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THE COMPONENTS OF THE SINGULAR LOCUS OF A COMPONENT OF A SPRINGER FIBER OVER x2 = 0

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For x ∈ End(𝕂n) satisfying x2 = 0 let ℱx be the variety of full flags stable under the action of x (Springer fiber over x). The full classification of the components of ℱx according to their smoothness was provided in [4] in terms of both Young tableaux and link patterns. Moreover in [2] the purely combinatorial algorithm to compute the singular locus of a singular component of ℱx is provided. However, this algorithm involves the computation of the graph of the component, and the complexity of computations grows very quickly, so that in practice it is impossible to use it. In this paper, we construct another algorithm, giving all the components of the singular locus of a singular component ℱσ of ℱx in terms of link patterns constructed straightforwardly from the link pattern of σ.

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References

  1. L. Fresse, Singular components of Springer fibers in the two-column case, Ann. Inst. Fourier 59 (2009), 2429–2444.

    Article  MathSciNet  Google Scholar 

  2. L. Fresse, On the singular locus of certain subvarieties of Springer fibers, Math. Res. Lett. 19 (2012), 753–768.

    Article  MathSciNet  Google Scholar 

  3. L. Fresse, A. Melnikov, On the singularity of the irreducible components of a Springer fiber in sl(n), Selecta Math. (N.S.) 16 (2010), no. 3, 393–418.

    Article  MathSciNet  Google Scholar 

  4. L. Fresse, A. Melnikov, Some charcterizations of singular components of Springer fibers in the two-column case, J. Alg. Represent. Theory 14 (2011), 1063–1086.

    Article  Google Scholar 

  5. L. Fresse, A. Melnikov, S. Sakas-Obeid, On the structure of smooth components of Springer fibers, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2301–2315.

    Article  MathSciNet  Google Scholar 

  6. W. Fulton, Young Tableaux with Applications to Representation Theory and Geometry, Cambridge Univ. Press, London, 1997.

  7. F. Y. C. Fung, On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory, Adv. Math. 178 (2003), no. 2, 244–276.

    Article  MathSciNet  Google Scholar 

  8. N. Perrin, E. Smirnov, Springer fiber components in the two column case for type A and D are normal, Bull. Soc. Math. France 140 (2012), no. 3, 309–333.

    Article  MathSciNet  Google Scholar 

  9. V. Lakshmibai, C.S. Seshadri, Singular locus of a Schubert variety, Bull. Amer. Math. Soc. (N.S.) 11 (1984), no. 2, 363–366.

    Article  MathSciNet  Google Scholar 

  10. A. Melnikov, B-orbits of nilpotent order 2 and link patterns, Indag. Math. (N.S.) 24 (2013), 443–473.

    Article  MathSciNet  Google Scholar 

  11. N. Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology 16 (1977), no. 2, 203–204.

    Article  MathSciNet  Google Scholar 

  12. N. Spaltenstein, Classes unipotentes et sous-groups de Borel, Lect. Notes Math., Vol. 946, Springer-Verlag, Berlin, 1982.

  13. Д. А. Тимашёв, Обобщение разложения Броа, Изв. РАН. Сер. матем. 58 (1994), вьш. 5, 110–123. Engl. transl.: D. A. Timashev, Generalization of the Bruhat decomposition, Russian Acad. Sci. Izv. Math. 45 (1995), no. 2, 339–352.

  14. J. Tymoczko, The geometry and combinatorics of Springer fibers, in: Around Langlands Correspondences, Contemp. Math. 691, Amer. Math. Soc., Providence, RI, 2017, pp. 359–376.

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Correspondence to RONIT MANSOUR.

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In memory of Ernest Borisovich Vinberg

RONIT MANSOUR is supported by ISF grant 797/14.

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MANSOUR, R., MELNIKOV, A. THE COMPONENTS OF THE SINGULAR LOCUS OF A COMPONENT OF A SPRINGER FIBER OVER x2 = 0. Transformation Groups 27, 597–633 (2022). https://doi.org/10.1007/s00031-020-09621-0

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  • DOI: https://doi.org/10.1007/s00031-020-09621-0

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