Constructive invariant theory for tori

Wehlau, David

doi:10.5802/aif.1364

Wehlau, David

Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 1055-1066.

Résumé
Abstract

Considérons une représentation rationnelle d’un tore algébrique $T$ sur un espace vectoriel $V$ . Soit ${f_{1}, \dots, f_{p}}$ un ensemble générateur homogène minimal pour l’anneau des invariants $k [V]^{T}$ . De nouvelles bornes supérieures sont établies pour le nombre $N_{V, T} : = \max {\deg f_{i}}$ . Ces bornes sont exprimées en termes du volume de l’enveloppe convexe des poids de $V$ et d’autres données géométriques. De plus on décrit un algorithme pour construire un ensemble partiel (essentiellement unique) ${f_{1}, \dots, f_{s}}$ dont les éléments sont des monômes et tel que $k [V]^{T}$ soit intègre sur $k [f_{1}, \dots, f_{s}]$ .

Consider a rational representation of an algebraic torus $T$ on a vector space $V$ . Suppose that ${f_{1}, \dots, f_{p}}$ is a homogeneous minimal generating set for the ring of invariants, $k [V]^{T}$ . New upper bounds are derived for the number $N_{V, T} : = \max {\deg f_{i}}$ . These bounds are expressed in terms of the volume of the convex hull of the weights of $V$ and other geometric data. Also an algorithm is described for constructing an (essentially unique) partial set of generators ${f_{1}, \dots, f_{s}}$ consisting of monomials and such that $k [V]^{T}$ is integral over $k [f_{1}, \dots, f_{s}]$ .

MR Zbl

DOI : 10.5802/aif.1364

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     author = {Wehlau, David},
     title = {Constructive invariant theory for tori},
     journal = {Annales de l'Institut Fourier},
     pages = {1055--1066},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {43},
     number = {4},
     year = {1993},
     doi = {10.5802/aif.1364},
     mrnumber = {95c:14068},
     zbl = {0789.14009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1364/}
}

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Wehlau, David. Constructive invariant theory for tori. Annales de l'Institut Fourier, Tome 43 (1993) no. 4, pp. 1055-1066. doi : 10.5802/aif.1364. http://www.numdam.org/articles/10.5802/aif.1364/

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