Degenerate flag varieties and the median Genocchi numbers

By Evgeny Feigin. 2011

75 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/95329ff84a4998becf8f3f03df29234ecb4689dd

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Degenerate flag varieties of type A and C are Schubert varieties (2014, background): “Another corollary of the theorem is that the median Genocchi number hn = χ(F l a n+1) (see [6]) has another interpretation: it is the number of elements τ ∈ SymJ2n which are smaller than σ in the (induced) Bruhat order.”
Quiver Grassmannians and degenerate flag varieties (2011, methodology): “The cells from [Feigin 2011] are labeled by collections L1, .”