Quasi-Invariant and Super-coinvariant Polynomials for the Generalized Symmetric Group
The aim of this work is to extend the study of super-coinvariant polynomials, introduced in [2, 3], to the case of the generalized symmetric group , defined as the wreath product of the symmetric group by the cyclic group. We define a quasi-symmetrizing action of on , analogous to those defined in  in the case of . The polynomials invariant under this action are called quasi-invariant, and we define super-coinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasi-invariant polynomials with no constant term. Our main result is the description of a Gröbner basis for the ideal generated by quasi-invariant polynomials, from which we dedece that the dimension of the space of super-coinvariant polynomials is equal to where is the -th Catalan number.
Résumé. Le but de ce travail est d’étendre l’étude des polynômes super-coinvariants (définis dans ), au cas du groupe symétrique généralisé , défini comme le produit en couronne du groupe symétrique par le groupe cyclique. Nous définissons ici une action quasi-symétrisante de sur , analogue à celle définie dans  dans le cas de . Les polynômes invariants sous cette action sont dits quasi-invariants, et les polynômes super-coinvariants sont les polynômes orthogonaux aux polynômes quasi-invariants sans terme constant (pour un certain produit scalaire). Notre résultat principal est l’obtention d’une base de Gröbner pour l’idéal engendré par les polynômes quasi-invariants. Nous en déduisons alors que la dimension de l’espace des polynômes super-coinvariants est où est le -ième nombre de Catalan.
Let denote the alphabet in variables and denote the space of polynomials with complex coefficients in the alphabet . Let denote the wreath product of the symmetric group by the cyclic group . This group is sometimes known as the generalized symmetric group (cf. ). It may be seen as the group of matrices in which each row and each column has exactly one non-zero entry (pseudo-permutation matrices), and such that the non-zero entries are -th roots of unity. The order of is . When , reduces to the symmetric group , and when , is the hyperoctahedral group , i.e. the group of signed permutations, which is the Weyl group of type (see  for example for further details). The group acts classically on by the rule (1.1) where is the transpose of the matrix and is considered as a row vector. Let denote the set of -invariant polynomials. Let us denote by the set of such polynomials with no constant term. We consider the following scalar product on : (1.2) where stands for and stands for . The space of -coinvariant polynomials is then defined by where denotes the ideal generated by a subset of .
A classical result of Chevalley  states the following equality: (1.3) which reduces when to the theorem of Artin  that the dimension of the harmonic space (cf. ) is .
Our aim is to give an analogous result in the case of quasi-symmetrizing action. The ring of quasi-symmetric functions was introduced by Gessel  as a source of generating functions for -partitions  and appears in more and more combinatorial contexts [5, 18, 19]. Malvenuto and Reutenauer  proved a graded Hopf duality between and the Solomon descent algebras and Gelfand et. al.  defined the graded Hopf algebra of non-commutative symmetric functions and identified it with the Solomon descent algebra.
In [2, 3], Aval et. al. investigated the space of super-coinvariant polynomials for the symmetric group, defined as the orthogonal (with respect to (1.2)) of the ideal generated by quasi-symmetric polynomials with no constant term, and proved that its dimension as a vector space equals the -th Catalan number: (1.4) Our main result is a generalization of the previous equation in the case of super-coinvariant polynomials for the group .
In Section 2, we define and study a “quasi-symmetrizing” action of on . We also introduce invariant polynomials under this action, which are called quasi-invariant, and polynomials orthogonal to quasi-invariant polynomials, which are called super-coinvariant. The Section 3 is devoted to the proof of our main result (Theorem 2.4), which gives the dimension of the space of super-coinvariant polynomials for : we construct an explicit basis for from which we deduce its Hilbert series.
2. A quasi-symmetrizing action of
We use vector notation for monomials. More precisely, for , we denote the monomial (2.1) For a polynomial , we further denote as the coefficient of the monomial in .
Our first task is to define a quasi-symmetrizing action of the group on , which reduces to the quasi-symmetrizing action of Hivert (cf. ) in the case . This is done as follows. Let be a subalphabet of with variables and be a vector of positive () integers. If is a vector whose entries are distinct variables multiplied by roots of unity, the vector is obtained by ordering the elements in with respect to the variable order. Now the quasi-symmetrizing action of is given by (2.2) where is the weight of , i.e. the product of its non-zero entries, is the matrix obtained by taking the modules of the entries of , and the oefficient is defined as follows:
Example 2.1.If and , and we denote by the complex number , then for example
It is clear that this defines an action of the generalized symmetric group on , which reduces to Hivert’s quasi-symmetrizing action (cf. , Proposition 3.4) in the case .
