Hessenberg varieties

Authors:
F. De Mari, C. Procesi and M. A. Shayman

Journal:
Trans. Amer. Math. Soc. **332** (1992), 529-534

MSC:
Primary 14L30; Secondary 14M17

DOI:
https://doi.org/10.1090/S0002-9947-1992-1043857-6

MathSciNet review:
1043857

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Numerical algorithms involving Hessenberg matrices correspond to dynamical systems which evolve on the subvariety of complete flags in satisfying the condition , , where is an endomorphism of . This paper describes the basic topological features of the generalization to subvarieties of , a complex semisimple algebraic group, which are indexed by certain subsets of negative roots. In the special case where the subset consists of the negative simple roots, the variety coincides with the torus embedding associated to the decomposition into Weyl chambers.

**[1]**A. Bialynicki-Birula,*Some theorems on actions of algebraic groups*, Ann. of Math. (2)**98**(1973), 480-497. MR**0366940 (51:3186)****[2]**-,*Some properties of the decomposition of algebraic varieties determined by actions of a torus*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**24**(1976), 667-674. MR**0453766 (56:12020)****[3]**M. W. Davis,*Some aspherical manifolds*, Duke Math J.**55**(1987), 105-139. MR**883666 (88j:57044)****[4]**G. De Concini and C. Procesi,*Complete symmetric varieties*. II, Adv. Stud. Pure Math.**6**(1985), 481-513.**[5]**F. De Mari,*On the topology of the Hessenberg varieties of a matrix*, Ph.D. thesis, Washington Univ., St. Louis, Missouri, 1987.**[6]**F. De Mari and M. A. Shayman,*Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix*, Acta Appl. Math.**12**(1988), 213-235. MR**973945 (89i:05009)****[7]**-,*Lie algebraic generalizations of Hessenberg matrices and the topology of Hessenberg varieties*, Realization and Modelling in System Theory: Proceedings of the International Symposium MTNS-89 (M. A. Kaashoek, J. H. van Schuppen and A. C. M. Ran, eds.) (to appear).**[8]**I. M. Gel'fand and V. V. Serganova,*Combinatorial geometries and torus strata on homogeneous compact manifolds*, Russian Math. Surveys**42**(1987), 133-168. MR**898623 (89g:32049)****[9]**J. E. Humphreys,*Linear algebraic groups*, Springer-Verlag, New York, 1975. MR**0396773 (53:633)****[10]**C. Procesi,*The toric variety of Weyl chambers*, preprint.**[11]**J. R. Stembridge,*Eulerian numbers, tableaux, and the Betti numbers of a toric variety*, preprint. MR**1158793 (93f:05103)****[12]**T. A. Springer,*Linear algebraic groups*, Birkhäuser, Boston, Mass., 1981.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
14L30,
14M17

Retrieve articles in all journals with MSC: 14L30, 14M17

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1043857-6

Article copyright:
© Copyright 1992
American Mathematical Society