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Criteria for rational smoothness of some symmetric orbit closures
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
2012 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 229, no 1, 183-200 p.Article in journal (Refereed) Published
Abstract [en]

Let G be a connected reductive linear algebraic group over C with an involution theta. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities t = {theta(omega(-1))omega | omega is an element of W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph" whose vertices form a subset of t. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on t is rank symmetric. less thanbrgreater than less thanbrgreater thanIn the special case K = Sp(2n) (C), G = SL(2n) (C), we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one", needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.

Place, publisher, year, edition, pages
Elsevier , 2012. Vol. 229, no 1, 183-200 p.
Keyword [en]
Rational smoothness, Symmetric orbit closure, Bruhat graph
National Category
URN: urn:nbn:se:liu:diva-73094DOI: 10.1016/j.aim.2011.09.002ISI: 000297184800007OAI: diva2:466459
Available from: 2011-12-16 Created: 2011-12-16 Last updated: 2012-12-18

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Hultman, Axel
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Mathematics and Applied MathematicsThe Institute of Technology
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