A proof of the Popov conjecture for tori

Wehlau, David L.

doi:10.1090/S0002-9939-1992-1074757-9

A proof of the Popov conjecture for tori
HTML articles powered by AMS MathViewer

by David L. Wehlau PDF

Proc. Amer. Math. Soc. 114 (1992), 839-845 Request permission

Abstract:

We prove a lemma which reduces much of the invariant theory of torus representations to the theory of faithful stable torus representations (Lemma 2). Using this reduction we obtain a structure theorem (Theorem 1) for equidimensional representations of tori. This theorem shows that the weights of an equidimensional torus representation are arranged in a very special manner within the lattice of characters. Understanding this arrangement allows us to prove that equidimensional representations of tori must be cofree (the Popov conjecture for tori).

References

Equidimensional representations of simple algebraic groups

128

O. M. Adamovich and E. O. Golovina, Simple linear Lie groups having a free algebra of invariants, Selecta Math. Soviet. 3 (1983/84), no. 2, 183–220. Selected reprints. MR 742478
V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213. MR 575790, DOI 10.1016/0021-8693(80)90141-6
Victor G. Kac, Vladimir L. Popov, and Ernest B. Vinberg, Sur les groupes linéaires algébriques dont l’algèbre des invariants est libre, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 12, Ai, A875–A878 (French, with English summary). MR 419468
Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984 (German). MR 768181, DOI 10.1007/978-3-322-83813-1
Peter Littelmann, Koreguläre und äquidimensionale Darstellungen, J. Algebra 123 (1989), no. 1, 193–222 (German, with English summary). MR 1000484, DOI 10.1016/0021-8693(89)90043-4
David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
V. L. Popov, Representations with a free module of covariants, Funkcional. Anal. i Priložen. 10 (1976), no. 3, 91–92 (Russian). MR 0417197
Gerald W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), no. 2, 167–191. MR 511189, DOI 10.1007/BF01403085
Gerald W. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 (1978/79), no. 1, 1–12. MR 516601, DOI 10.1007/BF01406465
Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821

Linear algebraic groups

Equidimensional representations and the Popov conjecture