Odd and even major indices and one-dimensional characters for classical Weyl groups

We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.


Introduction
In recent years a new statistic on the symmetric group has been introduced and studied in relation with vector spaces over finite fields equipped with a certain quadratic form ( [20]). This statistic combines combinatorial and parity conditions and is now known as the odd inversion number, or odd length ( [9], [11]). Analogous statistics have later been defined and studied for the hyperoctahedral and even hyperoctahedral groups ( [29], [30], [10]), and more recently for all Weyl groups ( [11]). A crucial property of this new statistic is that its signed (by length) generating function over the corresponding Weyl group always factors explicitly ( [11], [32]).
Another line of research in the last 20 years has been the definition and study of analogues of the major index statistic for the other classical Weyl groups, namely for the hyperoctahedral and even hyperoctahedral groups (see, e.g., [1], [4], [6], [13], [14], 1 2010 Mathematics Subject Classification: Primary 05A15; Secondary 05E15, 20F55. [15], [23], [24], [31]) and for finite Coxeter groups ( [26]). It is now generally recognized that, among these, the ones with the best properties are those first defined by Adin and Roichman in [1] for the hyperoctahedral group and by Biagioli and Caselli in [6] for the even hyperoctahedral group.
Our purpose in this work is to define odd (and even) analogues of these major index statistics for the classical Weyl groups and show that their generating function twisted by the one-dimensional characters of the corresponding Weyl group always factors in an explicit way. More precisely, we show that certain multivariate refinements of these generating functions always factor explicitly. As consequences of our results we obtain odd and even analogues of Carlitz's identity [12], which involves overpartitions, of the Gessel-Simion Theorem (see, e.g., [2,Theorem 1.3]), of several other results appearing in the literature ([2, Theorems 5.1, 6.1, 6.2] and [5,Theorem 4.8]) and an extension, and refinement of a result of Wachs ([33]).
The organization of the paper is as follows. In the next section we recall some definitions and results that are used in the sequel. In §3 we define and study odd and even analogues of the major index and descent statistics of the symmetric group. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs. In §4 we define odd and even analogues of the major index statistics introduced in [1] and [6] for the classical Weyl groups of types B and D, respectively, and of the usual descent statistics on these groups. More precisely, we compute a multivariate refinement of the generating functions of these statistics twisted by the one-dimensional characters of the corresponding groups and show that they always factor explicitly. Finally, in §5, we show that, under some mild and natural hypotheses, there is no "odd major index" that is equidistributed with the odd length in the symmetric or hyperoctahedral groups.

Preliminaries
In this section we recall some notation, definitions, and results that are used in the sequel.
As N we denote the set of non-negative integers and as P the set of positive integers. If  [i] q . We also find it convenient to let p n := (1 + (−1) n )/2.
The cardinality of a set X is denoted by |X| and the power set of X by P(X). For n, k ∈ N we let [n] k := {A ∈ P([n]) : |A| = k}.
Next we recall some basic results in the theory of Coxeter groups which are useful in the sequel. The reader can consult [7] or [19] for further details. Let (W, S) be a Coxeter system. The length of an element z ∈ W with respect to S is denoted as ℓ(z). If J ⊆ S and w ∈ W we let W J := {w ∈ W : ℓ(ws) > ℓ(w) ∀ s ∈ J}, D(w) := {s ∈ S : ℓ(ws) < ℓ(w)} and, more generally, for any A ⊆ W we let A J := A ∩ W J . When the group W is finite, there exists a unique element w 0 of maximal length.
For any n ∈ P let S n be the group of bijections of the set [n]. For σ, τ ∈ S n we let στ := σ • τ (composition of functions). It is well known (see e.g. [7]) that this is a Coxeter group with set of generators {s 1 , s 2 , ..., s n−1 }, s i being, in one line notation, 12...(i+1)i...n.
. . , a k } < , and σ ∈ S(A) we let τ be the only element of S |A| defined by σ(a i ) = a τ (i) for all i ∈ [k]. We call τ the flattening of σ and write F (σ) = τ . Moreover, we define: The elements of S B n are the bijective functions σ : [±n] → [±n] satisfying −σ(i) = σ(−i), for all i ∈ [n]. We use the window notation. So, for example, the element [−2, 1] ∈ S B 2 represents the function σ : It is well known that (S B n , S B ) is a Coxeter system of type B n and that, identifying S B with [0, n−1], the following holds (see, e.g., [7, §8.1]). Given σ ∈ S B n we let We let S D n be the subgroup of S B n defined by S D n := {σ ∈ S B n : | Neg(σ)| ≡ 0 (mod 2)}, s 0 := (1, −2)(2, −1), and S D := {s 0 , s B 1 , ..., s B n−1 }. It is then well known that (S D n , S D ) is a Coxeter system of type D n , and that the following holds (see, e.g., [7, §8.2]).
For simplicity we often write B n and D n respectively in place of S B n and S D n . We refer to [7,Chapter 8] for further details on the combinatorics of the groups S B n and S D n . The descent number and the major index are the functions des : S n → N and maj : S n → N defined respectively by des(σ) := |D(σ)|, and maj(σ) := i∈D(σ) i, for all σ ∈ S n .

