g) = col(τ ) and Des A (g) = Des A (σ). It follows that by Theorem 4.2. Now we claim that In fact let U (k r−1 , . . . , In other words the elements in U (k r−1 , . . . , k 0 ) are those in U having the first k r−1 entries colored with r − 1, then k r−2 entries colored with r − 2 and so on up to the last k 0 entries which are colored with 0. It is clear that the map τ → |τ | is a bijection between U (k r−1 , . . . , k 0 ) and S(k r−1 , . . . , k 0 ). Then by Lemma 4.3, where the last sum is taken over all r-tuples k r−1 , . . . , k 0 such that k r−1 + · · · + k 0 = n and at most one of the k i 's is odd. So, if n = 2m is even the k i 's are all even, we let k i = 2t i , and If n = 2m + 1 is odd there is exactly one of the k i which is odd. We split the sum according to this odd entry: if k i is odd we let k i = 2t i + 1 and if it is even we let k i = 2t i . Then If n is even the result follows immediately from Equations (13), (14), and (15). If n is odd the result follows similarly using the further observation that [r] q [2m + 1] q r = [(2m + 1)r] q . Now we face the problem of the quotient groups G(r, 1, s, n). We need a preliminary result.
Lemma 4.5. Let g ∈ G(r, 1, s, n) and g 0 , . . . , g s−1 be the s lifts of g in G(r, n). We have
where we have used the fact that r/s is an integer and hence (n − 1)nr/s is even.
Proof. It is a consequence of [5,11] that g∈G(r,1,s,n) and therefore we may unify this equation and Corollary 4.6 by obtaining where ǫ = ±1. We now want to describe the polynomial g∈G(r,p,s,n) ǫ inv(|g|) q fmaj(g) . To this end we observe that an element g ∈ G(r, 1, s, n) belongs to G(r, p, s, n) if and only if fmaj(g) ≡ 0 mod p (see [11,Lemma 5.2]). Therefore we have g∈G(r,p,s,n) We observe that Notice that F (q) q p = 1 p p−1 j=0 F (ζ j p q). Hence from (16) and (17) we have g∈G(r,p,s,n) The last factor can be manipulated as follows: where m = n/2 if n is even and m = (n − 1)/2 if n is odd. Hence we obtain that g∈G(r,p,s,n) Now we are ready to prove our main result.

Carlitz's Identities
In this section we give a general method to compute the trivariate distribution of des (or fdes), fmaj and col over G(r, p, s, n). This unifies and generalizes all related results cited in the introduction, and provides two different generalizations of Carlitz's identity for the group G(r, p, s, n).
Moreover, if f ∈ N n , for any k > 0 we let be the sum of the residues of the f i 's modulo k. Proof. This result is similar to several others appearing in the literature and in particular is a special case of the bijection appearing in [11,Theorem 8.3] and so we simply describe how the bijection is defined for the reader's convenience. If f = (f 1 , . . . , f n ) ∈ N n (p), then g is the unique element in G(r, p, s, n) having a liftg ∈ G(r, p, n) satisfying: Letting µ be the partition obtained by reordering the entries in (f 1 , . . . , f n ) in nonincreasing order one can show that µ − λ(g) (componentwise difference) is still a partition whose parts are all congruent to the same multiple of r/s mod r (i.e. µ is g-compatible in the notation of [11]). The partition λ ∈ P n and the integer h ∈ [0, s − 1] are therefore uniquely determined by the requirement that µ = λ(g) + r · λ + (hr/s, . . . , hr/s).
All the other statements are straightforward consequences, since λ i (g) ≡ c i (g) mod r/s for any liftg of g, and so also col(g) = n i=1 R r/s (λ i (g)).
We will make use of the following observation several times: if F is a set and m : F → N is such that the set {f ∈ F | m(f ) ≤ k} is finite for all k ∈ N then we always have where φ is any map defined on F taking values in a (polynomial) ring.
We can now state the main result of this section. s which is equivalent to max(f ) ≤ kr/s since max(f ) is an integer. We want to make use of Equation (18) with φ(f ) = q |f | a col r s (f ) and so we compute Therefore, by Equation (18), Now we want to compute this polynomial using the bijection described in Lemma 5.1. Observe that if f → (g, λ, h) then m(f ) = des(g) + sλ 1 + h. Therefore we have t des(g)+sλ1+h q fmaj(g)+r|λ|+nhr/s a col(g) = g∈G(r,p,s,n) and the result follows.
Letting s = p = 1 in the previous result we obtain [ The special case with r = 2, p = s = 1 of Corollary 5.3 is the main result of [12], and for p = s = 1 we obtain [13,Theorem 10 (iv)].
Now we see how a simple modification of the same ideas lead to the generalization of other identities that use different flag-descents.  = g∈G(r,p,s,n) t fdes(g) q fmaj(g) a col(g) where Q r/s (k) is determined by k = r/s · Q r/s (k) + R r/s (k).
A proof of this result can be done by paralleling that of Theorem 5.2. In this case one should simply consider the statistics m(f ) = max(f ) and follow the rest mutatis mutandis. By letting a = 1 in Equation 19, one obtain a second Carlitz's identity type, with flag-descents.

