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


Author: Sophie Morier-Genoud
Journal: Trans. Amer. Math. Soc. 360 (2008), 215-235
MSC (2000): Primary 14M25, 16W35, 14M15
DOI: https://doi.org/10.1090/S0002-9947-07-04216-X
Published electronically: June 22, 2007
MathSciNet review: 2342001
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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).


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  • 1. V. Alexeev, M. Brion.
    Toric degenerations of spherical varieties.
    ArXiv math.AG/0403379 MR 2072676 (2005a:14001)
  • 2. A. Berenstein, S. Fomin and A. Zelevinsky.
    Parametrization of canonical bases and totally positive matrices.
    Adv. Math., 122, (1996), 49-149. MR 1405449 (98j:17008)
  • 3. A. Berenstein, A. Zelevinsky.
    Canonical bases for the quantum group of type $ A_r$, and piecewise-linear combinatorics.
    Duke Math., 143 (1996), 473-502. MR 1387682 (97g:17007)
  • 4. A. Berenstein and A. Zelevinsky.
    Tensor product multiplicities, Canonical bases and Totally positive varieties.
    Invent. Math., 143 (2001), 77-128. MR 1802793 (2002c:17005)
  • 5. P. Caldero.
    Toric degenerations of Schubert varieties.
    Transf. Groups, Vol. 7, No. 1, 51-60, 2002. MR 1888475 (2003a:14073)
  • 6. P. Caldero.
    On the $ q$-commutations in $ U_q(n)$ at roots of one.
    J. Algebra, Vol. 210, (1998), 557-576. MR 1662288 (99i:17014)
  • 7. P. Caldero, R. Marsh, S. Morier-Genoud.
    Realisation of Lusztig cones.
    Represent. Theory 8 (2004), 458-478. MR 2110356 (2006f:17012)
  • 8. R. Chirivi.
    LS Algebras and applications to Schubert varieties,
    Transform. Groups 5, No. 3, 245-264, 2000. MR 1780934 (2001h:14060)
  • 9. N. Gonciulea, V. Lakshmibai.
    Degenerations of flag and Schubert varieties to tori varieties.
    Transform. Groups 1, no.3, 215-248, 1996. MR 1417711 (98a:14065)
  • 10. M. Kashiwara.
    On Crystal Bases.
    Canad. Math. Soc., Conference Proceed., 16, 155-195, 1995. MR 1357199 (97a:17016)
  • 11. P. Littelmann.
    Cones, crystals and patterns.
    Transformation Groups, 3, No. 2, 145-179, 1998. MR 1628449 (99e:17009)
  • 12. G. Lusztig.
    Introduction to quantum groups.
    Progress in Mathematics, 110, Birkhäuser, 1993. MR 1227098 (94m:17016)
  • 13. G. Lusztig.
    Braid group action and canonical bases.
    Adv. Math., 122, 237-261, 1996. MR 1409422 (98g:17019)
  • 14. S. Morier-Genoud.
    Relèvement géométrique de la base canonique et involution de Schützenberger.
    C.R. Acad. Sci. Paris. Ser. I337, 371-374, 2003. MR 2015078 (2004i:22016)
  • 15. R.W. Richardson.
    Intersections of double cosets in algebraic groups.
    Indag. Math., 3, 69-77, 1992. MR 1157520 (93b:20081)
  • 16. M. P. Schützenberger.
    Promotion des morphismes d'ensembles ordonnés.
    Discrete Math., 2, 73-94, 1972. MR 0299539 (45:8587)

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Additional Information

Sophie Morier-Genoud
Affiliation: Département de Mathématiques, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France
Email: morier@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/S0002-9947-07-04216-X
Received by editor(s): April 26, 2005
Received by editor(s) in revised form: September 27, 2005
Published electronically: June 22, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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