Which Schubert varieties are local complete intersections?

Proceedings of the London Mathematical Society, Volume 107, Issue 5 (2013), Pages 1004–1052

Henning and Alexander

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.

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• Proceedings of the London Mathematical Society, Volume 107, Issue 5 (2013), Pages 1004–1052
• arXiv

Presentations

• FPSAC 2012 Poster

Additional material

• FPSAC 2012 Extended abstract

Later work (last updated 2 October 2026)

• M. Albert and R. Brignall, Enumerating indices of Schubert varieties defined by inclusions, J. Combin. Theory Ser. A 123 (2014). Enumerates the permutations indexing Schubert varieties defined by inclusions, Av(4231, 35142, 42513, 351624), and suggests that its method could also apply to the classes in this paper; the class is on PermPAL.
• A. Woo and B. Wyser, Combinatorial results on (1,2,1,2)-avoiding GL(p,ℂ) × GL(q,ℂ)-orbit closures on GL(p+q,ℂ)/B, Int. Math. Res. Not. IMRN (2015). Gives criteria for when these orbit closures are lci, and, by analogy with this paper, conjectures that lci-ness of all such orbit closures is characterized by pattern avoidance.
• H. Abe and S. Billey, Consequences of the Lakshmibai–Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, Adv. Stud. Pure Math. 71 (2016). A survey of pattern characterizations in Schubert calculus, including the characterization of lci Schubert varieties given here.
• A. Fink, J. Rajchgot and S. Sullivant, Matrix Schubert varieties and Gaussian conditional independence models, J. Algebraic Combin. 44 (2016). Applies matrix Schubert varieties to algebraic statistics, using the descriptions of the Schubert varieties defined by inclusions from Gasharov–Reiner and from this paper.
• M. Ikeda, Enumeration of permutations indexing local complete intersection Schubert varieties, PhD thesis, University of Idaho (2016). Extends Albert and Brignall’s method to the permutations indexing lci Schubert varieties; its generating function was later found to be incorrect (see the Combinatorial Exploration entry below).
• S. Gao and A. Yong, Minimal equations for matrix Schubert varieties, J. Commut. Algebra 16 (2024). Finds explicit minimal generators for every Schubert determinantal ideal; this paper and Hsiao had done so only in the complete intersection case.
• M. Albert, C. Bean, A. Claesson, É. Nadeau, J. Pantone and H. Úlfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, Mem. Amer. Math. Soc. (2026). Among many other classes, finds the correct algebraic generating function for the permutations indexing lci Schubert varieties, correcting Ikeda’s thesis; the class is on PermPAL.
• A. Woo and A. Yong, Schubert geometry and combinatorics, Handbook of Combinatorial Algebraic Geometry (2026). A survey chapter on classifying the singularities of Schubert varieties combinatorially.
• M.-H. Dou, Contact rigidity and comparison kernels for type A lci Schubert varieties, preprint (2026). Results on the singular loci and Kazhdan–Lusztig polynomials of type A lci Schubert varieties, starting from the pattern characterization given here.
• All citing papers on Google Scholar (32)