Symmetric Dellac configurations and symplectic/orthogonal flag varieties

TL;DR

This paper explores the topology of degenerate flag varieties through combinatorics of symmetric Dellac configurations, introducing new algebraic varieties and computing their Poincaré polynomials.

Contribution

It defines three new classes of algebraic varieties related to degenerate flag varieties and establishes their cellular decompositions and Poincaré polynomial formulas.

Findings

Constructed cellular decompositions for the new varieties.

Connected the topology of these varieties to symmetric Dellac configurations.

Computed Poincaré polynomials using combinatorial statistics.

Abstract

The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag varieties of types A and C. The construction is based on the quiver Grassmannian realization of the degenerate flag varieties and odd symplectic and odd and even orthogonal groups. We study basic properties of our varieties; in particular, we construct cellular decomposition in all the three cases above (as well as in the case of even symplectic group). We also compute the Poincar\'e polynomials in terms of certain statistics on the set of symmetric Dellac configurations.