Representation Theory of Groups and D-Modules

-e 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. -is 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 π is semisimple by the decomposition theorem [2]. -e simplest case is when the map π: X � specB⟶ Y � specA 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 G. -e irreducible submodules of the direct image are in one-to-one correspondence with the irreducible representations of G (see [1]). In the case of the invariants of the symmetric group, B � C[x1, . . . , xn]A � C[x1, . . . , xn]S n � C[y1, . . . , yn], an explicit basis of the A-module structure of B is given by Hn � x|αi ≤ n − i, 1≤ i≤ n 􏼈 􏼉. In what follows, we endowed B with a differential structure by using directly the action of the Weyl algebra associated to A after a localization. We use the representation theory of symmetric groups to exhibit the generators of its simple components. -e 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 π+OX.


Introduction
e 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. is 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 π is semisimple by the decomposition theorem [2]. e simplest case is when the map π: X � specB ⟶ Y � specA 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 G. e irreducible submodules of the direct image are in one-to-one correspondence with the irreducible representations of G (see [1]). In the case of the invariants of the symmetric group, B � C[x 1 , . . . , x n ]A � C[x 1 , . . . , x n ]S n � C[y 1 , . . . , y n ], an explicit basis of the A-module structure of B is given by H n � x α |α i ≤ n − i, 1 ≤ i ≤ n . In what follows, we endowed B with a differential structure by using directly the action of the Weyl algebra associated to A after a localization. We use the representation theory of symmetric groups to exhibit the generators of its simple components. e 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][4][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 π + O X .

Preliminaries: Specht Polynomials and Specht Modules.
In this section, we recall some general facts about the actions of symmetric group on polynomial ring. e symmetric group S n is the group of permutations of the set of variables x 1 , . . . , x n . Let g ∈ C[x 1 , . . . , x n ] be a polynomial, and σ ∈ S n ; we define (σg) x 1 , . . . , x n � g σx 1 , . . . , σx n . (1) . . λ r > 0 and λ 1 + λ 2 + · · · + λ r � n. (2) Let λ � (λ 1 , . . . , λ r ) be a partition; we arrange the variables x 1 , . . . , x n in an array, with r rows and λ 1 columns, containing a variable in the first λ i positions of the ith row; each variable occurs exactly once in the array. For example, one such array for the partition (4, 2, 1) of 7 is and such an array is called a λ-tableau. ere are n! λ-tableaux for each partition λ of n. We shall denote such tableaux by t. Suppose that the variables a 1 , . . . , a l occur in jth column of λ-tableau t, with a i in the ith row. We form the difference product Δ(a 1 , . . . , a l ) � i<k (a i − a k ), if l > 1, and if l � 1, Δ(a 1 ) � 1. Multiplying these difference products for all the columns of t, we obtain a polynomial which we denote by f(t). For σ ∈ S n , let σt be the tableau obtained from t by replacing x i in t by σx i . en, σf(t) � f(σt). It follows that the set of all linear combinations of the n! polynomials f(t), obtained from the λ-tableaux t, is a cyclic C[S n ]-module generated by any f(t). We denote this module by S λ . A λ-tableau is said to be standard if the variables occur in increasing order (x i > x j if i > j ) along each row from left to right and down each column. Peel proved the following in [4].
We call f(t) the Specht polynomial corresponding to the λ-tableau t, we call S λ the Specht module corresponding to the partition λ, and f(t) is a standard Specht polynomial if t is a standard tableau. Theorem 2. S λ for λ⊢n forms a complete list of irreducible S n -module over the complex field.

Geometric Interpretation of the Specht Polynomials
In this section, we establish a decomposition theorem and give a geometric interpretation of the Specht polynomials.
We denote by D Y � C〈y 1 , . . . , y n , z/zy 1 , . . . , z/zy n 〉 the ring of differential operators associated to O Y � C[y 1 , . . . , y n ]. Since O X is a simple D X -module [6], the direct image π + (O X ) of O X under the map π: C n ⟶ C n /S n is semisimple [2]. We would like to study O X as a D Y -module without the machinery of the direct image structure but by the direct actions of D Y on O X . By localization, O X can be turned into a D Y -module, as the following lemma states.
We get the following equation: Since Δ ≠ 0, it follows that and it is clear that O X is a D Y -module. What are the simple components O X as D Y -module and their multiplicities?
We have that where

Simple Components and eir 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 2 International Journal of Mathematics and Mathematical Sciences Let us consider the multiplicative closed set where

Lemma 2. ere exists an injective map
Proof. e S n -module C[S n ] acts on itself by multiplication, and this multiplication yields an injective map ere exists an injective map Proof. Since D Y � C〈y 1 , . . . , y n , zy 1 , . . . , zy n , Δ −2 〉, we only need to show that every element of C[S n ] commute with y 1 , . . . , y n , zy 1 , . . . , zy n .  D) is also a differential ring. In the same way, σ − 1 Dσ � D for every σ ∈ S n . erefore, σD � Dσ and σ commute with D.
□ Corollary 1 Proof where the sum is taken over all the partitions of n and R λ are simple rings. We have the following corresponding decomposition of the identity element of C[S n ]: where r λ is the identity element of R λ , r 2 λ � 1 and r λ r μ � 0 if λ ≠ μ, and the set r λ λ⊢n is the set of central idempotents of Let n be a positive integer, λ be a partition of n, Tab(λ) be the set of standard tableau of shape λ, and Tab(n) � ∪ λ⊢n Tab(λ). We have r λ � t∈Tab(λ) e t where e t is the primitive idempotent associated to the standard tableau t ∈ Tab(λ) (see [10]).
and by Corollary 1, we get We also have, by [10,Proposition 3.29], that C[S n ] � ⊕ λ⊢n End C (S λ ) where S λ is the Specht module associated with the partition λ⊢n. But C S n � ⊕ λ⊢n r λ C S n and r λ C S n � End C C f λ , where f λ � dimS λ . We recall that each standard tableau t i is associated with an idempotent e i .

