Multiple Flag Varieties of Finite Type

We gave a classification of P and Q with a finite number of K-orbits of a double flag variety G/P*K/Q for a symmetric pair (G, K) when G=GL_{m+n} and K=GL_{m}*GL_{n}, and a description of K-orbits when the number of K-orbits of G/P*K/Q is finite. We solved the problem by providing a correspondence between the K-orbits and the quiver representations.

Problem: Given a reductive algebraic group G, find all k-tuples of parabolic subgroups (P_1,...,P_k) such that the product of flag varieties G/P_1 x ... x G/P_k has finitely many orbits under the diagonal action of G. In this case we call G/P_1 x ... x G/P_k a multiple flag variety of finite type. (If P_1 is a Borel subgroup, the partial product G/P_2 x ... x G/P_k is a spherical variety.) In this paper we solve this problem in the case of the symplectic group G = Sp(2n). W...

Let $G$ be the split orthogonal group of degree $2n+1$ over an arbitrary field $\mathbb{F}$ of ${\rm char}\,\mathbb{F}\ne 2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a multiple flag variety is called of finite type if it has a finite number of $G$-orbits with respect to the diagonal action of $G$ when $|\mathbb{F}|=\infty$.

Let $G$ be the split orthogonal group of degree $2n$ over an arbitrary infinite field $\mathbb{F}$ of chararcteristic not $2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a multiple flag variety is said to be of finite type if it has a finite number of $G$-orbits with respect to the diagonal action of $G$.

Let G be a reductive algebraic group over the complex number filed, and K = G^{\theta} be the fixed points of an involutive automorphism \theta of G so that (G, K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal K-action. The double flag variety G/P \times K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P...

These notes are the written version of my lectures at the Banach Center mini-school "Schubert Varieties" in Warsaw, May 18-22, 2003. Their aim is to give a self-contained exposition of some geometric aspects of Schubert calculus.

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 them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decomposit...

We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets that we determine explicitly. Furthermore, our computations show that the number of diagonal $SL(n)$-orbits in any of these posets is at most 10 for any positive integer $n$. This is in contrast with the complexity 0 case, where, in some case...

Let $ G $ be a connected, simply connected semisimple algebraic group over the complex number field, and let $ K $ be the fixed point subgroup of an involutive automorphism of $ G $ so that $ (G, K) $ is a symmetric pair. We take parabolic subgroups $ P $ of $ G $ and $ Q $ of $ K $ respectively and consider the product of partial flag varieties $ G/P $ and $ K/Q $ with diagonal $ K $-action, which we call a \emph{double flag variety for symmetric pair}. It is said to be \e...

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 Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: th...