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.
Highlights
The Weyl algebra’s history has begun with the birth of quantum mechanics
Since the rational Olshanetsky-Perelomov system is a generalization of rational quantum Calogero-Moser system, we extend the obtained results in [12] to the rational Olshanetsky-Perelomov system for reflection group of type Bn
A root system R associated to a reflection group W is a finite set of nonzero vectors in h satisfying the five following conditions: (1) R spans h as a vector space; (2) R ∩ Rα = {α, −α} for all α ∈ R; (3) sα(R) = R for all α ∈ R; (4)
Summary
The Weyl algebra’s history has begun with the birth of quantum mechanics. Group theory has played a major role in the discovery of general laws of quantum theory. The theory of group representations is one of the best example of interaction between fundamental physics and pure mathematics. Having this at hand we study the polynomials representation of the Weyl algebra with special consideration for physical problems. Since symmetries are relevant in physics, our attention has been drawn to the reflection group W (Bn) of type Bn and its representation in connection with rational Olshanetsky-Perelomov operators. In what follows we study the action of invariant differential operators on the polynomial ring through the representation theory of reflection groups of type Bn. We show that the polynomial representation of the rational Olshanetsky-Perelomov systems for W is related to the representation theory of W. Since the rational Olshanetsky-Perelomov system is a generalization of rational quantum Calogero-Moser system, we extend the obtained results in [12] to the rational Olshanetsky-Perelomov system for reflection group of type Bn
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