Springer fibers and Schubert points

2017/01/31 by Martha Precup, Julianna Tymoczko · 5 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebra over a field #Algebraic structures and combinatorial models #Betti number #Combinatorics #Computer science #Flag (linear algebra) #Generalized flag variety #Grassmannian #Lie group #Mathematics #Partition (number theory) #Pure mathematics #Representation theory #Row #Schubert calculus #Schubert polynomial #Schubert variety #Statistics #Variety (cybernetics) #math.CO #msc:05E10 #msc:14M15

paper · pdf · doi:10.1016/j.ejc.2018.08.010

published in European Journal of Combinatorics 76, 10-26 (Elsevier BV) · 18 pages, 8 figures

openalex publication_date 2018/10/01 · arxiv created 2018/10/10 · arxiv updated 2018/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

Springer fibers are subvarieties of the flag variety parametrized by partitions; they are central objects of study in geometric representation theory. Schubert varieties are subvarieties of the flag variety that induce a well-known basis for the cohomology of the flag variety. This paper relates these two varieties combinatorially. We prove that the Betti numbers of the Springer fiber associated to a partition with at most three rows or two columns are equal to the Betti numbers of a specific union of Schubert varieties.

Citations

• Combinatorics of Coxeter Groups
• On certain graded Sn-modules and the q-Kostka polynomials
• SCHUBERT CELLS AND COHOMOLOGY OF THE SPACESG/P
• Trigonometric sums, green functions of finite groups and representations of Weyl groups
• A construction of representations of Weyl groups
• Gröbner geometry of Schubert polynomials