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Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes. (English) Zbl 1084.14049
This paper connects two degenerations related to the manifold \(F_n\) of complete flags in \({\mathbb C}^n\). N. Gonciulea and V. Lakshmibai [Transform. Groups 1, No. 3, 215–248 (1996; Zbl 0909.14028)] used standard monomial theory to construct a flat sagbi degeneration of \(F_n\) into the toric variety of the Gelfand-Tsetlin polytope, and recently A. Knutson and E. Miller [Ann. Math. (2) 161, No. 3, 215–248 (2005; Zbl 1089.14007)] constructed Gröbner degenerations of matrix Schubert varieties into linear spaces corresponding to monomials in double Schubert polynmials. The flag variety is the geometric invariant theory (GIT) quotient of the space \(M_n\) of \(n\) by \(n\) matrices by the Borel group \(B\) of lower triangular matrices. A matrix Schubert variety is an inverse image of a Schubert variety under this quotient. The main result in the paper under review is that this GIT quotient extends to the degenerations. The sagbi degeneration is a GIT quotient of the Gröbner degeneration.\smallskip The nature of this GIT quotient is quite interesting. The authors exhibit an action of the Borel group \(B\) on the product \(M_n\times{\mathbb C}\) of \(M_n\) with the complex line, so that the GIT quotient \(B\backslash\backslash(M_n\times{\mathbb C})\) remains fibred over \({\mathbb C}\) and is the total space of the sagbi degeneration. In this GIT quotient, the total space of the Gröbner degeneration (as in Knutson and Miller) of a matrix Schubert variety in \(M_n\times{\mathbb C}\) covers the total space of the Lakshmibai-Gonciulea degeneration of the corresponding Schubert variety. At the degenerate point, the matrix Schubert variety has become a union of coordinate planes, each of which covers a component of the sagbi degeneration of the Schubert variety indexed by a face of the Gelfand-Tsetlin polytope.
The authors use this to identify which faces of the Gelfand-Tsetlin polytope occur in a given degenerate Schubert variety, and to give a simple explanation of the classical Gelfand-Tsetlin decomposition of an irreducible polynomial representation of \(\text{GL}_n\) into one-dimensional weight spaces; in the degeneration, sections of a line bundle over \(F_n\) become sections of the defining line bundle on the toric variety for the Gelfand-Tsetlin polytope.

MSC:
14M15 Grassmannians, Schubert varieties, flag manifolds
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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References:
[1] Bergeron, N.; Billey, S., RC-graphs and Schubert polynomials, Exp. math., 2, 4, 257-269, (1993) · Zbl 0803.05054
[2] Buch, A.S.; Fulton, W., Chern class formulas for quiver varieties, Invent. math., 135, 3, 665-687, (1999) · Zbl 0942.14027
[3] Billey, S.C.; Jockusch, W.; Stanley, R.P., Some combinatorial properties of Schubert polynomials, J. algebraic combin., 2, 4, 345-374, (1993) · Zbl 0790.05093
[4] Buch, A.S., Grothendieck classes of quiver varieties, Duke math. J., 115, 1, 75-103, (2002) · Zbl 1052.14056
[5] Caldero, P., Toric degenerations of Schubert varieties, Transform. groups, 7, 1, 51-60, (2002) · Zbl 1050.14040
[6] Chirivı̀, R., LS algebras and application to Schubert varieties, Transform. groups, 5, 3, 245-264, (2000) · Zbl 1019.14019
[7] Eisenbud, D., Commutative algebra, with a view toward algebraic geometry, Graduate texts in mathematics, Vol. 150, (1995), Springer New York · Zbl 0819.13001
[8] Fomin, S.; Kirillov, A.N., Combinatorial Bn-analogues of Schubert polynomials, Trans. amer. math. soc., 348, 9, 3591-3620, (1996) · Zbl 0871.05060
[9] Fomin, S.; Kirillov, A.N., The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete math., 153, 1-3, 123-143, (1996), Proceedings of the Fifth Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993 · Zbl 0852.05078
[10] Fomin, S.; Stanley, R.P., Schubert polynomials and the nil-Coxeter algebra, Adv. math., 103, 2, 196-207, (1994) · Zbl 0809.05091
[11] Fulton, W., Flags, Schubert polynomials, degeneracy loci and determinantal formulas, Duke math. J., 65, 3, 381-420, (1992) · Zbl 0788.14044
[12] Fulton, W., Introduction to toric varieties, Annals of mathematical studies, Vol. 131, (1993), Princeton University Press
[13] Fulton, W., Young tableaux. with applications to representation theory and geometry, London mathematical society student texts, Vol. 35, (1997), Cambridge University Press Cambridge · Zbl 0878.14034
[14] Fulton, W., Universal Schubert polynomials, Duke math. J., 96, 3, 575-594, (1999) · Zbl 0981.14022
[15] Gonciulea, N.; Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties, Transform. groups, 1, 3, 215-248, (1996) · Zbl 0909.14028
[16] Guillemin, V.; Sternberg, S., The Gel’fand-cetlin system and quantization of the complex flag manifolds, J. funct. anal., 52, 1, 106-128, (1983) · Zbl 0522.58021
[17] Gelfand, I.M.; Tsetlin, M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. akad. nauk SSSR (N.S.), 71, 825-828, (1950) · Zbl 0037.15301
[18] A. Knutson, E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. (2) (2004) to appear. arXiv:math.AG/0110058v3.
[19] A. Knutson, E. Miller, M. Shimozono, Four positive formulae for quiver polynomials, 2003 preprint. arXiv:math.AG/0308142. · Zbl 1107.14046
[20] M. Kogan, Schubert geometry of flag varieties and Gel’fand-Cetlin theory, Ph.D. Thesis, Massachusetts Institute of Technology, 2000.
[21] Lascoux, A.; Schützenberger, M.-P., Polynômes de Schubert, C. R. acad. sci. Paris Sér. I math., 294, 13, 447-450, (1982) · Zbl 0495.14031
[22] Lascoux, A.; Schützenberger, M.-P., Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. acad. sci. Paris Sér. I math., 295, 11, 629-633, (1982) · Zbl 0542.14030
[23] Littelmann, P., Cones, crystals, and patterns, Transform. groups, 3, 2, 145-179, (1998) · Zbl 0908.17010
[24] Sturmfels, B., Gröbner bases and convex polytopes, AMS university lecture series, Vol. 8, (1996), American Mathematical Society Providence, RI · Zbl 0856.13020
[25] R. Vakil, A geometric Littlewood-Richardson rule. arXiv:math.AG/0302294. · Zbl 1163.05337
[26] R. Vakil, Schubert induction. arXiv:math.AG/0302296. · Zbl 1115.14043
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