相关论文: Degenerate flag varieties and the median Genocchi …
The goal of this paper is twofold. First, we review the recently developed geometric approach to the combinatorics of the median Genocchi numbers. The Genocchi numbers appear in this context as Euler characteristics of the degenerate flag…
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…
In two recent papers (\textit{Mathematical Research Letters,18(6):1163--1178,2011} and \textit{European J. Combin.,33(8):1913--1918,2012}), Feigin proved that the Poincar\'e polynomials of the degenerate flag varieties have a combinatorial…
Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of…
Let $\Fl_\lambda$ be a generalized flag variety of a simple Lie group $G$ embedded into the projectivization of an irreducible $G$-module $V_\lambda$. We define a flat degeneration $\Fl_\lambda^a$, which is a ${\mathbb G}^M_a$ variety.…
Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the…
We study finite dimensional approximations to degenerate versions of affine flag varieties using quiver Grassmannians for cyclic quivers. We prove that they admit cellular decompositions parametrized by affine Dellac configurations, and…
Let $\Fl^a_\la$ be the PBW degeneration of the flag varieties of type $A_{n-1}$. These varieties are singular and are acted upon with the degenerate Lie group $SL_n^a$. We prove that $\Fl^a_\la$ have rational singularities, are normal and…
We compute toric degenerations arising from the tropicalization of the full flag varieties $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ embedded in a product of Grassmannians. For $\mathcal{F}\ell_4$ and $\mathcal{F}\ell_5$ we compare toric…
We study the generating function of descent numbers for the permutations with descent pairs of prescribed parities, the distribution of which turns out to be a refinement of median Genocchi numbers. We prove the $\gamma$-positivity for the…
We study toric degenerations arising from Gr\"obner degenerations or the tropicalization of partial flag varieties. We produce a new family of toric degenerations of partial flag varieties whose combinatorics are governed by matching fields…
Let $\SF^a_\lambda$ be the degenerate symplectic flag variety. These are projective singular irreducible $\bG_a^M$ degenerations of the classical flag varieties for symplectic group $Sp_{2n}$. We give an explicit construction for the…
The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded…
Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in…
We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver…
In this article we investigate the regularity properties of linear degenerations of flag varieties. We classify the linear degenerations of (partial) flag varieties that are smooth. Furthermore, we study the singular locus of irreducible…
Recently, Lazar and Wachs (arXiv:1910.07651) showed that the (median) Genocchi numbers play a fundamental role in the study of the homogenized Linial arrangement and obtained two new permutation models (called D-permutations and…
We define symmetric Dellac configurations as the Dellac configurations that are symmetric with respect to their centers. The symmetric Dellac configurations whose lengths are even were previously introduced by Fang and Fourier under the…
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…
In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their…