Recognizing Schubert Cells
Sergey Fomin
and Andrei Zelevinsky
DOI: 10.1023/A:1008759501188
Abstract
We address the problem of distinguishing between different Schubert cells using vanishing patterns of generalized Plücker coordinates.
Pages: 37–57
Keywords: Schubert cell; Schubert variety; flag manifold; plücker coordinate; Bruhat cell; vanishing pattern
Full Text: PDF
References
1. N. Bourbaki, Groupes et alg`ebres de Lie, Hermann, Paris, Ch. IV-VI, 1968.
2. V.V. Deodhar, “Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative M\ddot obius function,” Invent. Math. 39 (1977), 187-198.
3. V.V. Deodhar, “On Bruhat ordering and weight-lattice ordering for a Weyl group,” Indagationes Math. 40 (1978), 423-435.
4. C. Ehresmann, “Sur la topologie de certains espaces homog`enes,” Ann. Math. 35 (1934), 396-443.
5. S. Fomin and A. Zelevinsky, “Double Bruhat cells and total positivity,” J. Amer. Math. Soc. 12 (1999), 335- 380.
6. W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991.
7. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
8. M. Geck and Sungsoon Kim, “Bases for the Bruhat-Chevalley order on all finite Coxeter groups,” J. Algebra 197 (1997), 278-310.
9. I.M. Gelfand and V.V. Serganova, “Combinatorial geometries and the strata of a torus on homogeneous compact manifolds,” Russian Math. Surveys 42 (1987), 133-168.
10. D.Y. Grigoriev, “An analogue of the Bruhat decomposition for the closure of the cone of a classical Chevalley group series,” Soviet Math. Dokl. 23(2) (1981), 393-397.
11. D.Y. Grigoriev, “Additive complexity in directed computations,” Theoret. Comput. Sci. 19 (1982), 39-67.
12. V. Lakshmibai and C.S. Seshadri, Geometry of G/P. V, J. Algebra 100 (1986), 462-557.
13. A. Lascoux and M.-P. Sch\ddot utzenberger, “Treillis et bases des groupes de Coxeter,” Electron. J. Combin. 3(2) (1996), Research paper 27.
14. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. Part II, North-Holland, 1977.
15. A. Ramanathan, “Equations defining Schubert varieties and Frobenius splitting of diagonals,” Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61-90.
16. V.V. Serganova, A. Vince, and A. Zelevinsky, “A geometric characterization of Coxeter matroids,” Annals of Combinatorics 1 (1997), 173-181.
17. C.S. Seshadri, Geometry of G/P-I. “Theory of standard monomials for minuscule representations,” in: C.P. Ramanujam-a tribute, Tata Inst. Fund. Res. Studies in Math. 8 (1978), 207-239.
2. V.V. Deodhar, “Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative M\ddot obius function,” Invent. Math. 39 (1977), 187-198.
3. V.V. Deodhar, “On Bruhat ordering and weight-lattice ordering for a Weyl group,” Indagationes Math. 40 (1978), 423-435.
4. C. Ehresmann, “Sur la topologie de certains espaces homog`enes,” Ann. Math. 35 (1934), 396-443.
5. S. Fomin and A. Zelevinsky, “Double Bruhat cells and total positivity,” J. Amer. Math. Soc. 12 (1999), 335- 380.
6. W. Fulton and J. Harris, Representation Theory, Springer-Verlag, New York, 1991.
7. W. Fulton, Young Tableaux, Cambridge University Press, 1997.
8. M. Geck and Sungsoon Kim, “Bases for the Bruhat-Chevalley order on all finite Coxeter groups,” J. Algebra 197 (1997), 278-310.
9. I.M. Gelfand and V.V. Serganova, “Combinatorial geometries and the strata of a torus on homogeneous compact manifolds,” Russian Math. Surveys 42 (1987), 133-168.
10. D.Y. Grigoriev, “An analogue of the Bruhat decomposition for the closure of the cone of a classical Chevalley group series,” Soviet Math. Dokl. 23(2) (1981), 393-397.
11. D.Y. Grigoriev, “Additive complexity in directed computations,” Theoret. Comput. Sci. 19 (1982), 39-67.
12. V. Lakshmibai and C.S. Seshadri, Geometry of G/P. V, J. Algebra 100 (1986), 462-557.
13. A. Lascoux and M.-P. Sch\ddot utzenberger, “Treillis et bases des groupes de Coxeter,” Electron. J. Combin. 3(2) (1996), Research paper 27.
14. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. Part II, North-Holland, 1977.
15. A. Ramanathan, “Equations defining Schubert varieties and Frobenius splitting of diagonals,” Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61-90.
16. V.V. Serganova, A. Vince, and A. Zelevinsky, “A geometric characterization of Coxeter matroids,” Annals of Combinatorics 1 (1997), 173-181.
17. C.S. Seshadri, Geometry of G/P-I. “Theory of standard monomials for minuscule representations,” in: C.P. Ramanujam-a tribute, Tata Inst. Fund. Res. Studies in Math. 8 (1978), 207-239.