  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.
