Singular Loci of Schubert Varieties

By Sara C. Billey, V. Lakshmibai. 2000

362 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/f027e1798378753c9629a7c09520b44b3f296117

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Citations in the corpus

Excited Young diagrams and equivariant Schubert calculus (2007, neutral): “Such a multiplicity has been studied in detail by many authors (see [4]).”
Special matchings and Kazhdan-Lusztig polynomials (2006, background): “, [22], [23], [21], [19], [1], [18], [2], [25], and the references cited there).”
Cluster structures for 2-Calabi–Yau categories and unipotent groups (2007, methodology): “th the examples from I.3. We omit the calculation involved in proving the isomorphisms between the subcluster algebras arising from Band the coordinate rings of the varieties under consideration. See [BL] for general background on Schubert varieties and (isotropic) Grassmannians. For a subset Jof si”
Chains in the Bruhat order (2005, neutral): “Chevalley’s formula (3.5) implies the following well-known statement; see, e.g., [ BL ].”
Maximal singular loci of Schubert varieties in $SL(n)/B$ (2001, background): “y [Che94] showed that all Schubert varieties in G/B are nonsingular in codimension one. Since that time, many beautiful results on determining singular points of Schubert varieties have surfaced (see [BL00]). By deﬁnition, the Schubert variety Xw is the closure of the Date: January 26, 2018. 1991 Ma”
Linear conditions imposed on flag varieties (2004, background): “The closure Cw is a Schubert variety, whose geometry and associated combinatorics has been extensively studied [BL].”
Singularities of Generalized Richardson Varieties (2010, background): “W/WP index the Schubert varieties which form the irreducible components of the singular locus of Xu, see [3, 4, 7, 8, 23, 31, 34, 37].”
Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties (2004, background): “Again, a reference for these facts is [BL00].”
Lectures on the Geometry of Flag Varieties (2004, background): “it determination of the singular locus of a Schubert variety, and of the corresponding generic singularities (i.e., the singularities along each irreducible component of the singular locus). The book [1] by Billey and Lakshmibai is a survey of this question, which was recently solved (independently ”
When is a Schubert variety Gorenstein (2004, methodology): “As mentioned in the introduction, all Schubert varieties are known to be normal [14, 30] and Cohen-Macaulay [31], so we can then use the calculation of the canonical sheaf of Schubert varieties by Ramanathan [31, 32] to determine which Schubert varieties are Gorenstein.”
K-theoretic analogues of factorial Schur P- and Q-functions (2011, background): “See for example [1], [10] (note that our convention here is different from there).”
Geometric Schur Duality of Classical Type (2014, methodology): “It is indexed by the permutation τ = (d− 1, · · · 1, d) in the Weyl group of type Bd in the notation in [BL00] and [B98].”
Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field (2016, background): “0t∈ [σ0,σ]} and J(σ0,σ) = {t∈ T : σ0t≤ σ} = I(σ0,σ)∪ {ti,j : i<jand σ0(i) >σ0(j)}. The following result due to Lakshmibai–Seshadri is well known and admits many generalizations. (Cf. [Spr98] or [BL00] for more details.) Proposition 5.2 ([LS84]). The dimension of the tangent space of Xσ at any ”
A combinatorial rule for (co)minuscule Schubert calculus (2006, background): “We focus on the latter, explaining the adjustments for the former as necessary.) These G/P’s and their Schubert varieties are of significant and fundamental interest in geometry and representation theory, see, e.g., [ BiLa00 , Chapter 9], [Kos61] and the references therein, as well as, e.g., [Pe06, ”