Let W⊂O(n) be a finite reflection group, p1(x),…,pn(x), x∈Rn, be a basis of algebraically independent W-invariant real homogeneous polynomials, and p¯:Rn→Rn:x→(p1(x),…,pn(x)) the orbit map, whose image S=p¯(Rn)⊂Rn is diffeomorphic with the orbit space Rn/W. With the given basis of invariant polynomials it is possible to build an n×n polynomial matrix, Pˆ(p), p∈Rn, such that Pˆab(p¯(x))=∇pa(x)⋅∇pb(x), ∀a,b=1,…,n. It is known that Pˆ(p) enables to determine S, and that the polynomial det(Pˆ(p)) satisfies a system of n differential equations that depends on an n-dimensional polynomial vector λ(p). If n is large, the explicit determination of Pˆ(p) and λ(p) are in general impossible to calculate from their definitions, because of computing time and computer memory limits. In this article, when W is one of the finite reflection groups of type Sn, An, Bn, Dn, ∀n∈N, for given choices of the basis of W-invariant polynomials p1(x),…,pn(x), generating formulas for Pˆ(p) and λ(p) are established. Proofs are based on induction principle and elementary algebra. Transformation formulas allow then to determine both the matrices Pˆ(p′) and the vectors λ(p′), corresponding to any other basis p′1(x),…,p′n(x), of W-invariant polynomials.

### Generating formulas for finite reflection groups of the infinite series Sn, An, Bn and Dn

#### Abstract

Let W⊂O(n) be a finite reflection group, p1(x),…,pn(x), x∈Rn, be a basis of algebraically independent W-invariant real homogeneous polynomials, and p¯:Rn→Rn:x→(p1(x),…,pn(x)) the orbit map, whose image S=p¯(Rn)⊂Rn is diffeomorphic with the orbit space Rn/W. With the given basis of invariant polynomials it is possible to build an n×n polynomial matrix, Pˆ(p), p∈Rn, such that Pˆab(p¯(x))=∇pa(x)⋅∇pb(x), ∀a,b=1,…,n. It is known that Pˆ(p) enables to determine S, and that the polynomial det(Pˆ(p)) satisfies a system of n differential equations that depends on an n-dimensional polynomial vector λ(p). If n is large, the explicit determination of Pˆ(p) and λ(p) are in general impossible to calculate from their definitions, because of computing time and computer memory limits. In this article, when W is one of the finite reflection groups of type Sn, An, Bn, Dn, ∀n∈N, for given choices of the basis of W-invariant polynomials p1(x),…,pn(x), generating formulas for Pˆ(p) and λ(p) are established. Proofs are based on induction principle and elementary algebra. Transformation formulas allow then to determine both the matrices Pˆ(p′) and the vectors λ(p′), corresponding to any other basis p′1(x),…,p′n(x), of W-invariant polynomials.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11390/1169691`