Governing Singularities of Schubert Varieties

Abstract

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 [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [Woo-Yong'05]; the description of the singular locus (which was independently proved by [Billey-Warrington '03], [Cortez '03], [Kassel-Lascoux-Reutenauer'03], [Manivel'01]) 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.

23 pages. Software available at the authors' webpages. Version 2 is the submitted version. It has a nomenclature change: "Bruhat-restricted pattern avoidance" is renamed "interval pattern avoidance"; the introduction has been reorganized

References in corpus (6)

• Generic singularities of Schubert varieties
• On multiplicities of points on Schubert varieties in Grassmannians
• Hilbert functions of points on Schubert varieties in the Grassmannian
• Multiplicities of Points on Schubert Varieties in Grassmannians
• Multiplicities of the Most Singular Point on Schubert varieties on Gl(n)/B for n=5,6
• Gorenstein locus of minuscule Schubert varieties

Cited by in corpus (22)

• Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field
• Describing West-3-stack-sortable permutations with permutation patterns
• Towards combinatorial invariance for Kazhdan-Lusztig polynomials
• Puzzles, positroid varieties, and equivariant K-theory of Grassmannians
• Matrix Schubert varieties and Gaussian conditional independence models
• Type A quiver loci and Schubert varieties
• Governing singularities of symmetric orbit closures
• Singularities of Richardson varieties
• Three combinatorial formulas for type A quiver polynomials and K-polynomials
• Singularities of Schubert Varieties within a Right Cell
• Equivariant Hirzebruch class for singular varieties
• A Gröbner Basis for Schubert Patch Ideals
• Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties
• Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties
• Polynomials for GL_p x GL_q orbit closures in the flag variety
• Kazhdan-Lusztig polynomials and drift configurations
• Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory
• On the minimal power of $q$ in a Kazhdan-Lusztig polynomial
• The Bruhat decomposition of real Grassmann manifolds
• Patch ideals and Peterson varieties
• Combinatorial results on (1,2,1,2)-avoiding $GL(p,\mathbb{C})\times GL(q,\mathbb{C})$-orbit closures on $GL(p+q,\mathbb{C})/B$
• On the Gorensteinization of Schubert Varieties via Boundary Divisors