Spherical tropical geometry: a survey of recent developments. (English) Zbl 1420.14146
Summary: This is a survey of some recent results on spherical tropical geometry.
MSC:
| 14T05 | Tropical geometry (MSC2010) |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 14M27 | Compactifications; symmetric and spherical varieties |
References:
| [1] | Brion, M.: Vers une généralisation des espaces symétriques. J. Algebra, 134, 115-143 (1990) · Zbl 0729.14038 · doi:10.1016/0021-8693(90)90214-9 |
| [2] | De Concini, C., Procesi, C.: Complete symmetric varieties. II. Intersection theory. Algebraic groups and related topics (Kyoto/Nagoya, 1983), 481-513, Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985 · Zbl 0596.14041 |
| [3] | Gaitsgory, D., Nadler, D.: Spherical varieties and Langlands duality. Mosc. Math. J., 10(1), 65-137, 271 (2010) · Zbl 1207.22013 |
| [4] | Gelfand, I. M., Kapranov, M. M., Zelevinsky, A. V.: Discriminants, resultants and multidimensional determinants. Reprint of the 1994 edition. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA, 2008 · Zbl 1138.14001 |
| [5] | Gubler, W.: A guide to tropicalizations. Algebraic and combinatorial aspects of tropical geometry, 125-189, Contemp. Math., 589, Amer. Math. Soc., Providence, RI, 2013 · Zbl 1318.14061 |
| [6] | Kaveh, K., Manon, C.: Gröbner theory and tropical geometry on spherical varieties. arXiv:1611.01841 · Zbl 1451.14176 |
| [7] | Kennedy, G.: http://u.osu.edu/kennedy.28/files/2014/11/sphericaltropical2-12rpr9e.pdf |
| [8] | Knop, F.: The asymptotic behavior of invariant collective motion. Invent. Math., 116 (1994) · Zbl 0802.58024 |
| [9] | Knop, F., Krötz, B., Sayag, E., et al.: Simple compactifications and polar decomposition of homogeneous real spherical spaces. Selecta Math. N.S.21 (3), 1071-1097 (2015) · Zbl 1346.14121 · doi:10.1007/s00029-014-0174-6 |
| [10] | Knop, F.: The Luna-Vust theory of spherical embeddings. Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), 225-249, Manoj Prakashan, Madras, 1991 · Zbl 0812.20023 |
| [11] | Luna, D., Vust, Th., Plongements d’espaces homogènes. Comment. Math. Helv., 58, 186-245 (1983) · Zbl 0545.14010 · doi:10.1007/BF02564633 |
| [12] | Maclagan, D., Sturmfels, B.: Introduction to tropical geometry. Graduate Studies in Mathematics, AMS (2015) · Zbl 1321.14048 · doi:10.1090/gsm/161 |
| [13] | Nash, E., Tropicalizing spherical embeddings. arXiv:1609.07455 |
| [14] | Popov, V. L.: Contractions of actions of reductive algebraic groups. Math. USSR-Sb., 58 (2), 311-335 (1987) · Zbl 0627.14033 · doi:10.1070/SM1987v058n02ABEH003106 |
| [15] | Sakellaridis, Y.: Spherical varieties and integral representations of L-functions. Algebra Number Theory, 6 (4), 611-667 (2012) · Zbl 1253.11059 · doi:10.2140/ant.2012.6.611 |
| [16] | Sturmfels, B.: Gröbner bases and convex polytopes. University Lecture Series, 8. American Mathematical Society, Providence, RI, 1996 · Zbl 0856.13020 |
| [17] | Sumihiro, H.: Equivariant completion. J. Math. Kyoto Univ., 14, 1-28 (1974) · Zbl 0277.14008 · doi:10.1215/kjm/1250523277 |
| [18] | Tevelev, J.: Compactifications of subvarieties of tori. Amer. J. Math., 129 (4), 1087-1104 (2007) · Zbl 1154.14039 · doi:10.1353/ajm.2007.0029 |
| [19] | Timashev, D.: Homogeneous spaces and equivariant embeddings. Encyclopaedia of Mathematical Sciences, 138. Invariant Theory and Algebraic Transformation Groups, 8. Springer, Heidelberg, 2011 · Zbl 1237.14057 |
| [20] | Vogiannou, T.: Spherical tropicalization. arXiv:1511.02203 · Zbl 1479.14078 |
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.
