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 $D_n$

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.