Abstract
In this paper, we study a decomposition -module structure of the polynomial ring. Then, we illustrate a geometric interpretation of the Specht polynomials. Using Brauer’s characterization, we give a partial generalization of the fact that factors of the discriminant of a finite map generate the irreducible factors of the direct image of under the map .
1. Introduction
The main purpose of this paper is to generalize results on modules over the Weyl algebra which appeared in [1]. Results in [1] have been obtained in a geometric context. This paper is partially expository in nature. Section 2.2 has been presented at the 9th International Conference on Mathematical Modeling in Physical Sciences to describe the action of the rational quantum Calogero–Moser system on polynomials. For the sake of clarity, we reformulate it here in a more algebraic context.
A prevailing idea in representation theory is that larger structures can be understood by breaking them up into their smallest pieces. Also the natural framework of algebraic geometry is one of the polynomials and, the development of modern algebra has given a particular status to polynomials. In this vein, we study polynomial rings as modules over a ring of invariant differential operators by elaborating its irreducible submodules. We know that the direct image of a simple module under a proper map is semisimple by the decomposition theorem [2]. The simplest case is when the map is finite; in such case, it is easy to give an elementary and wholly algebraic proof, using essentially the generic correspondence with the differential Galois group, which equals the ordinary group . The irreducible submodules of the direct image are in one-to-one correspondence with the irreducible representations of (see [1]). In the case of the invariants of the symmetric group, , an explicit basis of the -module structure of is given by . In what follows, we endowed with a differential structure by using directly the action of the Weyl algebra associated to after a localization. We use the representation theory of symmetric groups to exhibit the generators of its simple components.
The approach in this paper is different from the one in [1].
Secondly, we give a geometric interpretation of the ordinary Specht polynomials which are defined as combinatorial objects [3–5].
Finally, using Brauer’s characterization of characters, we give a partial generalization to arbitrary finite maps of the fact that factors of the discriminant of the finite map generate the irreducible factors of the direct image .
1.1. Preliminaries: Specht Polynomials and Specht Modules
In this section, we recall some general facts about the actions of symmetric group on polynomial ring. The symmetric group is the group of permutations of the set of variables . Let be a polynomial, and ; we define
Peel gave the construction of irreducible submodules of in the following way [4].
By a partition of , we mean a sequence such that
Let be a partition; we arrange the variables in an array, with rows and columns, containing a variable in the first positions of the th row; each variable occurs exactly once in the array. For example, one such array for the partition of 7 isand such an array is called a -tableau. There are -tableaux for each partition of . We shall denote such tableaux by . Suppose that the variables occur in th column of -tableau , with in the th row. We form the difference product if , and if . Multiplying these difference products for all the columns of , we obtain a polynomial which we denote by . For , let be the tableau obtained from by replacing in by . Then, . It follows that the set of all linear combinations of the polynomials , obtained from the -tableaux , is a cyclic -module generated by any . We denote this module by . A -tableau is said to be standard if the variables occur in increasing order ( if ) along each row from left to right and down each column. Peel proved the following in [4].
Theorem 1. = { is a standard -tableau} is a basis of .
We call the Specht polynomial corresponding to the -tableau , we call the Specht module corresponding to the partition , and is a standard Specht polynomial if is a standard tableau.
Theorem 2. for forms a complete list of irreducible -module over the complex field.
2. Geometric Interpretation of the Specht Polynomials
In this section, we establish a decomposition theorem and give a geometric interpretation of the Specht polynomials.
2.1. Notation
Let be the ring of differential operators associated to , and let be the ring of invariant under the symmetric group where
We denote by the ring of differential operators associated to . Since is a simple -module [6], the direct image of under the map is semisimple [2]. We would like to study as a -module without the machinery of the direct image structure but by the direct actions of on . By localization, can be turned into a -module, as the following lemma states.
Lemma 1. Let , , and and let be the localization of at . Then, is a -module.
Proof. Let us make clear the actions of on .
We have , and hence . We get the following equation:Since , it follows thatand it is clear that is a -module.
What are the simple components as -module and their multiplicities?
Example 1. For , and where andWe have thatwhere and are -simple modules.
2.2. Simple Components and Their Multiplicities
In this section, we state our first main result. We use the representation theory of symmetric groups to yield results on modules over the ring of differential operators. It is well known that
Let us consider the multiplicative closed set . It follows thatwhere and are the localization of and at , respectively. But and , whereby we get
Lemma 2. There exists an injective map
Proof. The -module acts on itself by multiplication, and this multiplication yields an injective map . Since is invariant under this action of , we get the expected injective map.
