Related papers: ${\mathbb G}_a^M$ degeneration of flag varieties

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank.…

We describe the effect of Feigin's flat degeneration of the type $\textrm{A}$ flag variety on its Schubert varieties. In particular, we study when they stay irreducible and in several cases we are able to encode reducibility of the…

Let $g$ be a semisimple Lie algebra over $\mathbb C$ and $k$ be a reductive in $g$ subalgebra. We say that a simple $g$-module $M$ is a $(g; k)$-module if as a $k$-module $M$ is a direct sum of finite-dimensional $k$-modules. We say that a…

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 conceptualize in the paper the linear degenerate symplectic flag varieties as symmetric degenerations within the framework of type $A$ equioriented quivers. First, in the larger context of symmetric degenerations, we give a…

Fix a flat and projective morphism $X\rightarrow\Sigma$ of schemes. We show, first, that any set of $\mathbb{P}^1$-fibrations on $X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix $C$. Second, $X$ is an \'etale…

Here we show that, given a finite homological system $({\cal P},\leq,\{\Delta_u\}_{u\in {\cal P}})$ for a finite-dimensional algebra $\Lambda$ over an algebraically closed field, the category ${\cal F}(\Delta)$ of $\Delta$-filtered modules…

We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous…

In this paper, given a semisimple algebraic group $\bf G$ of rank 2, we construct a special semiorthogonal decomposition in the derived category of coherent sheaves on the flag variety ${\bf G}/{\bf B}$. These decompositions are defined…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

Our main goal is to show that the Gelfand--Tsetlin toric degeneration of the type A flag variety can be obtained within a degenerate representation-theoretic framework similar to the theory of PBW degenerations. In fact, we provide such…

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of…

The motivation of this work is to construct an analog of compactified moduli of abelian varieties and toric pairs in the case of non-commutative algebraic group G. We introduce a class of "stable reductive varieties" which contain connected…

We define the class of admissible linear embeddings of flag varieties. The definition is given in the general language of algebraic geometry. We then prove that an admissible linear embedding of flag varieties has a certain explicit form in…

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…

We show that the moduli space of stable n-pointed rational curves can be flatly degenerated to a projective toric variety. We arrive at this by showing that the Chow quotients of the Grassmannians admit toric degenerations, which in turn,…

Let $k$ be an arbitrary field, $\Lambda$ be a $k$-algebra and $V$ be a $\Lambda$-module. When it exists, the universal deformation ring $R(\Lambda,V)$ of $V$ is a $k$-algebra whose local homomorphisms to $R$ parametrize the lifts of $V$ up…

Let $G$ be a reductive algebraic group over a $p$-adic field or number field $K$, and let $V$ be a $K$-linear faithful representation of $G$. A lattice $\Lambda$ in the vector space $V$ defines a model $\hat{G}_{\Lambda}$ of $G$ over…

Degeneration of modules is usually defined geometrically, but due to results of Zwara and Riedtmann we can also define it in terms of exact sequences. This definition also works over fields that are not algebraically closed. Let $k$ be a…

A $p$-divisible group over a field $K$ admits a slope decomposition; associated to each slope $\lambda$ is an integer $m$ and a representation $\gal(K) \ra \gl_m(D_\lambda)$, where $D_\lambda$ is the $\rat_p$-division algebra with Brauer…