# Article

Full entry | PDF   (0.2 MB)
Keywords:
conformally invariant differential operators; generalized (higher-spin) Dirac operators; representations of spin-groups; Littlewood-Richardson rule
Summary:
Polynomials on \$\Bbb R^n\$ with values in an irreducible \$\operatorname{Spin}_n\$-module form a natural representation space for the group \$\operatorname{Spin}_n\$. These representations are completely reducible. In the paper, we give a complete description of their decompositions into irreducible components for polynomials with values in a certain range of irreducible modules. The results are used to describe the structure of kernels of conformally invariant elliptic first order systems acting on maps on \$\Bbb R^n\$ with values in these modules.
References:
[1] Branson T.: Stein-Weiss operators and ellipticity. J. Funct. Anal. 151 (1997), 334-383. MR 1491546 | Zbl 0904.58054
[2] Bernardes G., Sommen F.: Monogenic functions of higher spin by Cauchy-Kowalevska extension of real-analytic functions. Complex Variables Theory Appl. 39 (1999), 4 305-325. MR 1727626
[3] Bureš J.: Special invariant operators. Comment. Math. Univ. Carolinae 37 (1996), 1 179-198. MR 1396170
[4] Bureš J.: The higher spin Dirac operators. Proc. Conf. Diff. Geometry and its Applications, 1998, pp.319-334. MR 1708920
[5] Bureš J.: The Rarita-Schwinger equation and spherical monogenic forms. Complex Variables Theory Appl. 43 (2000), 77-108. MR 1809813
[6] Bureš J.: Monogenic forms of the polynomial type. 2000, accepted in Proc. Conf. Clifford Anal. and its Appl., Prague, 2000. MR 1890434
[7] Bureš J., Souček V.: Eigenvalues of conformally invariant operators on spheres. Proc. 18th Winter School Geom. and Phys., Serie II., Suppl. 59, 1999, pp.109-122. MR 1692262
[8] Bureš J., Souček V., Sommen F., Van Lancker P.: Symmetric analogues of Rarita-Schwinger equations. 2000, accepted in Annals of Global Analysis and Geometry, Kluwer Publ.
[9] Bureš J., Souček V., Sommen F., Van Lancker P.: Rarita-Schwinger type operators in Clifford analysis. 2000, accepted in J. Funct. Analysis.
[10] Čap A., Slovák J., Souček V.: Invariant operators on manifolds with AHS structures, I.,II. Acta Math. Univ. Comenianae 66 (1997), 33-69, 203-220.
[11] Fegan H.D.: Conformally invariant first order differential operators. Quart. J. Math. 27 (1976), 371-378. MR 0482879 | Zbl 0334.58016
[12] Fulton W., Harris J.: Representation Theory. Springer-Verlag, 1991. MR 1153249 | Zbl 0744.22001
[13] Homma Y.: The higher spin Dirac operators on \$3\$-dimensional manifolds. preprint arXiv:math.DG/0006210. MR 1874992 | Zbl 1021.53026
[14] Homma Y.: Spinor-valued and Clifford algebra-valued harmonic polynomials. submitted to Geometry and Physics. Zbl 0972.43005
[15] Lawson H.B. jr., Michelson M.L.: Spin Geometry. Princeton University Press, 1989. MR 1031992
[16] Littelmann P.: A generalization of the Littlewood-Richardson rule. J. Algebra 130 (1990), 2 328-368. MR 1051307 | Zbl 0704.20033
[17] Morrey C.B. jr., Nirenberg L.: On the analycity of the solutions of linear elliptic systems of PDEs. Com. Pure Appl. Math. 10 (1957), 271-290. MR 0089334
[18] Samelson H.: Notes on Lie algebras. Van Nostrand Reinhold Mathematical Studies, 1969. MR 0254112 | Zbl 0708.17005
[19] Severa V.: Invariant differential operators on Spinor-valued differential forms. Dissertation, Charles University, Prague, 1998.
[20] Slovák J.: Invariant operators on conformal manifolds. Research Lecture Notes, University of Vienna, 1992.
[21] Slovák J.: Parabolic Geometries. DrSc Dissertation, Masaryk University Brno, 1998.
[22] Plechšmíd M.: Polynomial solutions for a class of higher spin equations. Dissertation, Charles University, Prague, 2001.

Partner of