GOVERNING SINGULARITIES OF SCHUBERT VARIETIES

By Alexander Woo, Alexander Yong. 2006

74 citations recorded by the source corpus.

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable” invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P=“singular”; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45–52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan–Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205–220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 335 (2003) 3915–3945, MR 2004f:14071; A. Cortez, Singularites generiques et quasi-resolutions des varietes de Schubert pour le groupe lineaire, Adv. Math. 178 (2003) 396–445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108, MR 2005f:14096; L. Manivel, Le lieu singulier des varietes de Schubert, Int. Math. Res. Not. 16 (2001) 849–871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan–Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b880f20a77542c0d6e716bb00320467ccdf30b7a

Citations in the corpus

• Geometric conditions for □-irreducibility of certain representations of the general linear group over a non-archimedean local field (2016, unknown): “ΩI(σ) = τ (t) r,s and ΩI(σ1) = δ (t) r,s. (4) The Bruhat intervals [σ1,σ] and [δ (t) r,s,τ (t) r,s] are isomorphic (or equivalently, ℓ(σ) − ℓ(σ1) = ℓ(τ (t) r,s)− ℓ(δ (t) r,s)). (In the terminology of [WY08] this means that [δ(t) r,s,τ (t) r,s] interval pattern embeds into [σ1,σ].) 4.3. Given σ∈ Sk a”