Standard monomial theory for desingularized Richardson varieties in the flag variety \(\mathrm{GL}(n)/B\). (English) Zbl 1296.14036
Standard Monomial Theory deals with the problem of producing a basis of \(H^0(X, L)\) for a Schubert variety \(X\) inside a homogeneous space \(G/P\) and a line bundle \(L\) on \(G/P\) that can be understood by some combinatorial objects. One particularly desired property of such a basis is compatibility with special subvarieties. For example, a standard monomial basis of \(H^0(G/P, L)\) should restrict to a basis of \(H^0(X, L)\).
The Bott-Samelson variety is a classic desingularization of a Schubert variety. The intersection of a Schubert variety with an opposite Schubert variety is called a Richardson variety. The author considers desingularizations \(\Gamma\) of these Richardson varieties in the case of a complete flag variety of type \(A_n\) as a fibre of a projection from a certain Bott-Samelson variety \(Z\). The article deals with a version of Standard Monomial Theory on these desingularizations.
In case of the Bott-Samelson variety, V. Lakshmibai, P. Littelmann and P. Magyar [in: Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28–August 8, 1997. Broer, A. (ed.) et al., Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28–August 8, 1997. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 319–364 (1998; Zbl 1013.17014)] had previously defined a family of globally generated line bundles \(L_m\) and given a basis for \(H^0(Z, L_m)\) by objects called standard tableaux. The author deals with the question whether these standard monomial bases are compatible with all choices of \(\Gamma\), i.e. whether they restrict to a basis of \(H^0(\Gamma, L_m)\). The main result is that this is true if \(L_m\) is very ample. Moreover, there is a concrete criterion for which basis elements become zero after restricting and have to be removed.
The Bott-Samelson variety is a classic desingularization of a Schubert variety. The intersection of a Schubert variety with an opposite Schubert variety is called a Richardson variety. The author considers desingularizations \(\Gamma\) of these Richardson varieties in the case of a complete flag variety of type \(A_n\) as a fibre of a projection from a certain Bott-Samelson variety \(Z\). The article deals with a version of Standard Monomial Theory on these desingularizations.
In case of the Bott-Samelson variety, V. Lakshmibai, P. Littelmann and P. Magyar [in: Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28–August 8, 1997. Broer, A. (ed.) et al., Representation theories and algebraic geometry. Proceedings of the NATO Advanced Study Institute, Montreal, Canada, July 28–August 8, 1997. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 514, 319–364 (1998; Zbl 1013.17014)] had previously defined a family of globally generated line bundles \(L_m\) and given a basis for \(H^0(Z, L_m)\) by objects called standard tableaux. The author deals with the question whether these standard monomial bases are compatible with all choices of \(\Gamma\), i.e. whether they restrict to a basis of \(H^0(\Gamma, L_m)\). The main result is that this is true if \(L_m\) is very ample. Moreover, there is a concrete criterion for which basis elements become zero after restricting and have to be removed.
