The decomposition of the spinor bundle of Grassmann manifolds. (English) Zbl 1152.14305
Summary: The decomposition of the spinor bundle of the spin Grassmann manifolds \(G_{m,n} = \text{SO}(m+n)/ \text{SO}(m)\times \text{SO}(n)\) into irreducible representations of \(\mathfrak{so}(m)\oplus\mathfrak{so}(n)\) is presented. A universal construction is developed and the general statement is proven for \(G_{2k+1,3}\), \(G_{2k,4}\), and \(G_{2k+1,5}\) for all \(k\). The decomposition is used to discuss properties of the spectrum and the eigenspaces of the Dirac operator.
MSC:
| 53C27 | Spin and Spin\({}^c\) geometry |
| 22E46 | Semisimple Lie groups and their representations |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Keywords:
spinor bundle; spin Grassmann manifolds; irreducible representations; eigenspaces of the Dirac operator; spectrum; symmetric spaces; even dimensional GrassmanniansReferences:
| [1] | DOI: 10.1007/978-3-540-74311-8 · doi:10.1007/978-3-540-74311-8 |
| [2] | Cahen, M. and Gutt, S., Proceedings of the Workshop on Clifford Algebra, Clifford Analysis and Their Applications in Mathematical Physics, Ghent, 1988 (unpublished), Vol. 62, pp. 209–242. · Zbl 0677.53057 |
| [3] | DOI: 10.1090/S0002-9939-01-06294-3 · Zbl 1003.43007 · doi:10.1090/S0002-9939-01-06294-3 |
| [4] | DOI: 10.1063/1.1611265 · Zbl 1063.81059 · doi:10.1063/1.1611265 |
| [5] | DOI: 10.1088/0305-4470/12/12/010 · Zbl 0445.22020 · doi:10.1088/0305-4470/12/12/010 |
| [6] | DOI: 10.2140/pjm.2000.194.83 · Zbl 1027.58027 · doi:10.2140/pjm.2000.194.83 |
| [7] | DOI: 10.4310/AJM.2002.v6.n2.a5 · Zbl 1127.53304 · doi:10.4310/AJM.2002.v6.n2.a5 |
| [8] | Fulton W., Graduate Texts in Mathematics 129, in: Representation Theory (1991) |
| [9] | Goette S., Equivariant -Invariants of Homogeneous Spaces (Äquivariante -Invarianten Homogener Räume) (1997) · Zbl 0894.58068 |
| [10] | DOI: 10.1090/S0002-9939-01-06158-5 · Zbl 0995.58005 · doi:10.1090/S0002-9939-01-06158-5 |
| [11] | DOI: 10.1090/gsm/034 · doi:10.1090/gsm/034 |
| [12] | DOI: 10.1090/S0002-9947-04-03722-5 · Zbl 1069.22006 · doi:10.1090/S0002-9947-04-03722-5 |
| [13] | Humphreys, J. E. Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics Vol. 9, 3rd ed. (Springer-Verlag, Berlin, 1980), p. 171. · Zbl 0447.17002 |
| [14] | DOI: 10.1088/0305-4470/15/4/017 · Zbl 0495.22014 · doi:10.1088/0305-4470/15/4/017 |
| [15] | DOI: 10.1016/0001-8708(89)90004-2 · Zbl 0681.20030 · doi:10.1016/0001-8708(89)90004-2 |
| [16] | DOI: 10.1016/0021-8693(87)90099-8 · Zbl 0622.20033 · doi:10.1016/0021-8693(87)90099-8 |
| [17] | DOI: 10.1016/0001-8708(90)90059-V · Zbl 0698.22013 · doi:10.1016/0001-8708(90)90059-V |
| [18] | Littlewood D. E., The Theory of Group Characters and Matrix Representations of Groups (1940) · JFM 66.0093.02 |
| [19] | DOI: 10.1007/BF01018698 · doi:10.1007/BF01018698 |
| [20] | Macdonald, I. G. Symmetric Functions and Hall Polynomials, 2nd ed. (Oxford Science, Oxford, 1998), p. 475. · Zbl 0899.05068 |
| [21] | McKay W. G., Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras, Lecture Notes in Pure and Applied Mathematics 69 (1981) · Zbl 0448.17001 |
| [22] | DOI: 10.1063/1.2186924 · Zbl 1111.58023 · doi:10.1063/1.2186924 |
| [23] | DOI: 10.1063/1.532324 · Zbl 0919.58062 · doi:10.1063/1.532324 |
| [24] | DOI: 10.1007/s00220-005-1391-9 · Zbl 1079.53068 · doi:10.1007/s00220-005-1391-9 |
| [25] | DOI: 10.2307/1970892 · Zbl 0249.22003 · doi:10.2307/1970892 |
| [26] | DOI: 10.1023/A:1006513104421 · Zbl 0935.58017 · doi:10.1023/A:1006513104421 |
| [27] | DOI: 10.1002/mana.19800980107 · Zbl 0473.53047 · doi:10.1002/mana.19800980107 |
| [28] | Tsukamoto C., Osaka J. Math. 18 pp 407– (1981) |
| [29] | Vasilevich D. V., Teor. Mat. Fiz. 66 pp 350– (1986) |
| [30] | Wolf J. A., J. Math. Mech. 14 pp 1033– (1965) |
| [31] | Wybourne B. G., Symmetry Principles and Atomic Spectroscopy (1970) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
