Which Schubert varieties are local complete intersections?

Abstract

Abstract. We characterize by pattern avoidance the Schubert varieties for GL n which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. Applications include formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties.

Singularities of Richardson varieties

“…This P factorizes with respect to f P ="logical and", and thus (3.1) again applies. Recently, a characterization of which X w ⊆ GL n /B are local complete intersections has been determined by H.Úlfarsson and the second author [UlfWoo11]. To further determine when X v w is a local complete intersection, one also seems to need a characterization of the lci locus of a Schubert variety.…”

Pattern avoidance and fiber bundle structures on Schubert varieties

“…Most notably, Lakshmibai and Sandhya prove that a Schubert variety X w is smooth if and only if w avoids the patterns 3412 and 4231 [LS90]. Pattern avoidance has also been used to determine when Schubert varieties are defined by inclusions, Gorenstein, factorial, have small resolutions and are local complete intersections [BW03, Deo90, GR02, BMB07, WY06,UW13]. For a survey of these results and many others see [AB].…”

Governing singularities of symmetric orbit closures

“…These are the K-orbit versions of Kazhdan-Lusztig ideals, which define the aforementioned Kazhdan-Lusztig varieties. The latter ideals have been of use in both computational and theoretical analysis of Schubert varieties (see, for example, [19,13] and the references therein). In the same vein, we mention a practical advantage of the Mars-Springer ideal over the patch ideal.…”