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

TL;DR

This paper develops explicit formulas for invariant polynomial bases and their associated matrices for certain finite reflection groups, facilitating analysis of orbit spaces and applications in phase transitions and singularity theory.

Contribution

It introduces a systematic method to construct generating formulas for the $\u00a8hat P$-matrices of groups of types $S_n$, $A_n$, $B_n$, and $D_n$, simplifying calculations for large rank.

Findings

Provides explicit formulas for $\u00a8hat P$-matrices for all $n$ in the series

Shows $\u00a8hat P$-matrices can be expressed as sums of block Hankel matrices

Offers transformation formulas for changing invariant polynomial bases

Abstract

Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),\ldots,p_n(x)$, $x\in\mathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $\overline p:\mathbb{R}^n\to\mathbb{R}^n:x\to (p_1(x),\ldots,p_n(x))$ induces a diffeomorphism between the orbit space $\mathbb{R}^n/W$ and the set ${\cal S}=\overline p(\mathbb{R}^n)\subset\mathbb{R}^n$. The border of ${\cal S}$ 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, $\widehat P(p)$, $p\in\mathbb{R}^n$, sometimes called $\widehat P$-matrix, such that $\widehat P_{ab}(p(x))=\nabla p_a(x)\cdot \nabla p_b(x)$, $\forall\,a,b=1,\ldots,n$. The border of ${\cal S}$ is contained in the algebraic surface $\det(\widehat P(p))=0$, sometimes called discriminant, and the polynomial $\det(\widehat P(p))$ 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 $\widehat P(p)$ 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 $\widehat P$-matrices and $\lambda$-vectors. These $\widehat P$-matrices can be written, almost completely, as sums of block Hankel matrices. Transformation formulas allow to determine easily both the $\widehat P$-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.