Advertisement

Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 425–446 | Cite as

The Monogenic Fischer Decomposition: Two Vector Variables

  • P. Van LanckerEmail author
Article

Abstract

In this paper we will present two proofs of the monogenic Fischer decomposition in two vector variables. The first one is based on the so-called “Harmonic Separation of Variables Theorem” while the second one relies on some simple dimension arguments. We also show that these decomposition are still valid under milder assumptions than the usual stable range condition. In the process, we derive explicit formula for the summands in the monogenic Fischer decomposition of harmonics.

Keywords

Clifford analysis Dirac operators Representations Fischer decomposition Spin groups 

Mathematics Subject Classification (2000)

30G35 42B37 35J05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brackx F., De Schepper H., Eelbode D., Souček V.: The Howe dual pair in Hermitean Clifford analysis. Rev. Mat. Iberoamericana 26(2), 449–479 (2010)zbMATHCrossRefGoogle Scholar
  2. 2.
    Brackx F., Delanghe R., Sommen F.: Clifford Analysis. Pitman, London (1982)zbMATHGoogle Scholar
  3. 3.
    Brackx, F., Eelbode, D., Raeymaekers, T., Van de Voorde, L.: Triple monogenic functions and higher spin Dirac operators (submitted)Google Scholar
  4. 4.
    Brackx, F., Eelbode, D., Van de Voorde, L.: Higher spin Dirac operators between spaces of simplicial monogenics in two vector variables (submitted)Google Scholar
  5. 5.
    Bureš J., Sommen F., Souček V., Van Lancker P.: Rarita–Schwinger type operators in Clifford analysis. J. Funct. Anal. 185, 425–456 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bureš J., Sommen F., Souček V., Van Lancker P.: Symmetric analogues of Rarita–Schwinger equations. Ann. Glob. Anal. Geom. 21(3), 215–240 (2001)Google Scholar
  7. 7.
    Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Separation of variables in Clifford analysis and its application to Rarita–Schwinger field. In: Simos, T.E., Psihoyios, G., Tsitouras, Ch. (eds.) Extended abstracts of the ICNAAM 2006 Conference, Official Conference of the European Society of Computational Methods in Sciences and Engineering, Crete, Greece, pp. 630–633 (2006)Google Scholar
  8. 8.
    Colombo, F., Sabadini, I., Sommen, F., Struppa, D.C.: Analysis of Dirac Systems and Computational Algebra. Progress in Mathematical Physics, vol. 39. Birkhäuser, Basel (2004)Google Scholar
  9. 9.
    Constales, D.: The relative position of L 2-domains in complex and Clifford analysis, Ph. D. thesis, State University Ghent (1989–1990)Google Scholar
  10. 10.
    Debarre O., Ton-That T.: Representations of \({SO(k, {\mathbb C})}\) on harmonic polynomials on a null cone. Proc. Am. Math. Soc. 112(1), 31–44 (1991)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Delanghe, R., Lavicka, R., Soucek, V.: The Fischer decomposition for Hodge-de Rham systems in Euclidean spaces. arXiv:1012.4994v1Google Scholar
  12. 12.
    Delanghe R., Sommen F., Souček V.: Clifford Analysis and Spinor Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)CrossRefGoogle Scholar
  13. 13.
    Eelbode, D., Smid, D., Van de Voorde, L.: A note on polynomial solutions for higher spin Dirac operators (in preparation)Google Scholar
  14. 14.
    Gilbert J., Murray M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, London (1991)zbMATHCrossRefGoogle Scholar
  15. 15.
    Goodman, R.: Multiplicity-Free Spaces and Schur–Weyl–Howe Duality. In: Eng-Chye, T., Chen-Bo, Z. (eds.) Representations of Real and P-Adic Groups. World Scientific Publishing Company, Singapore (2004–06)Google Scholar
  16. 16.
    Gürlebeck K., Sprössig W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, Chichester (1997)zbMATHGoogle Scholar
  17. 17.
    Helgason S.: Invariants and fundamental functions. Acta Math. 109, 241–258 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Howe, R., Tan, E.C., Willenbring, J.: Reciprocity Algebras and Branching for Classical Symmetric Pairs, Groups and analysis. London Mathematical Society. Lecture Notes Series in Mathematics, vol. 354, pp. 191–231. Cambridge University Press, Cambridge (2008)Google Scholar
  19. 19.
    Kashiwara M., Vergne M.: On the Segal–Shale–Weil representations and harmonic polynomials. Inventiones Math. 44, 1–47 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kostant B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sabadini I., Sommen F., Struppa D.C., Van Lancker P.: Complexes of dirac operators in clifford algebras. Math. Zeit. 239(2), 293–320 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Sommen F.: Clifford analysis in two and several vector variables. Appl. Anal. 73, 225–253 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Sommen F.: Functions on the spin group. Adv. Appl. Clifford algebras 6(1), 37–48 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Sommen F.: An algebra of abstract vector variables. Portugal. Math. 54(3), 287–310 (1997)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sommen F., Van Acker N.: SO(m)-invariant differential operators on Clifford algebra valued functions. Found. Phys. 23(11), 1491–1519 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sommen F., Van Acker N.: Monogenic differential operators. Results Math. 22(3-4), 781–798 (1992)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Sommen F., Van Acker N.: Functions of two vector variables. Adv. Appl. Clifford Algebras 4(1), 65–72 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Sommen, F., Van Acker, N.: Invariant differential operators on polynomial-valued functions, In: Clifford Algebras and their Applications in Mathematical Physics, Fund. Theories Phys. vol. 55, pp. 203–212. Kluwer Academic Publishers, Dordrecht (1993)Google Scholar
  29. 29.
    Stein E.W., Weiss G.: Generalization of the Cauchy–Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Van Lancker P., Sommen F., Constales D.: Models for irreducible representations of Spin(m). Adv. Appl. Clifford Algebras 11(S1), 271–289 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Weyl H.: The Classical Groups, their Invariants and Representations. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Engineering Sciences, Member of Ghent University AssociationUniversity College GhentGhentBelgium

Personalised recommendations