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Reducibility of the intersections of components of a Springer fiber. (English) Zbl 1172.14033
Summary: The description of the intersections of components of a Springer fiber is a very complex problem. Up to now only two cases have been described completely. The complete picture for the hook case has been obtained by N. Spaltenstein [Nederl. Akad. Wet., Proc., Ser. A 79, 452–456 (1976; Zbl 0343.20029)] and J. A. Vargas [Bol. Soc. Mat. Mex., II. Ser. 24, 1–14 (1979; Zbl 0458.14019)], and for two-row case by F. Y. C. Fung [Adv. Math. 178, No. 2, 244–276 (2003; Zbl 1035.20004)]. They have shown in particular that the intersection of a pair of components of a Springer fiber is either irreducible or empty. In both cases all the components are non-singular and the irreducibility of the intersections is strongly related to the non-singularity. As it has been shown in [J. Algebra 298, No. 1, 1–14 (2006; Zbl 1097.17006)], a bijection between orbital varieties and components of the corresponding Springer fiber in GL$$_n$$ extends to a bijection between the irreducible components of the intersections of orbital varieties and the irreducible components of the intersections of components of Springer fiber preserving their codimensions. Here we use this bijection to compute the intersections of the irreducible components of Springer fibers for two-column case. In this case the components are in general singular. As we show the intersection of two components is non-empty. The main result of the paper is a necessary and sufficient condition for the intersection of two components of the Springer fiber to be irreducible in two-column case. The condition is purely combinatorial. As an application of this characterization, we give first examples of pairs of components with a reducible intersection having components of different dimensions.

MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14L30 Group actions on varieties or schemes (quotients) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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