Specht polynomials and modules over the Weyl algebra (original) (raw)

Higher Specht Polynomials and Modules Over the Weyl Algebra

2021

In this paper, we study an irreducible decomposition structure of the D-module direct image π + (O C n) for the finite map π : C n → C n /(S n1 × • • • × S nr). We explicitly construct the simple component of π + (O C n) by providing their generators and their multiplicities. Using an equivalence categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore we study the action invariants differential operators on the higher Specht polynomials.

Specht Polynomials and Modules Over the Weyl Algebra II

Journal of Mathematical Sciences, 2023

In this paper, we study an irreducible decomposition structure of the D-module direct image + (O n) for the finite map ∶ ℂ n → ℂ n ∕(S n 1 × ⋯ × S n r). We explicitly construct the simple components of + (O ℂ n) by providing their generators and their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of. Furthermore, we study the action of invariant differential operators on the higher Specht polynomials.

Representation Theory of Groups and D -Modules

International Journal of Mathematics and Mathematical Sciences, 2021

In this paper, we study a decomposition D-module structure of the polynomial ring. en, 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 π: specB ⟶ specA generate the irreducible factors of the direct image of B under the map π. Peel gave the construction of irreducible submodules of C[x 1 ,. .. , x n ] in the following way [4]. By a partition of n, we mean a sequence λ � (λ 1 ,. .. , λ r) such that

A categorical approach to Weyl modules

Transformation Groups, 2010

Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in were obtained. In this paper, we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.

On the homogeneized Weyl Algebra

2011

The aim of this paper is to give relations between the category of finetely generated graded modules over the homogeneized Weyl algebra Bn, the finetely generated modules over the Weyl algebra An and the finetely generated graded modules over the Yoneda algebra B ! n of Bn. We will give these relations both at the level of the categories of modules and at the level of the derived categories.

INVARIANT DIFFERENTIAL OPERATORS AND THE GENERALIZED SYMMETRIC GROUP

In this paper we study the decomposition of the direct image of π + (O X) the polynomial ring O X as a D-module, under the map π : spec O X → spec O G(r,n) X , where O G(r,n) X is the ring of invariant polynomial under the action of the wreath product G(r, p) := Z/rZ ≀ S n. We first describe the generators of the simple components of π + (O X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants differential operators on the higher Specht polynomials.

On the homogenized Weyl Algebra

arXiv (Cornell University), 2012

Weyl, Demazure and fusion modules for the current algebra of

Advances in Mathematics, 2006

We construct a Poincaré-Birkhoff-Witt type basis for the Weyl modules [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191-223, math.QA/0004174] of the current algebra of sl r+1 . As a corollary we prove the conjecture made in [V. Chari, A. Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001) 191-223, math.QA/0004174; V. Chari, A. Pressley, Integrable and Weyl modules for quantum affine sl 2 , in: Quantum Groups and Lie Theory, Durham, 1999, in: London Math. Soc. Lecture Note Ser., vol. 290, Cambridge Univ. Press, Cambridge, 2001, pp. 48-62, math.QA/0007123] on the dimension of the Weyl modules in this case. Further, we relate the Weyl modules to the fusion modules defined in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimenmath.QA/9812093] of the current algebra and the Demazure modules in level one representations of the corresponding affine algebra. In particular, this allows us to establish substantial cases of the conjectures in [B. Feigin, S. Loktev, On generalized Kostka polynomials and the quantum Verlinde rule, in: Differential Topology, Infinite-dimensional Lie Algebras, and Applications, in: Amer. Math. Soc. Transl. Ser. 2, vol. 194, 1999, pp. 61-79, math.QA/9812093] on the structure and graded character of the fusion modules.

A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible Modules

Contemporary Mathematics, 2013

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h generated by elements x, y, which satisfy yx − xy = h, where h ∈ F[x]. When h = 0, the algebras A h are subalgebras of the Weyl algebra A 1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A h and determined its automorphism group Aut F (A h) and the subalgebra of invariants under Aut F (A h). Here we determine the irreducible A h-modules. In a sequel to this paper, we completely describe the derivations of A h over any field.

D-SIMPLE RINGS AND PRINCIPAL MAXIMAL IDEALS OF THE WEYL ALGEBRA

Glasgow Mathematical Journal, 2005

The differential operators S = ∂ ∂x + β ∂ ∂y + γ of A 2 (C) such that S generates a maximal left ideal of A 2 (C) were characterized in [4] by Bratti and Takagi. We generalize their work to An(K), n ≥ 2, K a field of characteristic zero. We also show that if S = ∂ ∂ x 1

Specht polynomials and modules over the Weyl algebra (original) (raw)

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