Skip to main content
SpringerLink
Log in

WEYL’S POLARIZATION THEOREM IN POSITIVE CHARACTERISTIC

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

Let V be an n-dimensional algebraic representation over an algebraically closed field K of a group G. For m > 0, we study the invariant rings K[Vm]G for the diagonal action of G on Vm. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in K[Vm]G by the process of polarization and restitution from K[Vn]G. In particular, this means that if K[Vn]G is generated in degree ≤ d, then so is K[Vm]G no matter how large m is.

There are several explicit counterexamples to Weyl’s theorem in positive characteristic. However, when G is a (connected) reductive affine group scheme over ℤ and V*is a good G-module, we show that Weyl's theorem holds in sufficiently large characteristic. As a special case, we consider the ring of invariants R(n, m) for the left-right action of SLn × SLn on m-tuples of n × n matrices. In this case, we show that the invariants of degree ≤ n6 suffice to generate R(n, m) if the characteristic is larger than 2n6 + n2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K. Akin, D. Buchsbaum, J. Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), 207-278.

    Article  MathSciNet  Google Scholar 

  2. G. Boffi, The universal form of the Littlewood-Richardson rule, Adv. in Math. 68 (1988), no. 1, 64-84.

    Article  MathSciNet  Google Scholar 

  3. H. Derksen, Polynomial bounds for rings of invariants, Proc. Amer. Math. Soc. 129 (2001), no. 4, 955-963.

    Article  MathSciNet  Google Scholar 

  4. H. Derksen, G. Kemper, Computational Invariant Theory, 2nd enlarged ed., with two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov, Encyclopaedia of Mathematical Sciences, Vol. 130, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VIII, Springer, Heidelberg, 2015.

  5. H. Derksen, V. Makam, Polynomial degree bounds for matrix semi-invariants, Adv. in Math. 310 (2017), 44-63.

    Article  MathSciNet  Google Scholar 

  6. H. Derksen, V. Makam, Degree bounds for semi-invariant rings of quivers, J. Pure Appl. Algebra 222 (2018), no. 10, 3282-3298.

    Article  MathSciNet  Google Scholar 

  7. H. Derksen, V. Makam, Generating invariant rings of quivers in arbitrary characteristic , J. Algebra 489 (2017), 435-445.

    Article  MathSciNet  Google Scholar 

  8. H. Derksen, J. Weyman, Semi-invariants of quivers and saturation of Littlewood- Richardson coefficients, J. American Math. Soc. 13 (2000), 467-479.

    Article  MathSciNet  Google Scholar 

  9. M. Domokos, Matrix invariants and the failure of Weyl's theorem, in: Polynomial Identities and Combinatorial Methods (Pantelleria, 2001), Lecture Notes in Pure Appl. Math. 235, Dekker, New York, 2003, pp. 215-236.

  10. M. Domokos, Finite generating system of matrix invariants, Math. Pannon 13 (2002), 175-181.

    MathSciNet  MATH  Google Scholar 

  11. M. Domokos, P. E. Frenkel, Mod 2 indecomposable orthogonal invariants, Adv. Math. 192 (2005), 209-217.

    Article  MathSciNet  Google Scholar 

  12. M. Domokos, S. G. Kuzmin, A. N. Zubkov, Rings of matrix invariants in positive characteristic, J. Pure Appl. Algebra 176 (2002), no. 1, 61-80.

    Article  MathSciNet  Google Scholar 

  13. M. Domokos, A. N. Zubkov, Semi-invariants of quivers as determinants, Trans- formation Groups 6 (2001), 9-24.

    Article  MathSciNet  Google Scholar 

  14. S. Donkin, Rational Representations of Algebraic Groups: Tensor Products and Filtrations, Lecture Notes in Mathematics, Vol. 1140, Springer, Berlin, 1985.

    Book  Google Scholar 

  15. S. Donkin, Skew modules for reductive groups, J. Algebra 113 (1988), 465-479.

    Article  MathSciNet  Google Scholar 

  16. S. Donkin, The normality of closures of conjugacy classes of matrices, Inv. Math. 101 (1990), 717-736.

    Article  MathSciNet  Google Scholar 

  17. S. Donkin, Invariants of several matrices, Inv. Math. 110 (1992), 389-401.

    Article  MathSciNet  Google Scholar 

  18. S. Donkin, Invariant functions on matrices, Math. Proc. Cambridge Math. Soc. 113 (1993), 23-43.

    Article  MathSciNet  Google Scholar 

  19. S. Donkin, On tilting modules for algebraic groups, Math. Z. 212 (1993), 39-60.

    Article  MathSciNet  Google Scholar 

  20. S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series, Vol. 253, Cambridge University Press, Cambridge, 1998.

    Book  Google Scholar 

  21. P. Doubilet, G. C. Rota, J. Stein, On the foundations of combinatorial theory: IX combinatorial methods in invariant theory, Studies in Appl. Math. 53 (1974), 185-216.

    Article  MathSciNet  Google Scholar 

  22. J. Draisma, G. Kemper, D. Wehlau, Polarization of separating invariants, Canad. J. Math. 60 (2008), 556-571.

    Article  MathSciNet  Google Scholar 

  23. E. Formanek, Generating the ring of matrix invariants, in: Ring Theory, Lecture Notes in Mathematics, Vol. 1197, Springer, Berlin, 1986, pp. 73-82.

