THE COMPONENTS OF THE SINGULAR LOCUS OF A COMPONENT OF A SPRINGER FIBER OVER x 2 = 0
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Springer Science+Business Media New York (2020)
THE COMPONENTS OF THE SINGULAR LOCUS OF A COMPONENT OF A SPRINGER FIBER OVER x2 = 0 RONIT MANSOUR∗
ANNA MELNIKOV
Department of Mathematics University of Haifa 3498838 Haifa, Israel
Department of Mathematics University of Haifa 3498838 Haifa, Israel
[email protected]
[email protected]
In memory of Ernest Borisovich Vinberg
Abstract. 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.
1. Introduction Throughout this paper, we set K to be an algebraically closed field of arbitrary characteristic. We set V to be a vector space of finite dimension n. A complete flag of V is a chain V0 ⊂ V1 ⊂ · · · ⊂ Vn of subspaces of V with dim Vi = i for all i = 0, 1, . . . , n. We denote the set of all the complete flags by F . 1.1. Springer fibers and their components Let x be a nilpotent endomorphism of V . A Springer fiber Fx is a subset of xstable complete flags, that is, flags (V0 , V1 , . . . , Vn ) such that x(Vi ) ⊂ Vi−1 for all i = 1, 2, . . . , n. Clearly, Fx is a closed subvariety of F (see [11]) and depends on the Jordan form of x only. Note that Fx is reducible unless x is zero or regular. Different aspects of Springer fibers were studied by many authors. In particular DOI: 10.1007/S00031-020-09621-0 Supported by ISF grant 797/14. Received September 28, 2018. Accepted June 16, 2020. Corresponding Author: Anna Melnikov, e-mail: [email protected] ∗
RONIT MANSOUR, ANNA MELNIKOV
many aspects of the study of Springer fibers and their connection to Schubert varieties are described in the survey [14]. The main objects of our interest are the irreducible components of a Springer fiber. We concentrate on the case of x satisfying x2 = 0. In this case the components are described in terms of link patterns. We provide the algorithm describing all the components of the singular locus of a singular component in terms of admissible pairs of a link pattern. 1.2. Parametrization of the irreducible components of Fx by Young tableaux A nilpotent endomorphism x : V 7→ V has a unique eigenvalue 0, so its Jordan form can be written as the list of lengths of its Jordan blocks and since the Jordan form is unique up to the order of Jordan blocks this list can be viewed as a partition of n. Put λ(x) := λ = (λ1 , . . . , λr ) ⊢ n to be this
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