Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

Morier-Genoud, Sophie

doi:10.1090/S0002-9947-07-04216-X

Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties
HTML articles powered by AMS MathViewer

by Sophie Morier-Genoud PDF

Trans. Amer. Math. Soc. 360 (2008), 215-235 Request permission

Abstract:

In the $\mathfrak {sl}_n$ case, A. Berenstein and A. Zelevinsky (1996) studied the Schützenberger involution in terms of Lusztig’s canonical basis. We generalize their construction and formulas for any semisimple Lie algebra. We use the geometric lifting of the canonical basis, on which an analogue of the Schützenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero (2002).

References

Laurent Bonavero and Michel Brion (eds.), Geometry of toric varieties, Séminaires et Congrès [Seminars and Congresses], vol. 6, Société Mathématique de France, Paris, 2002. Lectures from the Summer School held in Grenoble, June 19–July 7, 2000. MR 2072676
Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49–149. MR 1405449, DOI 10.1006/aima.1996.0057
Arkady Berenstein and Andrei Zelevinsky, Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics, Duke Math. J. 82 (1996), no. 3, 473–502. MR 1387682, DOI 10.1215/S0012-7094-96-08221-6
Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
Philippe Caldero, Toric degenerations of Schubert varieties, Transform. Groups 7 (2002), no. 1, 51–60. MR 1888475, DOI 10.1007/s00031-002-0003-4
Philippe Caldero, On the $q$-commutations in $U_q(\mathfrak {n})$ at roots of one, J. Algebra 210 (1998), no. 2, 557–576. MR 1662288, DOI 10.1006/jabr.1998.7603
Philippe Caldero, Robert Marsh, and Sophie Morier-Genoud, Realisation of Lusztig cones, Represent. Theory 8 (2004), 458–478. MR 2110356, DOI 10.1090/S1088-4165-04-00225-0
R. Chirivì, LS algebras and application to Schubert varieties, Transform. Groups 5 (2000), no. 3, 245–264. MR 1780934, DOI 10.1007/BF01679715
N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups 1 (1996), no. 3, 215–248. MR 1417711, DOI 10.1007/BF02549207
Masaki Kashiwara, On crystal bases, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 155–197. MR 1357199
P. Littelmann, Cones, crystals, and patterns, Transform. Groups 3 (1998), no. 2, 145–179. MR 1628449, DOI 10.1007/BF01236431
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
George Lusztig, Braid group action and canonical bases, Adv. Math. 122 (1996), no. 2, 237–261. MR 1409422, DOI 10.1006/aima.1996.0061
Sophie Morier-Genoud, Relèvement géométrique de la base canonique et involution de Schützenberger, C. R. Math. Acad. Sci. Paris 337 (2003), no. 6, 371–374 (French, with English and French summaries). MR 2015078, DOI 10.1016/j.crma.2003.07.001
R. W. Richardson, Intersections of double cosets in algebraic groups, Indag. Math. (N.S.) 3 (1992), no. 1, 69–77. MR 1157520, DOI 10.1016/0019-3577(92)90028-J
M. P. Schützenberger, Promotion des morphismes d’ensembles ordonnés, Discrete Math. 2 (1972), 73–94 (French). MR 299539, DOI 10.1016/0012-365X(72)90062-3