Toric degenerations of Schubert varieties

By Philippe Caldero. 2000

134 citations recorded by the source corpus.

LetG be a simply connected semisimple complex algebraic group. We prove that every Schubert variety ofG has a flat degeneration into a toric variety. This provides a generalization of results of [9], [7], [6]. Our basic tool is Lusztig's canonical basis and the string parametrization of this basis.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/26e727917864a069e16c5c93367b46b3b5a1f3f0

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Citations in the corpus

From triangulated categories to cluster algebras (2005, background): “This provides toric degenerations of the spectrum of cluster algebras of finite type, in the spirit of [6].”
Toric degenerations of spherical varieties (2004, background): “Note that Pλ = B if and only if λ is regular, i.e., in the interior of � + R. We next recall results from [ Cal02 ].”
Integrable systems, toric degenerations and Okounkov bodies (2012, methodology): “Several years ago, we had the idea to use the toric degeneration results for flag varieties given in Gonciulea– Lakshmibai [15], Kogan–Miller [30], and Caldero [9], in order to construct integrable systems for flag varieties of general reductive groups, thus generalizing the Gel’fand–Cetlin integrab”
On Schubert Varieties (2013, background): “Caldero [2] solved this problem in its entirety in 2002.”
ON THE GEOMETRY OF SPHERICAL VARIETIES (2012, background): “For spherical varieties, this was first developed by Kaveh [73] and Alexeev and Brion [3] inspired in many aspects by the work of Caldero [43] on toric degeneration of projective rational homogeneous spaces.”
Essential bases and toric degenerations arising from birational sequences (2015, methodology): “y generated monoid Γ ⊆ Λ+ ×NN and a Γ-ﬁltration of C[G//U], the form is useful to transform the Γ-ﬁltration into an N-ﬁltration. In this sense, the following lemma plays the same role as Lemma 3.2 in [C1], it is only adapted to the more general type of total orders introduced above. We say that a mo”
Toric degeneration of Schubert varieties and Gelfand¿Tsetlin polytopes (2003, background): “Schubert varieties as in [GL96, Chi00, Cal02 ] accessible to an audience unaccustomed to specialized language surrounding arbitrary finite type root systems and standard monomial theory.”
Crystal bases and Newton–Okounkov bodies (2011, methodology): “r any (k,a),(ℓ,b) ∈ Z≥0 ×Γ we have F(k,a)F(ℓ,b) ⊂ F(k+ℓ,a+b). We denote the graded of the ﬁltration F• by grR. As in [And13, Proposition 5.1], (see also [KaKh08, Section 5.6], [AlBr04, Section 2] and [Cal02]) one proves that the algebra R degenerates to grR: Theorem 7.3 (Degeneration of graded algeb”
Continuous crystal and Duistermaat-Heckman measure for Coxeter groups (2008, background): “[5] P. Caldero, Toric degenerations of Schubert varieties.”
Note on cohomology rings of spherical varieties and volume polynomial (2003, unknown): “In [Cal02], Caldero gives a flat degeneration of the flag variety G/B together with a line bundle Lλ (corresponding to a dominant weight λ) to the toric variety X0 together with a Q-divisor class L0 corresponding to the polytope ∆w 0 (λ).”
Canonical bases for cluster algebras (2014, background): “to the ordinary polarized toric variety for Ξ ⊂Rn. See §9. We expect this specializes to a uniform construction of many degenerations of representation theoretic objects to toric varieties, see e.g., [C02], [AB], and [KM05]. Applied to the Fock-Goncharov moduli spaces of G-local systems, this will g”
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS (2013, background): “osition into homogeneous parts for the Γ(M)-grading such that (4.2) gi = ¯gi + Xri j=1 gi,j, where deggi,j = qi,j > qi = deg¯gi. The set {qi,qi,j) | i = 1,...,m, j = 1,...,ri} ⊂ NN is ﬁnite, so by [C], Lemma 3.2, there exists a linear map e : RN → R such that e(NN) ⊆ N and e(qi) < e(qi,j). Let”