Degenerate flag varieties: Moment graphs and Schröder numbers

Giovanni Cerulli Irelli, Evgeny Feigin*, Markus Reineke

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We study geometric and combinatorial properties of the degenerate flag varieties of type A. These varieties are acted upon by the automorphism group of a certain representation of a type A quiver, containing a maximal torus T. Using the group action, we describe the moment graphs, encoding the zero- and one-dimensional T-orbits. We also study the smooth and singular loci of the degenerate flag varieties. We show that the Euler characteristic of the smooth locus is equal to the large Schröder number and the Poincaré polynomial is given by a natural statistics counting the number of diagonal steps in a Schröder path. As an application we obtain a new combinatorial description of the large and small Schröder numbers and their q-analogues.

Original languageEnglish
Pages (from-to)159-189
Number of pages31
JournalJournal of Algebraic Combinatorics
Volume38
Issue number1
DOIs
StatePublished - Aug 2013
Externally publishedYes

Funding

FundersFunder number
AG Laboratory HSE
Russian Foundation for Basic Research12-01-00070, 12-01-00944
Dynasty Foundation
Ministry of Education and Science of the Russian Federation12-05-0014, 11-01-0017, 2012-1.1-12-000-1011-016

    Keywords

    • Flag varieties
    • Moment graphs
    • Quiver Grassmannians
    • Schröder numbers

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