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 ⊂⋯⊂S n−1 in ℂn 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, |λ1|<⋯<|λ1|, 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 λ1,λn.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ammar, G. S. and Martin, C. F.: The geometry of matrix eigenvalue methods, Acta Applic. Math. 5 (1986), 239–278.
Ammar, G. S.: Geometric aspects of Hessenberg matrices, Contemp. Math. 68 (1987), 1–21.
Steinberg, R.: Desingularization of the unipotent variety, Invent. Math. 36 (1976), 209–224.
Springer, T. A.: A construction of representations of Weyl groups, Invent. Math. 44 (1978), 279–293.
Spaltenstein, N.: The fixed point set of a unipotent transformation on the flag manifold, Proc. Kon. akad. Wetensch. Amsterdam 79 (1978), 452–456.
Hotta, R. and Shimomura, N.: The fixed point subvarieties of unipotent transformations on generalized flag varieties and the Green functions-combinatorial and cohomological treatments centering GL(n), Math. Ann. 241 (1979), 193–208.
Shimomura, N.: A theorem on the fixed point set of a unipotent transformation on the flag manifold, J. Math. Soc. Japan 32 (1980), 55–64.
Spaltenstein, N.: Sous-groupes de Borel Contenant un Unipotent Donné, Lecture Notes in Math. 948, Springer, New York, 1982.
Shimomura, N.: The fixed point subvarieties of unipotent transformations on the flag varieties, J. Math. Soc. Japan 37 (1985), 537–556.
Helmke, U. and Shayman, M. A.: The biflag manifold and the fixed points of a unipotent transformation on the flag manifold, Linear Algebra Appl. 92 (1987), 125–159.
De Concini, C., Lusztig, G., and Procesi, C.: Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), 15–34.
Hiller, H.: The Geometry of Coxeter Groups, Pitman, London, 1982.
Foata, D.: Distributions Eulériennes et Mahoniennes sur le groupe des permutations, in M. Aigner (ed.), Higher Combinatorics, D. Reidel, Dordrecht, 1977.
Carlitz, L.: Eulerian numbers and polynomials of higher order, Duke Math. J. 27 (1960), 401–423.
Carlitz, L. and Scoville, R.: Generalized Eulerian numbers: combinatorial applications, J. Reine Angew. Math. 265 (1974), 110–137.
Dillon, J. and Roselle, D.: Eulerian numbers of higher order, Duke Math. J. 35 (1968), 247–256.
Lehmer, D.: Generalized Eulerian numbers, J. Combin. Theory Ser A 32 (1982), 195–215.
Rawlings, D.: The r-major index, J. Combin. Theory Ser. A 31 (1981), 175–183.
Stanley, R. P.: Enumerative Combinatorics, Vol I, Wadsworth and Brooks/Cole, Monterey, 1986.
Hermann, R.: Cartanian Geometry, Nonlinear Waves, and Control Theory, Part A, Math Sci Press, Brookline, 1979.
Shub, M. and Vasquez, A. T.: Some linearly induced Morse-Smale systems, the QR-algorithm and the Toda lattice, Contemp. Math. 64 (1987), 181–194.
Faibusovich, L. E.: Generalized Toda flows, Riccati equations on Grassmann manifolds and QR-algorithm, Functional Anal. Appl. 21 (1987), 88–89. (in Russian)
Smale, S.: Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.
Golub, G. H. and Van Loan, C. F.: Matrix Computations, Johns Hopkins Univ. Press, Baltimore, MD, 1984.
Tits, J.: Théorème de Bruhat et sous-groupes paraboliques, C.R. Acad. Sci. Paris, 254 (1962) 2910–2912.
Shayman, M. A.: Phase portrait of the matrix Riccati equation, SIAM J. Control Optim. 24 (1986), 1–65.
Shayman, M. A.: Riccati equations, linear flows on the flag manifold, and the Bruhat decomposition, in C. I. Byrnes and A. Lindquist (eds.), Frequency Domain and State Space Methods for Linear Systems, North-Holland, Amsterdam, 1986.
Durfee, A. H.: Algebraic varieties which are a disjoint union of subvarieties, in C. McCrory and T. Shirfin (eds.), Geometry and Topology, Manifolds, Varieties and Knots, Dekker, New York, 1987.
Bialynicki-Birula, A.: Some theorems on actions of algebraic groups, Annals of Math. 98 (1973), 480–497.
Comtet, L.: Analyse Combinatoire, Vols I & II, Presses Univ. de France, Paris, 1970.
Riordan, J.: An Introduction to Combinatorial Analysis, Wiley, New York, 1958.
Author information
Authors and Affiliations
Additional information
Research partially supported by the National Science Foundation under Grant ECS-8696108.
Rights and permissions
About this article
Cite this article
de Mari, F., Shayman, M.A. Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix. Acta Appl Math 12, 213–235 (1988). https://doi.org/10.1007/BF00046881
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00046881