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Korbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skewsymmetric bilinear form
 Transform. Groups
"... Abstract. We use equivariant localization and divided difference operators to determine formulas for the torusequivariant fundamental cohomology classes of Korbit closures on the flag variety G/B, where G = GL(n,C), and where K is one of the symmetric subgroups O(n,C) or Sp(n,C). We realize these ..."
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Abstract. We use equivariant localization and divided difference operators to determine formulas for the torusequivariant fundamental cohomology classes of Korbit closures on the flag variety G/B, where G = GL(n,C), and where K is one of the symmetric subgroups O(n,C) or Sp(n,C). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skewsymmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles. Suppose that V → X is a rank n vector bundle over a smooth complex variety X, and that V is equipped with a symmetric or skewsymmetric bilinear form γ taking values in the trivial bundle, along with a complete flag of subbundles F•. Let b ∈ Sn be an involution, assumed fixed pointfree if n is even and γ is skewsymmetric. Consider the degeneracy locus (1) Db = {x ∈ X  rank(γFi(x)×Fj(x)) ≤ rb(i, j) ∀i, j}, where rb(i, j) is a nonnegative integer depending on b, i, and j. The main result of this paper is a recursive procedure by which one may obtain a formula for the fundamental class
(English summary)
"... Let G be a simple algebraic group over an algebraically closed field k with characteristic p> 0. Let q = pr and by G(Fq) denote the finite Chevalley group consisting of the set of Fqpoints of G; r F equivalently, G(Fq) is the set of fixed points G(k) of the rth power of a standard Frobenius map ..."
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Let G be a simple algebraic group over an algebraically closed field k with characteristic p> 0. Let q = pr and by G(Fq) denote the finite Chevalley group consisting of the set of Fqpoints of G; r F equivalently, G(Fq) is the set of fixed points G(k) of the rth power of a standard Frobenius map F. Let T be a maximal torus of G and let λ ∈ Xr(T) be a prrestricted weight, with associated simple Gmodule L(λ). By a result of Steinberg, L(λ) is also a simple kG(Fq)module and all simple kG(Fq)modules are obtained in this way. In [E. T. Cline et al., Invent. Math. 39 (1977), no. 2, 143–163; MR0439856 (55 #12737)] it was shown that the restriction map H1 (G, L(λ)) → H1 (G(Fq), L(λ)) is always injective for such weights λ. In the paper under review, it is shown that this restriction map is an isomorphism whenever λ is small enough and p and q satisfy some mild constraints. By λ small enough, we mean that λ satisfies λ ≤ ωi in the dominance order on weights, where ωi is a fundamental dominant weight for G. (If G is classical of type A to D, then this implies either λ = ωi or λ = 0; for the exceptional types there are more options: for instance 2ω4 < ω1 + ω4 < ω2 when G is of type F4.)
1 GL(p,C)×GL(q,C)ORBIT CLOSURES ON THE FLAG VARIETY AND SCHUBERT STRUCTURE CONSTANTS FOR (p, q)PAIRS
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COMBINATORIAL RESULTS ON (1, 2, 1, 2)AVOIDING GL(p,C)×GL(q,C)ORBIT CLOSURES ON GL(p + q,C)/B
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Polynomials for GLp ×GLq orbit closures in the flag variety. Preprint, available at http://arxiv.org/abs/1308.2632
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1 FIRST COHOMOLOGY FOR FINITE GROUPS OF LIE TYPE: SIMPLE MODULES WITH SMALL DOMINANT WEIGHTS
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4. TITLE AND SUBTITLE Marines in the Interagency: Are We in the Right Places?
, 2011
"... Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regardin ..."
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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection
Results 1  10
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42