Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix
Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix
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DOI：
10.1007/bf00046881
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发表时间：
1988-07
期刊：
影响因子：
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通讯作者：
F. Mari;M. Shayman
中科院分区：
文献类型：
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作者：
F. Mari;M. Shayman
设A ∈gl（n，C），p为正整数. A的度的Hessenberg簇是完备旗流形的子簇Hess（p，A），它由满足条件ASi Sn− 1 in Sn组成，对所有i。证明了如果A有不同的特征值，则Hess（p，A）是光滑连通的. Hess（p，A）的奇Betti数为零，而偶Betti数由欧拉数的自然推广给出。在A的特征值具有不同模的情况下，|λ1|联系我们|λ1|这些结果被应用于确定U（n）的子流形的维数和拓扑，其中U（n）的子流形是由A0 =P-1AP为Hessenberg型的酉矩阵P组成的，并且在A0处初始化的QR迭代的对角元收敛于λ1，λn的给定置换.
LetA∈gl(n, C) and letpbe a positive integer. The Hessenberg variety of degreepforAis the subvariety Hess(p, A) of the complete flag manifold consisting of those flagsS1⊂⋯⊂Sn−1in ℂnwhich satisfy the conditionASi⊂Si+p,for alli. We show that ifAhas 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 ofAhave distinct moduli, |λ1|<⋯<|λ1|, these results are applied to determine the dimension and topology of the submanifold of U(n) consisting of those unitary matricesPfor whichA0=P-1APis in Hessenberg form and for which the diagonal entries of the QR-iteration initialized atA0converge to a given permutation of λ1,λn.