## $(W,R)$-matroids and thin Schubert-type cells attached to algebraic torus actions

HTML articles powered by AMS MathViewer

- by Yi Hu
- Proc. Amer. Math. Soc.
**123**(1995), 2607-2617 - DOI: https://doi.org/10.1090/S0002-9939-1995-1223514-1
- PDF | Request permission

## Abstract:

Given a projective variety acted on by an algebraic torus, we introduce the notion of (**W**,

**R**)-matroids using the fixed-point set

**W**and the set

**R**of equivalence classes of one-parameter subgroups. The (

**W**,

**R**)-matroids provide close links among the geometry of torus orbits and Schubert-type cells, the theory of momentum polyhedra, and the combinatorial geometries. On the way to establishing the main theme of the paper, we showed that there are only finitely many Bialynicki-Birula decompositions induced by infinitely many one-parameter subgroups.

## References

- M. F. Atiyah,
*Convexity and commuting Hamiltonians*, Bull. London Math. Soc.**14**(1982), no. 1, 1–15. MR**642416**, DOI 10.1112/blms/14.1.1 - A. Białynicki-Birula,
*Some theorems on actions of algebraic groups*, Ann. of Math. (2)**98**(1973), 480–497. MR**366940**, DOI 10.2307/1970915
A. Borovik and I. Gelfand, - Henry H. Crapo and Gian-Carlo Rota,
*On the foundations of combinatorial theory: Combinatorial geometries*, Preliminary edition, The M.I.T. Press, Cambridge, Mass.-London, 1970. MR**0290980** - V. I. Danilov,
*The geometry of toric varieties*, Uspekhi Mat. Nauk**33**(1978), no. 2(200), 85–134, 247 (Russian). MR**495499** - I. M. Gel′fand, R. M. Goresky, R. D. MacPherson, and V. V. Serganova,
*Combinatorial geometries, convex polyhedra, and Schubert cells*, Adv. in Math.**63**(1987), no. 3, 301–316. MR**877789**, DOI 10.1016/0001-8708(87)90059-4 - I. M. Gel′fand and V. V. Serganova,
*Combinatorial geometries and the strata of a torus on homogeneous compact manifolds*, Uspekhi Mat. Nauk**42**(1987), no. 2(254), 107–134, 287 (Russian). MR**898623** - V. Guillemin and S. Sternberg,
*Convexity properties of the moment mapping*, Invent. Math.**67**(1982), no. 3, 491–513. MR**664117**, DOI 10.1007/BF01398933 - Yi Hu,
*The geometry and topology of quotient varieties of torus actions*, Duke Math. J.**68**(1992), no. 1, 151–184. MR**1185821**, DOI 10.1215/S0012-7094-92-06806-2 - Yi Hu,
*On the homology of complements of arrangements of subspaces and spheres*, Proc. Amer. Math. Soc.**122**(1994), no. 1, 285–290. MR**1204377**, DOI 10.1090/S0002-9939-1994-1204377-6 - M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky,
*Chow polytopes and general resultants*, Duke Math. J.**67**(1992), no. 1, 189–218. MR**1174606**, DOI 10.1215/S0012-7094-92-06707-X - M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky,
*Quotients of toric varieties*, Math. Ann.**290**(1991), no. 4, 643–655. MR**1119943**, DOI 10.1007/BF01459264 - David Kazhdan and George Lusztig,
*Representations of Coxeter groups and Hecke algebras*, Invent. Math.**53**(1979), no. 2, 165–184. MR**560412**, DOI 10.1007/BF01390031 - David Kazhdan and George Lusztig,
*Schubert varieties and Poincaré duality*, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 185–203. MR**573434** - Bernd Sturmfels,
*On the matroid stratification of Grassmann varieties, specialization of coordinates, and a problem of N. White*, Adv. Math.**75**(1989), no. 2, 202–211. MR**1002208**, DOI 10.1016/0001-8708(89)90037-6

*W P-matroids and thin Schubert cells on Tits systems*, preprint, 1992. —,

*Matroids on chamber systems*, preprint, 1992.

## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**123**(1995), 2607-2617 - MSC: Primary 14L30; Secondary 14M25, 52B40
- DOI: https://doi.org/10.1090/S0002-9939-1995-1223514-1
- MathSciNet review: 1223514