Proposition 1. There exists an injective map
Proof. Since , we only need to show that every element of commute with .(i)It is clear that every element of commutes with .(ii)Let us show that every element of commutes with . Let be a derivation on the field (the field of fractions of ); then, is a differential field. Since (the field of fractions of ) is a Galois extension of , by [7, Theorem 6.2.6] there exists a unique derivation on which extends ; then, is also a differential ring. In the same way, for every . Therefore, and commute with .
Corollary 1.
Proof. see [[1], Corollary 2.6]. Before we state our first main result, let us recall some facts.
By Maschke’s theorem [[9], Chap XVIII], we know that is a semisimple ring, andwhere the sum is taken over all the partitions of and are simple rings. We have the following corresponding decomposition of the identity element of :where is the identity element of , and if , and the set is the set of central idempotents of .
Let be a positive integer, be a partition of , be the set of standard tableau of shape , and . We have where is the primitive idempotent associated to the standard tableau (see [10]).
Theorem 3. For every primitive idempotent ,(1) is a nontrivial -submodule of (2)The -module is simple(3)There exist a partition and a Specht polynomial associated to a standard tableau of shape such that
Proof. (1)Let be a primitive idempotent; by [10, Theorem 4.3], there exists a Specht polynomial such that is a scalar multiple of ; then, and . Since commutes with every element of and is a -module, it follows that is a -module.(2)Assume that where is the set of primitive idempotents of ; then, . Let with ; then, and , but then , and hence . Therefore, , and we get and by Corollary 1, we get We also have, by [10, Proposition 3.29], that where is the Specht module associated with the partition . But where . We recall that each standard tableau is associated with an idempotent . Let us show that . Let be an element of and be a central idempotent with . Then, induces an -homomorphism . Since is in the center of , , which means . Then, if and . We get The number of direct factors in the sum is . Let us show that if . Consider the following commutative diagram: where and are canonical projections and is the isomorphism of Corollary 1. It follows that is an isomorphism, and hence . Now we identify with the set of square matrices of order with coefficients in or with . Let be the square matrix of order with 1 at the position and 0 elsewhere and and then we identify the idempotent with in . Let ; we get ; in fact is the matrix with in the position and 0 elsewhere; if , we get that . This isomorphism implies that ; the restriction of to yields a map and this map is surjective; moreover, we have . Therefore, . Let us assume that is not a simple -module; then, may be written as where are simple -modules and . It follows that but , so we obtain that , which necessarily implies that is a simple -module.(3)By proof (i), there exists a Specht polynomial such that .
Corollary 2. With the above notations, if and have the same size (if there is a partition such that .
Proof. The -modules are simple and whenever there exists a partition such that . Since , we conclude by using the Schur lemma.
Proposition 2. For every young tableau , let be the associated Specht polynomial; then, we have(1).(2), where .
Proof. We have by the proof of Theorem 3 thatand are simple -modules. Since to each corresponds a partition and a -tableau such that , then . If and are two -tableaux, by Corollary 2, . Therefore, with .
2.3. Example
We consider the case
For , the Specht polynomials corresponding to standard tableaux are
Correspondingly, we have that
For , the Specht polynomials corresponding to standard tableaux are
Correspondingly, we have that
2.4. Geometric Interpretation of Specht Polynomials
Specht polynomials were introduced as combinatoric objects [3, 4]. In fact, the Specht polynomials were first used by Wilhem Specht to generate rational representations of the symmetric group [5].
We give a geometric interpretation of those polynomials as follows.
Let and . Let , and we define
Let be a partition of as a set. To such partition, we associated the subgroup of and the ring . Let and . The maps and are the obvious ones, and we get the following commutative diagram.
We clearly have the map , whenever is a refinement of . The Jacobian of isa product of Vandermonde determinants on each of the set of variables with subscript in . This is a Specht polynomial. Now defines a numerical partition such that and where each for some . This partition induces a Ferrers diagram where the first column has boxes, the second columns has boxes, and so on. Moreover, one gets an induced tableau by filling (in increasing order) the numbers in in the first column, the numbers in in the second column, and so on, where P1 is the subset of integers in the first columns, P2 is the subset of integers in the second, and so on. Conversely to every tableau corresponds a partition of {1, ⋯, n} given by letting the integers in the different columns form the partition Hence, to every tableau , we can associate a Specht polynomial . These are geometrical objects, since they are Jacobians of certain polynomials maps.
Proposition 3. From the previous section, we get the following fact:(a).(b)The submodules and are factors of rank one in the decomposition of as -module, i.e.,
3. A Generalization
Consider the map . We proved in [1, Theorem 2.10] that the irreducible -module factors of the direct image are generated by the Specht polynomials which are divisors of the Jacobian of .
