QuasiInvariant and Supercoinvariant Polynomials for the Generalized Symmetric Group
Abstract.
The aim of this work is to extend the study of supercoinvariant 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 quasisymmetrizing action of on , analogous to those defined in [12] in the case of . The polynomials invariant under this action are called quasiinvariant, and we define supercoinvariant polynomials as polynomials orthogonal, with respect to a given scalar product, to the quasiinvariant polynomials with no constant term. Our main result is the description of a Gröbner basis for the ideal generated by quasiinvariant polynomials, from which we dedece that the dimension of the space of supercoinvariant polynomials is equal to where is the th Catalan number.
Résumé. Le but de ce travail est d’étendre l’étude des polynômes supercoinvariants (définis dans [2]), au cas du groupe symétrique généralisé , défini comme le produit en couronne du groupe symétrique par le groupe cyclique. Nous définissons ici une action quasisymétrisante de sur , analogue à celle définie dans [12] dans le cas de . Les polynômes invariants sous cette action sont dits quasiinvariants, et les polynômes supercoinvariants sont les polynômes orthogonaux aux polynômes quasiinvariants 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 quasiinvariants. Nous en déduisons alors que la dimension de l’espace des polynômes supercoinvariants est où est le ième nombre de Catalan.
1. Introduction
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. [17]). It may be seen as the group of matrices in which each row and each column has exactly one nonzero entry (pseudopermutation matrices), and such that the nonzero 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 [14] 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 [6] states the following equality: (1.3) which reduces when to the theorem of Artin [1] that the dimension of the harmonic space (cf. [9]) is .
Our aim is to give an analogous result in the case of quasisymmetrizing action. The ring of quasisymmetric functions was introduced by Gessel [11] as a source of generating functions for partitions [18] and appears in more and more combinatorial contexts [5, 18, 19]. Malvenuto and Reutenauer [16] proved a graded Hopf duality between and the Solomon descent algebras and Gelfand et. al. [10] defined the graded Hopf algebra of noncommutative symmetric functions and identified it with the Solomon descent algebra.
In [2, 3], Aval et. al. investigated the space of supercoinvariant polynomials for the symmetric group, defined as the orthogonal (with respect to (1.2)) of the ideal generated by quasisymmetric 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 supercoinvariant polynomials for the group .
In Section 2, we define and study a “quasisymmetrizing” action of on . We also introduce invariant polynomials under this action, which are called quasiinvariant, and polynomials orthogonal to quasiinvariant polynomials, which are called supercoinvariant. The Section 3 is devoted to the proof of our main result (Theorem 2.4), which gives the dimension of the space of supercoinvariant polynomials for : we construct an explicit basis for from which we deduce its Hilbert series.
2. A quasisymmetrizing 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 quasisymmetrizing action of the group on , which reduces to the quasisymmetrizing action of Hivert (cf. [12]) 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 quasisymmetrizing action of is given by (2.2) where is the weight of , i.e. the product of its nonzero 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 quasisymmetrizing action (cf. [12], 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 quasisymmetric if and only if, for any and in , we have whenever . The space of quasisymmetric polynomials in variables is denoted by .
The polynomials invariant under the action (2.2) of are said to be quasiinvariant and the space of quasiinvariant polynomials is denoted by , i.e. Let us recall (cf. [12], 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 quasisymmetric, whence is also quasisymmetric.
The reverse implication is obvious. ∎
Let us now define supercoinvariant polynomials: with the scalar product defined in (1.2). This is the natural analogous to in the case of quasisymmetrizing actions and reduces to the space of superharmonic polynomials (cf. [3]) 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 [7] defined a class of quasisymmetric 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 [3]. To see this, we observe that if is the Gröbner basis of constructed in [3] (see also the next section), then the set is a Gröbner basis (any syzygy is reducible thanks to Buchberger’s first criterion, cf. [8]).
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. [3]) 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 [3] 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 nonzero 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. [8]) 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 [2] is constructed a family of polynomials indexed by transdiagonal vectors . This family is constructed by using recursive relations of the fundamental quasisymmetric functions and one of its property (cf. [2]) 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 [3] is given an explicit formula for : (3.5) using the number of Dyck paths with a given number of factors (cf. [13]).
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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