The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties.
We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group S_n, n=dim(V), but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.
A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties…
The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a finite-dimensional vector space V; or, equivalently, the closure of an orbit of the group GL(V) acting on the direct product of two full flag varieties. We obtain a similar result for triples consisting of two subspaces and a partial flag in V; this is equivalent to describing the closure of a GL(V)-orbit in the product of two Grassmannians and one flag variety. We give a rank criterion to check whether such a triple can be degenerated to another one, and we classify the minimal degenerations. Our…
For a complex semi-simple group G and its real form G0 we define a Poisson structure on the flag variety of G such that all the Bruhat cells (for a suitable choice of a Borel subgroup of G) as well as all the G0-orbits are Poisson submanifolds. We show that each intersection of a G0-orbit with a Bruhat cell is a regular Poisson submanifold and we compute the dimensions of its symplectic leaves.
Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat o…
This paper provides a unified combinatorial framework to study orbits in certain affine flag varieties via the associated Bruhat-Tits buildings. We first formulate, for arbitrary affine buildings, the notion of a chimney retraction. This simultaneously generalizes the two well-known notions of retractions in affine buildings: retractions from chambers at infinity and retractions from alcoves. We then present a recursive formula for computing the images of certain minimal galleries in the building under chimney retractions, using purely combinatorial tools associated to the underlying affine We…
Berenstein, Fomin and Zelevinsky defined functions on double Bruhat cells which they called generalized minors. By relating certain double Bruhat cells to configuration spaces of flags, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities. The weights of the tensor invariants can then be used to construct the quiver for the cluster structure on the configuration space of three flags. We also show a converse statement--that the weights of tensor invariants can by computed from the structure of the quiver. The weight identities ar…
The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of M_{m,n}(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GL_{m+n}(C). Three different presentations of the T-orbits of symplectic leaves in M_{m,n}(C) are obtained - (a) as pullbacks of Bruhat cells in GL_{m+n}(C) under a particular map; (b) in terms of rank conditions on rectangula…
We give a necessary and sufficient condition in terms of pattern avoidance for the conjugates of the bottom vertex in the Bruhat graph attached to an O_n orbit O in the flag variety for GL_n to have degree equal to the rank of this graph as a poset, showing that this condition is equivalent to the rational smoothness of the closure of O. We also give a necessary and sufficient condition in terms of pattern avoidance for the closure of O to be smooth.
We calculate Bernstein-Sato b-functions for $f_{G^3}^λ$, a $SL_n$-invariant section of a line bundle on $SL_n/B \times SL_n/B \times \mathbb{P}^{n - 1}$ whose zero-set is the complement of the open $G$-diagonal orbit. The proof uses a similar calculation by Kashiwara of the b-function for $f^λ$, a $B^-$-semiinvariant section of a line bundle on $SL_n/B$ whose zero-set is the complement of the big Bruhat cell.
Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A^n. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend on the ring in question. For n=3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.
Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that th…
Kontsevich and Soibelman defined the Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety can produce an example of such a category, whose corresponding Donaldson-Thomas invariants are encoded by a special formal automorphism of the cluster variety, known as the Donaldson-Thomas transformation. In this paper we prove a conjecture of Goncharov and Shen in the case of $\mathrm{GL}_n$, which describes the Donaldson-Thomas transformation of the double quotient of the double Bruhat cells $H \backslash \mathrm{GL}_n^{u,v}/H$ where $H$ is a maximal t…
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: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degener…
We obtain an analog of the compression of angles theorem in symmetric spaces for Bruhat--Tits buildings of the type $A$. More precisely, consider a $p$-adic linear space $V$ and the set $Lat(V)$ of all lattices in $V$. The complex distance in $Lat(V)$ is a complete system of invariants of a pair of points of $Lat(V)$ under the action of the complete linear group. An element of a Nazarov semigroup is a lattice in the duplicated linear space $V\oplus V$. We investigate behavior of the complex distance under the action of the Nazarov semigroup on the set $Lat(V)$.
For the space $Lat_n$ of all the lattices in a $p$-adic $n$-dimensional linear space we obtain an analog of matrix beta-functions; this beta-function has a degeneration to the Tamagawa zeta-function. We propose an analog of Berezin kernels for $Lat_n$. We obtain conditions of positive definiteness of these kernels and explicit Plancherel formula.
We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial objects counted by Motzkin numbers.
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 decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.
We first describe the tangent space to the affine flag manifold associated to a simple algebraic group over $\mathbb{C}$ at the distinguished point starting from standard definitions. We then construct projective lines in the affine flag manifold tangent to given root vectors associated to imaginary roots of the corresponding affine Kac-Moody algebra and describe in which Schubert varieties they lie.
Let $G$ be the group scheme $\operatorname{SL}_{d+1}$ over $\mathbb{Z}$ and let $Q$ be the parabolic subgroup scheme corresponding to the simple roots $α_{2},\cdots,α_{d-1}$. Then $G/Q$ is the $\mathbb{Z} $-scheme of partial flags $\{D_{1}\subset H_{d}\subset V\}$. We will calculate the cohomology modules of line bundles over this flag scheme. We will prove that the only non-trivial ones are isomorphic to the kernel or the cokernel of certain matrices with multinomial coefficients.
