Maximal singular loci of Schubert varieties in $SL(n)/B$

By Sara C. Billey, Gregory S. Warrington. 2001

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Schubert varieties in the flag manifold SL(n)/B play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety X w is nonsingular if and only if w avoids the patterns 4231 and 3412. They also gave a conjectural description of the singular locus of X w . In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety X w for any element w ∈ G n . In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from w by a cycle depending naturally on a 4231 or 3412 pattern in w. Our description of the irreducible components is computationally more efficient (O(n 6 )) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.

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

View Paper: https://www.semanticscholar.org/paper/fd033d2a5e20e2777f4fd52c9295173f04d5e4de

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Singularities of Generalized Richardson Varieties (2010, methodology): “ulations prove the following corollary to Lemma 4.24. Corollary 4.25. For 1 ≤ k ≤ n, we have g[k,n] = S[n+1,n− k +1]·q−( n−k+1 2 ). 24 Below are the 25 rank sets for G(2,4) listed by dimension. Thus g[2,4] = 6 + 8q + 7q2 + 3q3 + q4). Here (34,123) means the rank set consisting of two subspaces spann”
Small resolutions of minuscule Schubert varieties (2006, background): “way is to describe the singular locus, its irreducible components and the generic singularity in each of these components. This has been completely achieved for GLn only recently (see [Man1], [Man2], [BW], [KLR] and [Cor2]). For the general case, there are only partial results (for an account, see [”
Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field (2016, background): “d Hernandez and Bernard Leclerc for very useful correspondence and to Pascal Boyer for communicating to us the thesis of Taiwang Deng. We are indebted to Greg Warrington for clarifying some points in [BW03] and most importantly for his C code for computing the Kazhdan– Lusztig polynomials for the sy”
Lower bounds for Kazhdan-Lusztig polynomials from patterns (2002, background): “More examples of pattern avoidance appear in [LasSc, St, BiP, BiW2 , Ma, KLR, Co, Co2].”
GOVERNING SINGULARITIES OF SCHUBERT VARIETIES (2006, background): “Now let Φ be the embedding of v into w = 589716234 where the underlined positions indicate the embedding, which in symbols is given by φ1 = 1, φ2 = 4, φ3 = 5, φ4 = 6, φ5 = 8.”