Reviewer: Benjamin Schmidt (Columbus)
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14L17 | Affine algebraic groups, hyperalgebra constructions |
Citations:
Zbl 1013.17014References:
| [1] | M. Balan, Multiplicity on a Richardson variety in a cominuscule G/P, to appear in Trans. Amer. Math. Soc., preprint, http://arxiv.org/abs/1009.2873. · Zbl 0833.22001 |
| [2] | S. Billey, I. Coskun, Singularities of generalized Richardson varieties, Comm. Algebra 40 (2012), no. 4, 1466-1495. · Zbl 1274.14059 · doi:10.1080/00927872.2011.551903 |
| [3] | A. Björner, F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, Vol. 231, Springer, New York, 2005. · Zbl 1110.05001 |
| [4] | R. Bott, H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. · Zbl 0101.39702 · doi:10.2307/2372843 |
| [5] | M. Brion, Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 (2002), no. 1, 137-159. · Zbl 1052.14054 · doi:10.1016/S0021-8693(02)00505-7 |
| [6] | Brion, M., Lectures on the Geometry of Flag Varieties, 33-85 (2005), Basel · Zbl 1487.14105 · doi:10.1007/3-7643-7342-3_2 |
| [7] | M. Brion, V. Lakshmibai, A geometric approach to standard monomial theory, Represent. Theory 7 (2003), 651-680 (electronic). · Zbl 1053.14056 · doi:10.1090/S1088-4165-03-00211-5 |
| [8] | C. Contou-Carrère, Géométrie des groupes semi-simples, résolutions équivariantes et lieu singulier de leurs variétés de Schubert, Thèse d’ État, Université de Montpellier II, 1983. |
| [9] | M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. · Zbl 0312.14009 |
| [10] | V. Deodhar, On some geometric aspects of Bruhat orderings. I. A finer decomposition of Bruhat cells, Invent. Math. 79 (1985), no. 3, 499-511. · Zbl 0563.14023 · doi:10.1007/BF01388520 |
| [11] | P. Foth, S. Kim, Standard Monomial Theory of RR varieties, preprint, http:// arxiv.org/abs/1009.1645. · Zbl 1326.14119 |
| [12] | S. Gaussent, The fibre of the Bott-Samelson resolution, Indag. Math. (N.S.) 12 (2001), no. 4, 453-468. · Zbl 1065.14065 · doi:10.1016/S0019-3577(01)80034-3 |
| [13] | K. Goodearl, M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5753-5780. · Zbl 1179.53087 · doi:10.1090/S0002-9947-09-04654-6 |
| [14] | H. Hansen, On cycles in flag manifolds, Math. Scand. 33 (1973), 269-274. · Zbl 0301.14019 |
| [15] | M. Härterich, The T-equivariant Cohomology of Bott-Samelson varieties, preprint, http://arxiv.org/abs/math/0412337. · Zbl 0618.14026 |
| [16] | R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York, 1977. Russian transl.: Р. Xартсxорн, Алгeбpaичecкaя гeoмeтpия, Миp, M., 1981. · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0 |
| [17] | X. He, T. Lam, Projected Richardson varieties and affine Schubert varieties, preprint, http://arxiv.org/abs/1106.2586. · Zbl 1350.14035 |
| [18] | X. He, J. Lu, On intersections of certain partitions of a group compactification, Int. Math. Res. Not. 2011, no. 11, 2534-2564. · Zbl 1253.20049 |
| [19] | W. V. D. Hodge, Some enumerative results in the theory of forms, Proc. Cambridge Philos. Soc. 39 (1943), 22-30. · Zbl 0060.04107 · doi:10.1017/S0305004100017631 |
| [20] | W. Hodge, D. Pedoe, Methods of Algebraic Geometry. Vol. II, Cambridge University Press, Cambridge, 1952. Russian transl.: B. Xoдж, Д. Пидo, Meтoды aлгeбpaичecкoй гeoмeтpии, т. II, ИЛ, M., 1954. · Zbl 0048.14502 |
| [21] | A. Knutson, T. Lam, D. Speyer, Projections of Richardson Varieties, to appear in J. Reine Angew. Math., preprint, http://arxiv.org/abs/1008.3939. · Zbl 1345.14047 |
| [22] | A. Knutson, T. Lam, D. Speyer, Positroid varieties: juggling and geometry, preprint, http://arxiv.org/abs/1111.3660. · Zbl 1330.14086 |
