Degenerate flag varieties and the median Genocchi numbers

TL;DR

This paper explores degenerations of type A flag varieties, revealing their cell structures and connecting their combinatorics to median Genocchi numbers and Dellac configurations, providing new formulas and q-analogues.

Contribution

It explicitly describes degenerate flag varieties as subvarieties, constructs their cell decompositions, and links their combinatorics to median Genocchi numbers and Dellac configurations.

Findings

Number of cells equals normalized median Genocchi numbers for complete flags

Provides a new combinatorial definition of median Genocchi numbers

Computes Poincaré polynomials via Dellac configurations and introduces a q-analogue

Abstract

We study the $\bG_a^M$ degenerations $\Fl^a_\la$ of the type $A$ flag varieties $\Fl_\la$. We describe these degenerations explicitly as subvarieties in the products of Grassmanians. We construct cell decompositions of $\Fl^a_\la$ and show that for complete flags the number of cells is equal to the normalized median Genocchi numbers $h_n$. This leads to a new combinatorial definition of the numbers $h_n$. We also compute the Poincar\' e polynomials of the complete degenerate flag varieties via a natural statistics on the set of Dellac's configurations, similar to the length statistics on the set of permutations. We thus obtain a natural $q$-version of the normalized median Genocchi numbers.