Description of infinite orbits on multiple projective spaces. (English) Zbl 1441.14162
Summary: Let \(G\) be the general linear group of the degree \(n \geq 2\) over an algebraically closed field \(\mathbb{K}\). In this article, we describe the orbit decomposition of a product of copies of \(\mathbb{P}^{n - 1} \mathbb{K}\) under the diagonal action of \(G\). We also construct representatives of orbits. If \(m \geq 4\), the number of orbits is infinite, and we give a description of them.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14L30 | Group actions on varieties or schemes (quotients) |
References:
| [1] | Brion, M., Classification des espaces homogenes spheriques, Compos. Math., 63, 189-208 (1987) · Zbl 0642.14011 |
| [2] | Brion, M., Lectures on the geometry of flag varieties, (Topics in Cohomological Studies of Algebraic Varieties, Trends Math. (2005)), 33-85 · Zbl 1487.14105 |
| [3] | Kac, V., Infinite root systems, representations of graphs and invariant theory, Invent. Math., 56, 57-92 (1980) · Zbl 0427.17001 |
| [4] | Kobayashi, T.; Matsuki, T., Classification of finite multiplicity symmetric pairs, Transform. Groups, 19, 457-493 (2014) · Zbl 1298.22015 |
| [5] | Kobayashi, T.; Oshima, T., Finite multiplicity theorems for induction and restriction, Adv. Math., 248, 921-944 (2013) · Zbl 1317.22010 |
| [6] | Kobayashi, T.; Speh, B., Symmetry breaking for representations of rank one orthogonal groups, Mem. Am. Math. Soc., 238 (2015), v+110 pp · Zbl 1334.22015 |
| [7] | Kobayashi, T.; Speh, B., Symmetry Breaking for Representations of Rank One Orthogonal Groups II, Lecture Notes in Mathematics, vol. 2234 (2018), Springer, xv+342 pp., v+110 pp. · Zbl 1421.81003 |
| [8] | Magyar, P.; Weyman, J.; Zelevinsky, A., Multiple flag varieties of finite type, Adv. Math., 141, 97-118 (1999) · Zbl 0951.14034 |
| [9] | Magyar, P.; Weyman, J.; Zelevinsky, A., Symplectic multiple flag varieties of finite type, J. Algebra, 230, 245-265 (2000) · Zbl 0996.14023 |
| [10] | Matsuki, T., The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Jpn., 31, 331-357 (1979) · Zbl 0396.53025 |
| [11] | Matsuki, T., Orbits on affine symmetric spaces under the action of parabolic subgroups, Hiroshima Math. J., 12, 307-320 (1982) · Zbl 0495.53049 |
| [12] | Matsuki, T., Orbits on flag manifolds, (Proceedings of the International Congress of Mathematicians, Vols. I, II. Proceedings of the International Congress of Mathematicians, Vols. I, II, Kyoto, 1990 (1991), Math. Soc. Japan: Math. Soc. Japan Tokyo), 807-813 · Zbl 0745.22010 |
| [13] | Matsuki, T., An example of orthogonal triple flag variety of finite type, J. Algebra, 375, 148-187 (2013) · Zbl 1332.22017 |
| [14] | Matsuki, T., Orthogonal multiple flag varieties of finite type I: odd degree case, J. Algebra, 425, 450-523 (2015) · Zbl 1383.22008 |
| [15] | Tauchi, T., Dimension of the space of intertwining operators from degenerate principal series representations, Sel. Math. New Ser., 24, 3649-3662 (2018) · Zbl 1398.22016 |
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