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- W3100106044 abstract "Let $W$ be a rank $n$ irreducible finite reflection group and let $p_1(x),ldots,p_n(x)$, $xinmathbb{R}^n$, be a basis of algebraically independent $W$-invariant real homogeneous polynomials. The orbit map $overline p:mathbb{R}^ntomathbb{R}^n:xto (p_1(x),ldots,p_n(x))$ induces a diffeomorphism between the orbit space $mathbb{R}^n/W$ and the set ${cal S}=overline p(mathbb{R}^n)subsetmathbb{R}^n$. 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 $ntimes n$ polynomial matrix, $widehat P(p)$, $pinmathbb{R}^n$, 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 $S_n$, $A_n$, $B_n$, $D_n$, $forall,nin mathbb{N}$, 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." @default.
- W3100106044 created "2020-11-23" @default.
- W3100106044 creator A5016233724 @default.
- W3100106044 date "2019-09-20" @default.
- W3100106044 modified "2023-09-26" @default.
- W3100106044 title "Generating formulas for finite reflection groups of the infinite series $$S_n$$, $$A_n$$, $$B_n$$ and $$D_n$$" @default.
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- W3100106044 doi "https://doi.org/10.1007/s12215-019-00455-8" @default.
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