Type A
In this section we introduce and study odd and even analogues of the descent and major index statistics for the symmetric group. In particular, we obtain odd and even analogues of Carlitz's identity, of the Gessel-Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs.
For any J ⊆ [n − 1] we let the parabolic q-Eulerian polynomials and parabolic signed q-Eulerian polynomials, respectively, be where A n (q, x) and B n (q, x) are, respectively, the q-Eulerian polynomials and the signed q-Eulerian polynomials, as defined in [33].
Our goal is to compute the generating functions of odes, omaj, and of edes, emaj, twisted by the one-dimensional characters of the symmetric groups. Our first result is the odd and even analogue (for y = −1 and x = 1) of the Gessel-Simion Theorem (see, e.g. Proof. Let, for brevity, A o n (y, q, x) := σ∈Sn y ℓ(σ) q omaj(σ) x odes(σ) . We prove the first equation by induction on n 1. We have that A o 1 (y, q, x) = 1, and A o 2 (y, q, x) = 1 + yxq. So let n ≥ 3. For n ≡ 1 (mod 2) we find, by our inductive hypothesis, as desired. For n ≡ 0 (mod 2) we have, by our inductive hypothesis, as desired.
The proof for the even statistics is analogous, and is therefore omitted.
Proof. By Theorem 3.1 we have that and the result follows immediately. The proof of the second equation is identical.
Note that Corollary 3.2 can also be stated in terms of super-Schur functions. Given a partition λ and variables x 1 , ..., x m , y 1 , ..., y n we denote by s λ (x 1 , ..., x m /y 1 , ..., y n ) the super-Schur function (also known as hook Schur function, see [3]) associated to λ (we refer the reader to [27] for the definition and further information about super-Schur functions).
Then we have, by [8, Equation (6)], A second corollary of Theorem 3.1 is the odd-even analogue of the Gessel-Simion Theorem. Recall our definition of p n from §2.
A further corollary is the following. Proof. It follows immediately from Theorem 3.1 that (1 + q i ). But it is well known (see, e.g., [28]) that the polynomial Another unimodality result that arises from Corollary 3.2 is the following. Proof. Let P n,m (q) := {λ∈P:λ 1 n,ℓ(λ)=m} q |λ| . It is easy to see that deg(P n,m ) = nm and that q (n+1)m P n,m (q −1 ) = P n,m (q). Moreover, by Corollary 3.2 and Theorem 3.
For n ∈ P we define: So, for example, 21534 ∈ D(S 5 ) while 23541 / ∈ D(S 5 ). We call the elements of D(S n ) domino permutations. It is not hard to see that |D(S n )| = 2 ⌊ n 2 ⌋ n 2 ! and that, if n is even, The proposition below gives a sign-reversing involution that reduces the computation of the signed generating function over any quotient to the corresponding set of domino permutations.
Proposition 3.6. Let n ∈ P, and J ⊆ [n − 1]. Then Note that there is a bijection between D(S 2m ) and S m ×P([m]) obtained by associating The proof of the following result is a routine check, and is therefore omitted. The following result is a refinement, and extension, of [33, Theorem 1] (which is the case J = ∅, q 1 = q 2 = q 2 , x 1 = x/q and x 2 = x).
The result follows from Proposition 3.6.
We note the following consequences of Theorem 3.8.
Note that for symmetric groups of odd rank the bivariate signed generating function  Proof. This follows immediately by taking q 1 = q 2 = q 2 , x 1 = x q , and x 2 = x in Theorem 3.8.
We note that the previous "sign-balance" identities are examples of the phenomenon described in [18], namely that the signed enumeration on 2m objects by certain statistics is essentially equivalent to the ordinary enumeration on m objects by the same statistics.

Types B and D
In this section we define and study odd and even analogues of the descent and flag-major statistics on the classical Weyl groups of types B and D, and compute the generating functions of these statistics twisted by the one-dimensional characters of these groups.
For i ∈ [n − 1] we write, for brevity, "s i " rather than "s B i ". We also define for all i ∈ [n − 1]. Note that the involution of B n defined by σ → s * i σ, restricts to D n . We define six statistics on the hyperoctahedral group B n by letting for all σ ∈ B n . So, for example, if σ is as above then odes D (σ) = 1, oDmaj(σ) = 5, edes D (σ) = 2 and eDmaj(σ) = 2. We call oDmaj and eDmaj the odd D-major index and the even D-major index of σ ∈ B n , respectively.

The trivial character
We start with the trivial characters of B n and D n . Recall that we let p n := (1 + (−1) n )/2.
Proof. We only prove the formula for the odd statistics, the proof for the even ones being analogous. We proceed by induction on n ≥ 2, the result being easy to check for n = 2.
The corresponding result for D n is a consequence of the one in type B. Note that if (1) Proof. Let n ≥ 3 be odd. Then  (1). Let now n be even. Let σ ∈ B n and [τ, i, j] be its window notation (i, j ∈ [±n]). Then we have that Therefore and the result again follows from Theorem 4.1 and (1).
The proof for the even statistics is analogous and is therefore omitted.