International Journal of Mathematics and Mathematical Sciences 3
Let us show that C[S n ] � ⊕ λ⊢n (⊕ t i ,t j ∈Tab(λ) Hom D Y (e i O X , e j O X )). Let x be an element of C[S n ] and r λ be a central idempotent with λ⊢n. en, e number of direct factors in the sum are canonical projections and ϕ is the isomorphism of Corollary 1. It follows that ψ is an isomorphism, and hence r λ C[S n ] � Hom D Y (r λ O X , r λ O X ). Now we identify r λ C[S n ] with the set Mat f λ (C) of square matrices of order f λ with coefficients in C or with End C (C f λ ). Let E ij be the square matrix of order f λ with 1 at the position (i, j) and 0 elsewhere and E i � E i,i , and then we identify the idempotent e i ∈ r λ C[S n ] with Proof. We have by the proof of eorem 3 that and e i O X are simple D Y -modules. Since to each e i corresponds a partition λ⊢n and a λ-tableau t i such that We consider the case n � 3, 4 For n � 3, the Specht polynomials corresponding to standard tableaux are Correspondingly, we have that For n � 4, the Specht polynomials corresponding to standard tableaux are Correspondingly, we have that 4 International Journal of Mathematics and Mathematical Sciences

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 S n [5].
We give a geometric interpretation of those polynomials as follows. Let Let J � P 1 ∪ . . . ∪ P k be a partition of 1, 2, . . . , n { } as a set. To such partition, we associated the subgroup S J � S P 1 × · · · × S P k of S n and the ring e maps π ′ : X ⟶ Y J and π ″ : Y J ⟶ Y are the obvious ones, and we get the following commutative diagram.
We clearly have the map π IJ : Y I ⟶ Y J , whenever I is a refinement of J. e Jacobian of π ′ is a product of Vandermonde determinants on each of the set of variables with subscript in P i . is is a Specht polynomial. Now J defines a numerical partition λ � (λ 1 , . . . , λ k ) such that λ 1 ≤ λ 2 ≤ · · · ≤ λ k and λ 1 + λ 2 + · · · + λ k � n where each λ i � |P i s | for some i s . is partition induces a Ferrers diagram where the first column has λ 1 boxes, the second columns has λ 2 boxes, and so on. Moreover, one gets an induced tableau by filling (in increasing order) the numbers in P 1 in the first column, the numbers in P 2 in the second column, and so on, where P 1 is the subset of integers in the first columns, P 2 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 T, we can associate a Specht polynomial p T . ese are geometrical objects, since they are Jacobians of certain polynomials maps.

A Generalization
Consider the map π: X � specO X ⟶ Y � specO Y . We proved in [1, eorem 2.10] that the irreducible D-module factors of the direct image π + (O X ) are generated by the Specht polynomials which are divisors of the Jacobian of π.
We will now consider a general finite map π: X ⟶ Y. A consequence of that situation is that the simple submodules of π + O X are generated by divisors of the Jacobian of π. 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 k be an algebraically closed field of characteristic 0. Denote for a k-algebra B, by T B/K the k-linear derivations of B.
ere is a general correspondence between representations of the differential Galois group of a D X -module M, defined using a Picard-Vessiot extension and the category of modules generated by M (for the case of one variable see [8]).
Let L and K be two fields, say that a T K/k -module M is L-trivial if L ⊗ K M � L n as T L/k -modules. Denote by Mod L (T K/k ) the full subcategory of finitely generated T K/k -modules that are L-trivial.
If G is a finite group, let Mod(k[G]) be the category of finite-dimensional representations of k [G]. Let now k ⟶ K ⟶ L be a tower of fields such that K � L G . Proposition 4 (see [1]).
can be extended to a Galois correspondence. Fix K and L and consider intermediate fields L E K . Given two such fields E 1 E 2 , we have the categories Mod L (T E 1 /k ) and Mod L (T E 2 /k ). e map π: specE 1 ⟶ specE 2 induces an isomorphism E 2 ⊗ E 1 T E 2 /k � T E 1 /k , in particular a canonical lifting D E 2 ⟶ D E 1 . Corresponding to this ring homomorphism, we have the usual pair of adjoint functors. First the inverse image: is given by It is immediate by L ⊗ E 1 (E 1 ⊗ E 2 M) � L ⊗ E 2 M that the image of the inverse image lies in Mod L (T E 1 /k ). Secondly, we have the direct image functor π + , between the same categories, given by restricting the action on M to T E 2 /k using the canonical lifting T E 2 /k ⟶ T E 1 /k . e direct image is right adjoint to the inverse image.
us, the direct image landing in Mod L (T E 2 /k ) is clear, e.g., in the following way. By the proposition in the preceding section, it suffices to prove this for E 1 , since it then follows for any direct factor. Now the category Mod L (T L/k ) is closed under submodules and quotients, and L ⊗ E 2 E 1 is a submodule of L ⊗ E 2 L, which is a quotient of L ⊗ K L ∈ Mod L (T L/k ). So L ⊗ E 2 E 1 ∈ Mod L (T E 2 /k ). By the proposition, there are equivalences of categories Mod(k[H i ]) ⟶ Mod(T E i /k ), where E i � L H i , and we now want to express the direct and inverse images of D-modules in terms of the corresponding group representation categories and functors.