Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A proof of the Popov conjecture for tori

Author: David L. Wehlau
Journal: Proc. Amer. Math. Soc. 114 (1992), 839-845
MSC: Primary 14L30; Secondary 20G45
MathSciNet review: 1074757
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] O. M. Adamovich, Equidimensional representations of simple algebraic groups, Amer. Math. Soc. Transl. (2) 128 (1986), 25-29.
  • [2] O. M. Adamovich and E. O. Golovina, Simple linear Lie groups having a free algebra of invariants, Selecta Math. Soviet 3 (1983/84), 183-220. MR 742478 (86a:14044)
  • [3] V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), 190-213. MR 575790 (81i:17005)
  • [4] V. G. Kac, V. L. Popov, and E. B. Vinberg, Sur les groupes lineaires algébriques dont l'algèbre des invariants est libre, C. R. Acad. Sci. Paris Sér. I Math. 283 (1976), 865-878. MR 0419468 (54:7489)
  • [5] H. Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Math., Vieweg, Braunschweig, 1984. MR 768181 (86j:14006)
  • [6] P. Littelmann, Koreguläre und äquidimensionale Darstellungen, J. Algebra 123 (1989), 193-222. MR 1000484 (90e:20039)
  • [7] D. Mumford and J. Fogarty, Geometric invariant theory (2nd enlarged ed.), Ergeb. Math./ Grenzgeb., vol. 34, Springer-Verlag, Berlin, 1982. MR 719371 (86a:14006)
  • [8] V. L. Popov, Representations with a free module of covariants, Funct. Anal. Appl. 10 (1976), 242-244. MR 0417197 (54:5255)
  • [9] G. W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), 167-191. MR 511189 (80m:14032)
  • [10] -, Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 (1978), 1-12. MR 516601 (80c:14008)
  • [11] -, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37-135. MR 573821 (81h:57024)
  • [12] T. A. Springer, Linear algebraic groups, PM 9, Birkhäuser Verlag, 1977.
  • [13] D. L. Wehlau, Equidimensional representations and the Popov conjecture, Ph.D. Thesis, Brandeis University, 1989.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 14L30, 20G45

Retrieve articles in all journals with MSC: 14L30, 20G45

Additional Information

Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society