scholarly journals Differential operators and reection group of type Bn

2021 ◽  
Vol 2090 (1) ◽  
pp. 012097
Author(s):  
Ibrahim Nonkané ◽  
M. Latévi Lawson
Keyword(s):  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Reflection Group ◽  
Type B ◽  
Invariant Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential Operators ◽  
Finite Reflection Group ◽  
Invariant Differential

Abstract In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

scholarly journals Differential operators and reection group of type Dn

2021 ◽  
Vol 2090 (1) ◽  
pp. 012098
Author(s):  
Ibrahim Nonkané ◽  
Latévi M. Lawson
Keyword(s):  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Invariant Polynomials ◽  
Invariant Differential Operators ◽  
Differential Structure ◽  
Invariant Differential

Abstract In this note, we study the actions of rational quantum Olshanetsky-Perelomov systems for finite reflections groups of type D n . we endowed the polynomial ring C[x 1,..., xn ] with a differential structure by using directly the action of the Weyl algebra associated with the ring C[x 1,..., xn ] W of invariant polynomials under the reflections groups W after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the reflections groups. We use the higher Specht polynomials associated with the representation of the reflections group W to exhibit the generators of its simple components.

scholarly journals Differential operators and Higher Specht polynomials

2021 ◽  
Vol 2090 (1) ◽  
pp. 012096
Author(s):  
Ibrahim Nonkané ◽  
Léonard Todjihounde
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Symmetric Groups ◽  
Symmetric Polynomial ◽  
Symmetric Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential ◽  
Moser System

Abstract In this note, we study the action of the rational quantum Calogero-Moser system on polynomials rings. This a continuation of our paper [Ibrahim Nonkan 2021 J. Phys.: Conf. Ser. 1730 012129] in which we deal with the polynomial representation of the ring of invariant differential operators. Using the higher Specht polynomials we give a detailed description of the actions of the Weyl algebra associated with the ring of the symmetric polynomial C[x 1,..., xn]Sn on the polynomial ring C[x 1,..., xn ]. In fact, we show that its irreducible submodules over the ring of differential operators invariant under the symmetric group are its submodules generated by higher Specht polynomials over the ring of the symmetric polynomials. We end up studying the polynomial representation of the ring of differential operators invariant under the actions of products of symmetric groups by giving the generators of its simple components, thus we give a differential structure to the higher Specht polynomials.

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