Abstract
This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO ( n ) in SU ( n ) , the asymptotic of the restrictions of representations of SL ( n ) to SO ( n ) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.
Highlights
This paper develops a new method for studying the cohomology of orthogonal flag varieties
A restriction variety is the intersection of OF (k1, . . . , kh; n) with a Schubert variety in the ordinary flag variety F (k1, . . . , kh; n) defined by a flag satisfying certain tangency conditions with respect to Q
To each non-vanishing branching coefficient, in [BS], Berenstein and Sjamaar associate an inequality satisfied by the K -moment polytope of a K-coadjoint orbit
Summary
We recall the preliminaries about the geometry of quadric hypersurfaces and orthogonal Grassmannians. If Q is a quadric hypersurface of corank r in Pn−1, the largest dimensional linear space on Q has dimension n−r−2 2. Denote the Fano variety of s-dimensional projective linear spaces contained in a quadric hypersurface Q ∈ Pn−1 of corank r by Fsr,n(Q). The orthogonal Grassmannian OG(k, n) is isomorphic to one irreducible component of the Fano variety Fk0−1,n(Q) of (k − 1)-dimensional projective linear spaces on a smooth quadric hypersurface. The description of the Schubert varieties in OG(k, 2m) requires minor modifications to account for the fact that the space of m-dimensional isotropic subspaces have two irreducible components. Note that strictly speaking the intersection of the quadric hypersurface with Fm⊥−1 consists of the union of two m-dimensional isotropic subspaces one in each irreducible component. This is seen by considering the projection from OF (k1, . . . , kh; n) to OG(kh, n)
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