Degenerate flag varieties: moment graphs and Schr\"oder numbers

TL;DR

This paper explores the geometric and combinatorial aspects of degenerate flag varieties of type A, revealing connections to Schr"oder numbers through moment graphs, Euler characteristics, and Poincaré polynomials.

Contribution

It provides a new combinatorial description of Schr"oder numbers via the geometry of degenerate flag varieties and their moment graphs.

Findings

Euler characteristic of the smooth locus equals the large Schr"oder number

Poincaré polynomial counts diagonal steps in Schr"oder paths

New combinatorial interpretations of Schr"oder numbers and q-analogues

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 Schr\"oder number and the Poincar\'e polynomial is given by a natural statistics counting the number of diagonal steps in a Schr\"oder path. As an application we obtain a new combinatorial description of the large and small Schr\"oder numbers and their q-analogues.