Multivariate generating functions
In this section we make further use of the bijection [11,Theorem 8.3] to compute new multivariate distributions for the groups G(r, p, s, n). We first concentrate on the case of the groups G(r, n) to make the used arguments more clear to the reader. We need to state the particular case of this bijection needed for our purposes, namely the special case of 2-partite partitions.
We recall that a 2-partite partition of length n (see [14]) is a 2 × n matrix with non negative integer . . . f (2) n satisfying the following conditions: One may think of a 2-partite partition as a generic multiset of pairs of nonnegative integers (the columns of f ) of cardinality n. If f is a 2-partite partition we denote by f (1) and f (2) the first and the second row of f respectively. To be consistent with the notation of [11, §8], we denote by B(n) the set of 2-partite partitions of length n and we let Note that B(1, n) = B(n) and that B(r, n) = B(r, 1, 1, n).
We can easily verify that m i (f ) ≤ k i if and only if max(f (i) ) ≤ rk i . Therefore, using an idea already appearing in [14], one can show that

Now we use a natural bivariate extension of Equation (18) to obtain
On the other hand, by Proposition 6.1 used with s = 1, we have and the proof is complete.
Now we want to extend this result to the quotients G(r, 1, s, n). .
Proof. Say that f ∈ B(r, s, 1, n) is of type l, with l ∈ On the other hand, by Proposition 6.1, we have .
The corresponding result for the the full class of projective reflection groups G(r, p, s, n) can now be easily deduced from Theorem 6.3. We remind that G(r, p, s, n) is defined if and only if p|r, s|r, and sp|rn. In particular if r, p, s are fixed then G(r, p, s, n) is defined only if sp/GCD(sp, r) divides n.
. 6.1. Hilbert series of diagonal invariant algebras. We can now make a specialization of this result. We set a 1 = a 2 = 1 in Theorem 6.4 .
Multiplying both sides of the previous identity by (1 − t 1 )(1 − t 2 ) and then taking the limit as t 1 , t 2 → 1 we obtain There is a nice algebraic interpretation of the previous identity that goes back to the work of Adin and Roichman [1]. We let S p [X, Y ] be the subalgebra of the algebra of polynomials in 2n variables x 1 , . . . , x n , y 1 , . . . , y n generated by 1 and the monomials whose degrees in both the x's and the y's variables is divided by p. Then we can observe (see [11, §8]) that the factor is the bivariate Hilbert series of the invariant algebra of the "tensorial action" of the group G(r, s, p, n) 2 = G(r, s, p, n) × G(r, s, p, n) (note the interchanging of the roles of p and s) on the ring of polynomials S p [X, Y ]. Furthermore by [11,Corollary 8.6] we can deduce that g∈G(r,p,s,n) q where ∆G(r, s, p, n) denotes the diagonal embedding of G(r, s, p, n) in G(r, s, p, n) 2 . We can now conclude that Equation (21) provides the following interpretation for the generating function of the Hilbert series of the diagonal invariant algebras of the groups G(r, p, s, n). This identity was apparently known only in the case of the symmetric group [14,21].

Orderings
In the literature, when considering statistics based on descents and/or inversions on the groups G(r, n) at least two distinct orderings have been considered on the set of colored integers. In this last section we would like to clarify some of the relationships between two of these orderings concerning the results shown in this paper.
The order < ′ is the "good" one to give a combinatorial interpretation of the length function in G(r, n) in terms of inversions (see e.g. [19], [4]), while the order < is often used in the study of some algebraic aspects such as the invariant theory of B n and G(r, n) (see [1], [11]). Here we give a motivation of these choices. So we denote • Des ′ G (g) = {i ∈ [0, n − 1] | g(i) > ′ g(i + 1)}; • des ′ G (g) = |Des ′ G (g)|; • fmaj ′ (g) = r · i∈Des ′ G (g) i + col(g).
If one considers the special case of Theorem 5.2 with p = s = 1, and relate it to [8, Theorem 5.1] one can deduce that the two polynomials g∈G(r,n) t des G (g) q fmaj(g) a col(g) and g∈G(r,n) t des ′ G (g) q fmaj ′ (g) a col(g) are equal. This can also be easily proved bijectively.
It follows from Proposition 7.1 that the two statistics fmaj and fmaj ′ are equidistributed on G(r, n). Nevertheless the two polynomials g∈G(r,n) χ(g)q fmaj(g) and g∈G(r,n) χ(g)q fmaj ′ (g) , where χ is any linear character of G(r, n), do not coincide in general. So, in order to obtain the results appearing in §4 we must consider the order <. We should also mention that the polynomial g∈G(r,n) χ(g)q fmaj ′ (g) does not factor nicely at all in general.
There is a more subtle difference if one considers the multivariate distributions g∈G(r,n) t desG(g) 1 t desG(g −1 ) 2 q fmaj(g) 1 q fmaj(g −1 ) 2 a col(g) 1 a col(g −1 ) 2 , and the analogue computed with the order < ′ . In fact, by comparing Theorem 6.2 and [9, Theorem 7.1] we deduce that these two polynomials coincide for r = 1, 2, and are distinct if r > 2. The case r = 1 being trivial we can justify this coincidence for r = 2 with a bijective proof. We split the set Des G (g) into four subsets.