On the choice of a basis of invariant polynomials of a Finite Reflection Group. Generating Formulas for $\hat{P}$-matrices of groups of the infinite series $S_n$, $A_n$, $B_n$ and $D_n$

Abstract

Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\dots ,p_n(x)$, $x\in {\mathbb{R}}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline{p}:{\mathbb{R}}^n\to {\mathbb{R}}^n:x\to (p_1(x),\dots ,p_n(x))$ induces a diffeomorphism between the orbit space ${\mathbb{R}}^n/W$ and the set $\mathcal{S}=\overline{p}({\mathbb{R}}^n)\subset {\mathbb{R}}^n$. The border of $\mathcal{S}$ is the $\overline{p}$ image of the set of reflecting hyperplanes of $W$. With a given basic set of invariant polynomials it is possible to build an $n\times n$ polynomial matrix, $\hat{P}(p)$, $p\in {\mathbb{R}}^n$, sometimes called $\hat{P}$-matrix, such that ${\hat{P}}_{ab}(p(x))=\nabla p_a(x)\cdot \nabla p_b(x)$, $\forall a,b=1,\dots ,n$. The border of $\mathcal{S}$ is contained in the algebraic surface $\det (\hat{P}(p))=0$, sometimes called discriminant, and the polynomial $\det (\hat{P}(p))$ satisfies a system of differential equations that depends on an $n$-dimensional polynomial vector $\lambda (p)$. Possible applications concern phase transitions and singularities. If the rank $n$ is large, the matrix $\hat{P}(p)$ is in general difficult to calculate. In this article I suggest a choice of the basic invariant polynomials for all the reflection groups of type $S_n$, $A_n$, $B_n$, $D_n$, $\forall n\in \mathbb{N}$, for which I give generating formulas for the corresponding $\hat{P}$-matrices and $\lambda$-vectors. These $\hat{P}$-matrices can be written, almost completely, as sums of block Hankel matrices. Transformation formulas allow to determine easily both the $\hat{P}$-matrix and the $\lambda$-vector in any other basis of invariant polynomials. Examples of transformations into flat bases, $a$-bases, and canonical bases, are considered.

58 pages

References in corpus (2)

• Geometric invariant theory approach to the determination of ground states of D-wave condensates in isotropic space
• P-matrices in orbit spaces and invariant theory

Cited by in corpus (1)

• Orbit spaces of Weyl groups acting on compact tori: a unified and explicit polynomial description