The alternating character
The computation for the character σ → (−1) ℓ B (σ) of S B n is considerably more involved. We begin with the following reduction result. For n ∈ P we let    For shortness, we define the following two monomials, for any σ ∈ B n : σ o (x, y) := x omaj(σ) y odes(σ) and σ e (x, y) := x emaj(σ) y edes(σ) . We need to prove a further reduction result.
Proof. Note that, by Proposition 4.3, For the even statistics note that the assignment σ → s * |σ(1)| σ defines an involution of B 2m which preserves the functions emaj and edes, so the result follows.
Proof. Let 1 ∈ S. Then 0 ∈ D(σ) for all σ ∈ B n such that Neg(σ) = S. We have that, by Lemma 4.5, o (x, y), and the result follows by Lemma 4.6. If 1 ∈ S the result follows by analogous computations.
Note that if |S| is even (resp., odd) the first (resp., second) case in the above lemma cannot occur. We can now prove one of our main results. Recall that we have defined {σ∈Bn: σ(n)<0 neg(σ)≡ǫ (mod 2)} where ǫ ∈ {0, 1} and O n (x, y, z 1 , for shortness and we begin by proving (2) for ǫ = 1. Note that This already proves (2) if ǫ = 1 and n is even, by Lemma 4.6. Suppose now that n is odd.
We now prove (2) for ǫ = 0. Suppose first that n is even. Then proceeding as in the previous case and using Lemma 4.6 we have that by Lemma 4.6 and (2) again follows.
We now prove (3) for ǫ = 1. We have that Finally, if n is even, then we obtain similarly that This proves (3) for ǫ = 0 and this concludes the proof.
The following is the even analogue of the previous theorem. Its proof is similar and is therefore omitted.
Theorem 4.9. Let n 2. Then As a consequence of Theorems 4.8 and 4.9 we can now easily obtain the generating function of the odd flag-major index, the odd descent number, and the odd negative number, twisted by the alternating character of the hyperoctahedral group, and the corresponding even ones. The following result is the odd and even analogue, for y = z = 1, We note that Corollary 4.10 implies that σ∈Bn (−1) ℓ B (σ) x omaj(σ) y odes(σ) = 0 which is also implied by [25,Theorem 3.2].
does not factor nicely, in general, for example, if n = 5 then one obtains ). As another simple consequence of Theorems 4.8 and 4.9 we also obtain the generating function of oDmaj, odes D and oneg D , and the corresponding even one for the only nontrivial one-dimensional character of the even hyperoctahedral group. This result is the odd and even analogue, for y = z = 1, of [5,Theorem 4.8].
We note that, as in the case of the hyperoctahedral group, does not seem to factor, in general.

The other characters
We conclude by computing the generating function of the statistics studied in this section twisted by the remaining one-dimensional characters of the hyperoctahedral group.
As in the case of the alternating character, the following corollary can be deduced directly from Theorems 4.8 and 4.9. The next result gives the odd and even analogue, for y = z = 1, of [2, Theorem 6.1].
Corollary 4.12. Let n 2. Then does not factor explicitly in general.
To calculate the generating function of the statistics studied in this section twisted by the remaining character we begin with a reduction result.  Proof. Let σ ∈ B n and i ∈ [n − 1] be such that i ≡ 1 (mod 2) and |σ(i)| > |σ(i + 1)|.
The second formula is proved analogously.
The second equality follows analogously, using the reduction of Proposition 4.13.
The joint distribution of ofmaj and efmaj twisted by "negative" character does not seem to factor.

Final comments
It is clear that the most desirable property that one would like an "odd major index" to possess is that it is equidistributed with the odd length. It is easy to see that, if we require, as seems reasonable, such an odd major index to be a function of the descent set, such an odd major index does not exist in general. For example, one has that π∈S 5 i∈D(π) x i = 1 + 4x 4 + 9x 3 + 6x 3 x 4 + 9x 2 + 16x 2 x 4 + 11x 2 x 3 + 4x 2 x 3 x 4 + 4x 1 and one can check that there are no j 1 , j 2 , j 3 , j 4 ∈ N such that π∈S 5 i∈D(π) x j i = 1 + 12x + 23x 2 + 48x 3 + 23x 4 + 12x 5 + x 6 = π∈S 5 x L(π) where L(π) := |{(i, j) ∈ [n] 2 : i < j, π(i) > π(j), i ≡ j (mod 2)}| is the odd length of the symmetric group.

Acknowledgements
The first author would like to thank Sylvie Corteel for useful and interesting conversations that led to the proof of Proposition 3.5. This material is partly based upon work supported by the Swedish Research Council under grant no. 2016-06596 while the first author was in residence at Institut Mittag-Leffler in Djursholm, Sweden during Spring 2020.
The first author is partially supported by the MIUR Excellence Department Project CUP E83C18000100006.