|
Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties
Author(s):
Sophie
Morier-Genoud
Journal:
Trans. Amer. Math. Soc.
360
(2008),
215-235.
MSC (2000):
Primary 14M25, 16W35, 14M15
Posted:
June 22, 2007
MathSciNet review:
2342001
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In the  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:
-
- 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  , 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  -commutations in  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)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
14M25, 16W35, 14M15
Retrieve articles in all Journals with
MSC (2000):
14M25, 16W35, 14M15
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:
10.1090/S0002-9947-07-04216-X
PII:
S 0002-9947(07)04216-X
Received by editor(s):
April 26, 2005
Received by editor(s) in revised form:
September 27, 2005
Posted:
June 22, 2007
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
