Constructive invariant theory for tori

Wehlau, David

Wehlau, David

Annales de l'Institut Fourier, Tome 43 (1993), p. 1055-1066 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

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 $k [f_{1}, \dots, f_{s}]$ .

Publié le : 1993-01-01

MR 95c:14068 | Zbl 0789.14009

DOI : https://doi.org/10.5802/aif.1364

@article{AIF_1993__43_4_1055_0,
     author = {Wehlau, David},
     title = {Constructive invariant theory for tori},
     journal = {Annales de l'Institut Fourier},
     volume = {43},
     year = {1993},
     pages = {1055-1066},
     doi = {10.5802/aif.1364},
     mrnumber = {95c:14068},
     zbl = {0789.14009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1993__43_4_1055_0}
}

Wehlau, David. Constructive invariant theory for tori. Annales de l'Institut Fourier, Tome 43 (1993) pp. 1055-1066. doi : 10.5802/aif.1364. http://gdmltest.u-ga.fr/item/AIF_1993__43_4_1055_0/

Bibliographie

[B] A. Brondsted, An Introduction to Convex Polytopes, Springer-Verlag, Berlin-Heidelberg-New York, 1983. | MR 84d:52009 | Zbl 0509.52001

[EW] G. Ewald, U. Wessels, On the ampleness of invertible sheaves in complete projective toric varieties, Results in Math., (1991), 275-278. | MR 92b:14028 | Zbl 0739.14031

[Ga] F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea Publishing Company, New York, 1959. | Zbl 0085.01001

[Go] P. Gordan, Invariantentheorie, Chelsea Publishing Company, New York, 1987.

[K] G. Kempf, Computing Invariants, S. S. Koh (Ed.) Invariant Theory, Lect. Notes Math., 1278, 81-94, Springer-Verlag, Berlin-Heidelberg-New York, 1987. | MR 89h:20057 | Zbl 0633.14007

[N1] E. Noether, Der endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., 77 (1916), 89-92. | JFM 45.0198.01

[N2] E. Noether, Der endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p., Nachr. v. d. Ges. Wiss. zu Göttingen, (1926), 485-491. | JFM 52.0106.01

[O] T. Oda, Convex Bodies and Algebraic Geometry, Ergeb. Math. und Grenzgeb., Bd. 15, Springer-Verlag, Berlin-Heidelberg-New York, 1988. | Zbl 0628.52002

[P] V.L. Popov, Constructive Invariant Theory, Astérisque, 87/88 (1981), 303-334. | MR 83i:14040 | Zbl 0491.14004

[R] H.J. Ryser, Maximal Determinants in Combinatorial Investigations, Can. Jour. Math., 8 (1956), 245-249. | MR 18,105a | Zbl 0071.35903

[S] B. Schmid, Finite Groups and Invariant Theory, M.-P. Malliavin (Ed.) Topics in Invariant Theory (Lect. Notes Math. 1478), 35-66, Springer-Verlag, Berlin-Heidelberg-New York, 1991. | MR 94c:13002 | Zbl 0770.20004

[St] R.P. Stanley, Combinatorics and Commutative Algebra, Progress in Mathematics, 41, Birkhäuser, Boston-Basel-Stuttgart, 1983. | MR 85b:05002 | Zbl 0537.13009

[W] D. Wehlau, The Popov Conjecture for Tori, Proc. Amer. Math. Soc., 114 (1992), 839-845. | MR 92f:14049 | Zbl 0754.20013