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ROW CONVEX TABLEAUX AND BOTT–SAMELSON VARIETIES

Published online by Cambridge University Press:  23 October 2014

PHILIP FOTH
Affiliation:
Champlain St. Lawrence College, Quebec, Canada G1V 4K2 email [email protected]
SANGJIB KIM*
Affiliation:
Department of Mathematics, Ewha Womans University, Seoul 151-892, South Korea email [email protected]
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Abstract

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By using row convex tableaux, we study the section rings of Bott–Samelson varieties of type A. We obtain flat deformations and standard monomial type bases of the section rings. In a separate section, we investigate a three-dimensional Bott–Samelson variety in detail and compute its Hilbert polynomial and toric degenerations.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Bott, R. and Samelson, H., ‘The cohomology ring of GT’, Proc. Natl Acad. Sci. USA 41 (1955), 490493.Google Scholar
Bott, R. and Samelson, H., ‘Applications of the theory of Morse to symmetric spaces’, Amer. J. Math. 80 (1958), 9641029.CrossRefGoogle Scholar
Conca, A., Herzog, J. and Valla, G., ‘Sagbi bases with applications to blow-up algebras’, J. reine angew. Math. 474 (1996), 113138.Google Scholar
Demazure, M., ‘Désingularisation des variétés de Schubert généralisées’, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 5388.CrossRefGoogle Scholar
Doubilet, P., Rota, G.-C. and Stein, J., ‘On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory’, Stud. Appl. Math. 53 (1976), 185216.Google Scholar
Grossberg, M. and Karshon, Y., ‘Bott towers, complete integrability, and the extended character of representations’, Duke Math. J. 76 (1994), 2358.Google Scholar
Iskovskikh, V. and Prokhorov, Yu., Algebraic Geometry V: Fano varieties, Encyclopædia of Mathematical Sciences, 47 (Springer, Berlin, 1999).Google Scholar
Kogan, M. and Miller, E., ‘Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes’, Adv. Math. 193(1) (2005), 117.Google Scholar
Lakshmibai, V., Littelmann, P. and Magyar, P., ‘Standard monomial theory for Bott–Samelson varieties’, Compositio Math. 130(3) (2002), 293318.CrossRefGoogle Scholar
Lakshmibai, V. and Magyar, P., ‘Standard monomial theory for Bott–Samelson varieties of G L (n)’, Publ. Res. Inst. Math. Sci. 34(3) (1998), 229248.Google Scholar
Magyar, P., ‘Schubert polynomials and Bott–Samelson varieties’, Comment. Math. Helv. 73(4) (1998), 603636.CrossRefGoogle Scholar
Miller, E. and Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227 (Springer, New York, 2005).Google Scholar
Pasquier, B., ‘Vanishing theorem for the cohomology of line bundles on Bott–Samelson varieties’, J. Algebra 323(10) (2010), 28342847.Google Scholar
Sturmfels, B., Gröbner Bases and Convex Polytopes, University Lecture Series, 8 (American Mathematical Society, Providence, RI, 1996).Google Scholar
Taylor, B. D., ‘A straightening algorithm for row convex tableaux’, J. Algebra 236 (2001), 155191.CrossRefGoogle Scholar