We will now consider a general finite map . A consequence of that situation is that the simple submodules of are generated by divisors of the Jacobian of . A natural question is in what generality this is true. We will prove a similar though weaker result in general. To describe this generalization, let us recall some facts established in [1].
Let be an algebraically closed field of characteristic 0. Denote for a -algebra , by the -linear derivations of .
There is a general correspondence between representations of the differential Galois group of a -module , defined using a Picard–Vessiot extension and the category of modules generated by (for the case of one variable see [8]).
Let and be two fields, say that a -module is -trivial if as -modules. Denote by the full subcategory of finitely generated -modules that are -trivial.
If is a finite group, let be the category of finite-dimensional representations of . Let now be a tower of fields such that .
Proposition 4 (see [1]). The functoris fully faithful and defines an equivalence of categories
The quasi-inverse of is the functor
The above equivalencecan be extended to a Galois correspondence. Fix and and consider intermediate fields . Given two such fields , we have the categories and . The map induces an isomorphism , in particular a canonical lifting . Corresponding to this ring homomorphism, we have the usual pair of adjoint functors. First the inverse image:is given by
It is immediate by that the image of the inverse image lies in . Secondly, we have the direct image functor , between the same categories, given by restricting the action on to using the canonical lifting . The direct image is right adjoint to the inverse image.
Thus, the direct image landing in is clear, e.g., in the following way. By the proposition in the preceding section, it suffices to prove this for , since it then follows for any direct factor. Now the category is closed under submodules and quotients, and is a submodule of , which is a quotient of . So . By the proposition, there are equivalences of categories , where , and we now want to express the direct and inverse images of -modules in terms of the corresponding group representation categories and functors.
Proposition 5 . (see [1]).(i) Define as the restriction associated to the injection . Then,(ii)Define as the coinduction associated to the injection , defined in the following way:
Then,
The study of the decomposition factors of can be reduced to the behavior of the direct image over the complement to the branch locus or even over the generic point. Let and be the inclusions.
Proposition 6 (see [1]). Let be a finite map. Then,(i) is semisimple as a -module.(ii)If is a decomposition into simple (non-zero) -modules, then , is a decomposition of into simple (nonzero) -modules.
For simplicity, we assume that we are in a generic situation, working with fields. Consider a factorization
Generically, this corresponds to extensions of fields , and we have isomorphisms . Assume that is Galois with group . The field corresponds to a subgroup , and to . Maps may be factored through the quotient with the commutator subgroup and give rise to one-dimensional representations. Corresponding functions exist using the functor , characterized by the property that . Hence, contains as direct factor the -submodule , where the sum runs over all homomorphisms . This is actually the sum of all one-dimensional -submodules of . Each gives rise to submodules . As is seen in the example below, these functions may sometimes be thought of as powers of discriminants (corresponding to the extension ). In the case of the symmetric group, the Specht polynomials were such functions for the sign representation .
An immediate consequence of Brauer’s characterization of characters [11, Theorem 20], saying that any character of a finite group is a linear sum of monomial characters with integer coefficients, is then the generic case of the following theorem.
Theorem 4. Let be an affine finite map. The Grothendieck group generated by the submodules of , is also generated by , where is a rank 1 submodule of and is an intermediate factorization (38).
Proof. The corresponding generic result follows from Brauer’s theorem and the correspondence of irreducible modules with group representations. Then, the result follows by Proposition 6.
3.1. Example: Cyclic Affine Covering
Consider the affine finite map given by the map on rings that identifies . Then, we claim that, as (left) -modules,
We use the description of the direct image in [1, Lemma 2.3]; it is thus a question of finding the -module generated by .
The extension of function fields is a Galois extension with Galois group, the cyclic group of order . Abelian groups only have one-dimensional irreducible representations. Hence, by the theory in [1], we know that splits as a sum of rank 1 simple -modules.
But we may of course also easily see this using the action of . The relation between and is described by noting that
This implies that all the factors in (39) above are -modules:
Hence, , and this module, as rank 1 and torsionfree -module, is irreducible.
Data Availability
No data were used to support this study.
Disclosure
This research is part of the author’s presentation at Stockholm University Licentiate seminar: http://su.diva-portal.org/smash/record.jsf?pid=diva2%3A611460&dswid=−9965.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author is deeply thankful to Professor Rikard Bøgvad from Stockholm University for instructive comments during the writing of this paper. This study was financially supported by the International Science Program (ISP).