  24. W. Fulton, Young Tableaux, With Applications to Representation Theory and Geometry , London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.

  25. A. Garg, L. Gurvits, R. Oliveira, A. Widgerson, A deterministic polynomial time algorithm for non-commutative rational identity testing, Foundations of Computer Science (FOCS), IEEE, 2016, 109-117.

  26. F. D. Grosshans, Vector invariants in arbitrary characteristic, Transformation Groups 12 (2007), 499-514.

    Article  MathSciNet  Google Scholar 

  27. W. Haboush, Reductive groups are geometrically reductive, Ann. of Math. 102 (1975), 67-85.

    Article  MathSciNet  Google Scholar 

  28. M. Hashimoto, Good filtrations of symmetric algebras and strong F-regularity of invariant subrings, Math. Z. 236 (2001), 605-623.

    Article  MathSciNet  Google Scholar 

  29. D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473-534.

    Article  MathSciNet  Google Scholar 

  30. D. Hilbert, Über die vollen Invariantensysteme, Math. Ann. 42 (1893), 313-370.

    Article  MathSciNet  Google Scholar 

  31. M. Hochster, J. L. Roberts. Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. in Math. 13 (1974), 115-175.

    Article  MathSciNet  Google Scholar 

  32. G. Ivanyos, Y. Qiao, K.V. Subrahmanyam, Non-commutative Edmonds’ problem and matrix semi-invariants, Comput. Complexity 26 (2017), no. 3, 717-763.

    Article  MathSciNet  Google Scholar 

  33. J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107. American Mathematical Society, Providence, RI, 2003.

  34. G. Kempf, The Hochster-Roberts theorem of invariant theory, Michigan Math. J. 26 (1979), no 1, 19-32.

    Article  MathSciNet  Google Scholar 

  35. F. Knop, On Noether’s and Weyl's bound in positive characteristic, in: Invariant Theory in All Characteristics, CRM Proc. Lecture Notes, Vol. 35, Amer. Math. Soc., Providence, RI, 2004, pp. 175-188.

  36. H. Kraft, C. Procesi, Classical Invariant Theory : A primer, http://www.unibas.math.ch.

  37. U. Kulkarni, A homological interpretation of Jantzen’s sum formula, Transformation Groups 11 (2006), no. 3, 517-538.

    Article  MathSciNet  Google Scholar 

  38. O. Mathieu, Filtrations of G-modules, Ann. Scient. Ec. Norm. Sup (2) 23 (1990), 625-644.

  39. K. Mulmuley, Geometric complexity theory V: E_cient algorithms for Noether normalization , J. Amer. Math. Soc. 30 (2017), no. 1, 225-309.

    Article  MathSciNet  Google Scholar 

  40. M. Nagata, Invariants of a group in an affine ring, J. Math. Kyoto Univ. 3 (1963/1964), 369-377.

    MathSciNet  MATH  Google Scholar 

  41. A. E. Parker, The global dimension of Schur algebras for GL2 and GL3, J. Algebra 241 (2001), no. 1, 340-378.

    Article  MathSciNet  Google Scholar 

  42. C. Procesi, The invariant theory of n × n matrices, Adv. in Math. 19 (1976), 306-381.

    Article  MathSciNet  Google Scholar 

  43. Ю. П. Размыслов, Тождества со следами полных матричных алгебр над полем характеристики нулъ, Изв. АН СССР, Сер. матем. 38 (1974),вьш. 4, 723–745. Engl. transl.: Y. Razmyslov, Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR Izv. 8 (1974), no. 4, 727–760.

  44. D. Richman, On vector invariants over finite fields, Adv. in Math. 81 (1990), no. 1, 30{65.

  45. C. S. Seshadri, Geometric reductivity over arbitrary base, Adv. in Math. 26 (1977), no. 3, 225-274.

    Article  MathSciNet  Google Scholar 

  46. A. Schofield, M. van der Bergh, Semi-invariants of quivers for arbitrary dimension vectors, Indag. Mathem., N.S 12 (2001), 125-138.

  47. B. Totaro, Projective resolutions of representations of GLn, J. Reine Angew. Math. 482 (1997), 1{13.

  48. D. Wehlau, The Noether number in invariant theory, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), no. 2, 39-62

    MathSciNet  MATH  Google Scholar 

  49. J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts in Mathematics, Vol 149, Cambridge University Press, Cambridge, 2003.

    Book  Google Scholar 

  50. H. Weyl, The Classical Groups, their Invariants and Representations, Princeton Mathematical Series, Vol. 1, Princeton Univ. Press, Princeton, 1946.

  51. А. Н. Зубков, Матричные инварианты над бесконечным полем конечной характеристики, Сиб. Матем. журн. 34 (1993), ном. 6, 68–74. Engl. transl.: A. N. Zubkov, Matrix invariants over an infinite field of finite characteristic, Siberian Math. J. 34 (1993), no. 6, 1059–1065.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48104, USA

    H. DERKSEN

  2. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    V. MAKAM

Authors
  1. H. DERKSEN
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. MAKAM
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. MAKAM.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

DERKSEN, H., MAKAM, V. WEYL’S POLARIZATION THEOREM IN POSITIVE CHARACTERISTIC. Transformation Groups 26, 1241–1260 (2021). https://doi.org/10.1007/s00031-020-09559-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-020-09559-3

Search

Navigation