Let us now study its invariant and coinvariant polynomials. We need to recall some definitions.
A composition of a positive integer is an ordered list of positive integers () whose sum is . For a vector , let represent the composition obtained by erasing zeros (if any) in . A polynomial is said to be quasi-symmetric if and only if, for any and in , we have whenever . The space of quasi-symmetric polynomials in variables is denoted by .
The polynomials invariant under the action (2.2) of are said to be quasi-invariant and the space of quasi-invariant polynomials is denoted by , i.e. Let us recall (cf. , Proposition 3.15) that . The following proposition gives a characterization of .
Proposition 2.2.One has where .
Proof.Let be an element of . Let us denote by the -th root of unity and by the element of whose matrix is with the in place . Then we observe that the identities imply that every exponents appearing in are multiples of . Thus there exists a polynomial such that . To conclude, we note that implies that is quasi-symmetric, whence is also quasi-symmetric.
The reverse implication is obvious. ∎
Let us now define super-coinvariant polynomials: with the scalar product defined in (1.2). This is the natural analogous to in the case of quasi-symmetrizing actions and reduces to the space of super-harmonic polynomials (cf. ) when .
Remark 2.3.It is clear that any polynomial invariant under (2.2) is also invariant under (1.1), i.e. . By taking the orthogonal, this implies that . These observations somewhat justify the terminology.
Our main result is the following theorem which is a generalization of equality (1.4).
Theorem 2.4.The dimension of the space is given by (2.3)
Remark 2.5.In the case of the hyperoctahedral group , C.-O. Chow  defined a class of quasi-symmetric functions of type in the alphabet . His approach is quite different from ours. In particular, one has the equality: In the study of the coinvariant polynomials, it is not difficult to prove that the quotient is isomorphic to the quotient studied in . To see this, we observe that if is the Gröbner basis of constructed in  (see also the next section), then the set is a Gröbner basis (any syzygy is reducible thanks to Buchberger’s first criterion, cf. ).
The next section is devoted to give a proof of Theorem 2.4 by constructing an explicit basis for the quotient .
3. Proof of the main theorem
Our task is here to construct an explicit monomial basis for the quotient space . Let us first recall (cf. ) the following bijection which associates to any vector a path in the plane with steps going north or east as follows. If , the path is For example the path associated to is
We distinguish two kinds of paths, thus two kinds of vectors, with respect to their “behavior” regarding the diagonal . If the path remains above the diagonal, we call it a Dyck path, and say that the corresponding vector is Dyck. If not, we say that the path (or equivalently the associated vector) is transdiagonal. For example is Dyck and is transdiagonal.
We then have the following result which generalizes Theorem 4.1 of  and which clearly implies the Theorem 2.4.
Theorem 3.1.The set of monomials is a basis for the quotient .
To prove this result, the goal is here to construct a Gröbner basis for the ideal . We shall use results of [2, 3].
Recall that the lexicographic order on monomials is (3.1) if and only if the first non-zero part of the vector is positive.
For any subset of and for any positive integer , let us introduce . If we denote by the unique reduced monic Gröbner basis (cf. ) of an ideal , then the simple but crucial fact in our context is the following.
Proposition 3.2.With the previous notations, (3.2)
Proof.This is a direct consequence of Buchberger’s criterion. Indeed, if for every pair in , the syzygy reduces to zero, then the syzygy also reduces to zero in by exactly the same computation. ∎
Let us recall that in  is constructed a family of polynomials indexed by transdiagonal vectors . This family is constructed by using recursive relations of the fundamental quasi-symmetric functions and one of its property (cf. ) says that the leading monomial of is: . Since is a Gröbner basis of , the following result is a consequence of Propositions 2.2 and 3.2.
Proposition 3.3.The set is a Gröbner basis of the ideal .
To conclude the proof of Theorem 3.1, it is sufficient to observe that the monomials not divisible by a leading monomial of an element of , i.e. by a for transdiagonal, are precisely the monomials appearing in the set .
As a corollary of Theorem 3.1, one gets an explicit formula for the Hilbert series of . For , let denote the projection (3.3) where is the vector space of homogeneous polynomials of degree together with zero.
Let us denote by the Hilbert series of , i.e. (3.4)
Let us recall that in  is given an explicit formula for : (3.5) using the number of Dyck paths with a given number of factors (cf. ).
The Theorem 3.1 then implies the
Corollary 3.4.With the notations of (3.5), the Hilbert series of is given by from which one deduces the close formula
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Appendix 1 by Sergey Fomin.