Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix |
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Authors: | Filippo de Mari Mark A Shayman |
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Institution: | (1) Department of Mathematics, Washington University, 63130 St. Louis, MO, U.S.A.;(2) Present address: Forschungsschwerpunkt Dynamische Systeme, Universität Bremen, 2800 Bremen 33, F.R.G.;(3) Electrical Engineering Department, University of Maryland, 20742 College Park, MD, U.S.A.;(4) Systems Research Center, University of Maryland, 20742 College Park, MD, U.S.A. |
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Abstract: | Let A gl(n, C) and let p be a positive integer. The Hessenberg variety of degree p for A is the subvariety Hess(p, A) of the complete flag manifold consisting of those flags S
1
![sub](/content/r6323188467u7087/xxlarge8834.gif) ![ctdot](/content/r6323188467u7087/xxlarge8943.gif) S
n–1
in n which satisfy the condition AS
i
S
i+p
,for all i. We show that if A has distinct eigenvalues, then Hess(p, A) is smooth and connected. The odd Betti numbers of Hess(p, A) vanish, while the even Betti numbers are given by a natural generalization of the Eulerian numbers. In the case where the eigenvalues of A have distinct moduli, | 1|< <| 1|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matrices P for which A
0=P
-1
AP is in Hessenberg form and for which the diagonal entries of the QR-iteration initialized at A
0 converge to a given permutation of 1, n.Research partially supported by the National Science Foundation under Grant ECS-8696108. |
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Keywords: | 05A10 14M15 32M10 65F15 |
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