The topology of Gelfand-Zeitlin fibers
| Autor: | Carlson, Jeffrey D., Lane, Jeremy |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We prove several new results about the topology of fibers of Gelfand--Zeitlin systems on unitary and orthogonal coadjoint orbits, at the same time finding a unifying framework recovering and shedding light on essentially all known results. We find completely explicit descriptions of the diffeomorphism type of the fiber in many instances a direct factor decomposition of the fiber, and a torus factor corresponding to the action given by the Thimm trick. The new description also gives us a weak local normal form for a coadjoint orbit, which we use to define a topological toric degeneration, new in the orthogonal case. We also compute the first three homotopy groups (new in the orthogonal case) and cohomology rings of a fiber (new in both cases). All these descriptions can be read in a straightforward manner from the combinatorics of the associated Gelfand--Zeitlin pattern. Comment: Explicit coordinates for fibers, local normal forms for coadjoint orbits, new description of torus action and equivalence with Thimm trick action, and topological toric degeneration all added.Existing material substantially rearranged and condensed (especially material on homotopy groups and homology) for reasons of space. 39 pages |
| Databáze: | arXiv |
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