Mathematical Research Letters

Volume 18 (2011)

Number 6

Degenerate flag varieties and the median Genocchi numbers

Pages: 1163 – 1178

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n6.a8

Author

Evgeny Feigin (Department of Mathematics, National Research University Higher School of Economics, Vavilova str., 117312, Moscow, Russia)

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 Grassmannians. 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 Poincaré 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.

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