| [23] | V. Kreiman, Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondance, J. Algebraic Combin. 27 (2008), no. 3, 351-382. · Zbl 1142.14034 · doi:10.1007/s10801-007-0093-0 |
| [24] | Kreiman, V.; Lakshmibai, V., Richardson varieties in the Grassmannian, 573-597 (2004), Baltimore, MD · Zbl 1092.14059 |
| [25] | V. Lakshmibai, P. Littelmann, Richardson varieties and equivariant K-theory, J. Algebra 260 (2003), no. 1, 230-260. · Zbl 1071.14049 · doi:10.1016/S0021-8693(02)00634-8 |
| [26] | V. Lakshmibai, P. Littelmann, P. Magyar, Standard monomial theory and applications, Notes by R. W. T. Yu, in: Representation Theories and Algebraic Geometry, Montreal, PQ, 1997, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 514, Kluwer, Dordrecht, 1998, pp. 319-364. · Zbl 1013.17014 |
| [27] | V. Lakshmibai, P. Littelmann, P. Magyar, Standard monomial theory for Bott- Samelson varieties, Compositio Math. 130 (2002), no. 3, 293-318. · Zbl 1061.14051 · doi:10.1023/A:1014396129323 |
| [28] | V. Lakshmibai, P. Magyar, Standard monomial theory for Bott-Samelson varieties of GL(n), Publ. Res. Inst. Math. Sci. 34 (1998), no. 3, 229-248. · Zbl 0967.14031 · doi:10.2977/prims/1195144694 |
| [29] | V. Lakshmibai, C. S. Seshadri, Geometry of G/P. V, J. Algebra 100 (1986), no. 2, 462-557. · Zbl 0618.14026 · doi:10.1016/0021-8693(86)90089-X |
| [30] | V. Lakshmibai, C. S. Seshadri, Standard monomial theory, in: Proceedings of the Hyderabad Conference on Algebraic Groups, Hyderabad, 1989, Manoj Prakashan, Madras, 1991, 279-322. · Zbl 0785.14028 |
| [31] | N. Lauritzen, J. Thomsen, Line bundles on Bott-Samelson varieties, J. Algebraic Geom. 13 (2004), no. 3, 461-473. · Zbl 1080.14056 · doi:10.1090/S1056-3911-03-00358-8 |
| [32] | P. Littelmann, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math. 116 (1994), no. 1-3, 329-346. · Zbl 0805.17019 · doi:10.1007/BF01231564 |
| [33] | P. Littelmann, Paths and root operators in representation theory, Ann. of Math. (2) 142 (1995), no. 3, 499-525. · Zbl 0858.17023 · doi:10.2307/2118553 |
| [34] | P. Littelmann, Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc. 11 (1998), no. 3, 551-567. · Zbl 0915.20022 · doi:10.1090/S0894-0347-98-00268-9 |
| [35] | P. Magyar, Schubert polynomials and Bott-Samelson varieties, Comment. Math. Helv. 73 (1998), no. 4, 603-636. · Zbl 0951.14036 · doi:10.1007/s000140050071 |
| [36] | R. Richardson, Intersections of double cosets in algebraic groups, Indag. Math. (N.S.) 3 (1992), no. 1, 69-77. · Zbl 0833.22001 · doi:10.1016/0019-3577(92)90028-J |
| [37] | R. Richardson, T. Springer, Combinatorics and geometry of K-orbits on the flag manifold, in: Linear Algebraic Groups and their Representations, Los Angeles, CA, 1992, Contemp. Math., Vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109-142. · Zbl 0840.20039 |
| [38] | R. Stanley, Some combinatorial aspects of the Schubert calculus, in: Combinatoire et Représentation du Groupe Symétrique, Actes Table Ronde CNRS, Univ. Louis Pasteur Strasbourg, Strasbourg, 1976, Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 217-251. · Zbl 0359.05006 |
| [39] | B. Taylor, A straightening algorithm for row-convex tableaux, J. Algebra 236 (2001), no. 1, 155-191. · Zbl 0967.05065 · doi:10.1006/jabr.2000.8495 |
| [40] | S. Upadhyay, Initial ideals of tangent cones to Richardson varieties in the orthogonal Grassmannian via a orthogonal-bounded-RSK-correspondence, preprint, http://arxiv.org/abs/0909.1424. · Zbl 1273.13053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
