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conditions imposed on flag varieties
 Department of Mathematics, University of Iowa, Iowa City
"... Abstract. We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenber ..."
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Cited by 7 (4 self)
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Abstract. We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(C) and show that they have no odddimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions. 1.
Paving Hessenberg varieties by affines
, 2004
"... Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all ..."
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Cited by 3 (1 self)
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Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of this paving and their dimensions can be identified by a combinatorial condition on roots. We use this paving to prove these Hessenberg varieties have no odddimensional homology. 1.
Topics I. Overview of the computation II. New developments
, 2004
"... IV. A dimensionreducing trick I. Overview of the computation Reviewing the setup from our lecture notes of last year [S1], we let R denote a crystallographic root system with V the ambient real Euclidean space, along with the usual choices (simple roots, positive roots,...), and W the Weyl group. G ..."
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IV. A dimensionreducing trick I. Overview of the computation Reviewing the setup from our lecture notes of last year [S1], we let R denote a crystallographic root system with V the ambient real Euclidean space, along with the usual choices (simple roots, positive roots,...), and W the Weyl group. Given a reduced expression for the longest element of the Weyl group, say w0 = si1 · · · sil, there is an induced ordering β ∨ 1,...,β ∨ l of the positive coroots; namely, β ∨ l = α ∨ il, β ∨ l−1 = slα ∨ il−1,..., β ∨ 1 = sl · · · s2α ∨ i1. Now for each ν ∈ V, let A(ν) denote the following element of the group algebra RW: A(ν): = (1 + 〈ν,β ∨ 1 〉si1) · · · (1 + 〈ν,β ∨ l 〉sil). Elementary Facts 1–3. (1) A(ν) is independent of the choice of reduced expression. (2) If w0ν = −ν, then A(ν) is Hermitian in every unitary representation of W. (3) A(ν) is invertible if and only if 〈ν,β ∨ 〉 ̸ = ±1 for all roots β.Let V0 denote the subspace of V fixed by −w0.
IMPOSING LINEAR CONDITIONS ON FLAG VARIETIES
, 2004
"... Abstract. We study subvarieties of the flag variety defined by certain linear conditions. These subvarieties are called Hessenberg varieties and arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of He ..."
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Abstract. We study subvarieties of the flag variety defined by certain linear conditions. These subvarieties are called Hessenberg varieties and arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(C) and show that they have no odddimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We also characterize these affine pieces by fillings of certain Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules. We give an equivalent formulation in terms of roots, and open questions about Hessenberg varieties. 1.