Orbit Closures Research Articles

Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study Sym(∞)-orbit closures of non-necessarily closed points in the Zariski spectrum of the infinite polynomial ring C[xij:i∈N,j∈[n]]. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by Sym(∞) orbits of polynomials in R[xij:i∈N,j∈[n]]. For n=1 we prove a quantifier elimination type result which fails for n>1.

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

Quotient quiver subtraction

We develop the diagrammatic technique of quiver subtraction to facilitate the identification and evaluation of the SU(n) hyper-Kähler quotient (HKQ) of the Coulomb branch of a 3dN=4 unitary quiver theory. The target quivers are drawn from a wide range of theories, typically classified as “good” or “ugly”, which satisfy identified selection criteria. Our subtraction procedure uses quotient quivers that are “bad”, differing thereby from quiver subtractions based on Kraft-Procesi transitions. The simple diagrammatic procedure identifies one or more resultant quivers, the union of whose Coulomb branches corresponds to the desired HKQ. Examples include quivers whose Coulomb branches are moduli spaces of free fields, closures of nilpotent orbits of classical and exceptional type, and slices in the affine Grassmanian. We calculate the Hilbert Series and Highest Weight Generating functions for HKQ examples of low rank. For certain families of quivers, we are able to conjecture HWGs for arbitrary rank. We examine the commutation relations between quotient quiver subtraction and other diagrammatic techniques, such as Kraft-Procesi transitions, quiver folding, and discrete quotients.

Specification and ω-chaos in non-compact systems

In this paper, we demonstrate conditions under which a Lindelöf dynamical system exhibits ω-chaos. In particular, if a system exhibits a generalized version of the specification property and has at least three points with mutually separated orbit closures, then the system exhibits dense ω-chaos.

Equivariant 𝒟-modules on 2×2×𝓃 hypermatrices

We study D \mathcal {D} -modules and related invariants on the space of 2 × 2 × n 2\times 2\times n hypermatrices for n ≥ 3 n\geq 3 , which has finitely many orbits under the action of G = G L 2 ( C ) × G L 2 ( C ) × G L n ( C ) G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C}) . We describe the category of coherent G G -equivariant D \mathcal {D} -modules as the category of representations of a quiver with relations. We classify the simple equivariant D \mathcal {D} -modules, determine their characteristic cycles and find special representations that appear in their G G -structures. We determine the explicit D \mathcal {D} -module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases n = 3 n=3 and n = 4 n=4 , they are completely uniform for n ≥ 5 n\geq 5 .

Partial order on involutive permutations and double Schubert cells

As shown by Melnikov, the orbits of a Borel subgroup acting by conjugation on upper-triangular matrices with square zero are indexed by involutions in the symmetric group. The inclusion relation among the orbit closures defines a partial order on involutions. We observe that the same order on involutive permutations also arises while describing the inclusion order on [Formula: see text]-orbit closures in the direct product of two Grassmannians. We establish a geometric relation between these two settings.

Orbits of Linear Series on the Projective Line

Abstract We compute the equivariant fundamental class of the orbit closure of a linear series on the projective line. We also describe the boundary of the orbit closure and how the orbits specialize in one parameter families.

A New Orbit Closure in Genus 8?

We provide numerical evidence that the GL 2 ( R ) -orbit closure of the unfolding of the (3, 4, 13)-triangle in the moduli space of Abelian differentials is a previously unknown 4-dimensional variety. Furthermore, we report on an exhaustive exploration that supports the idea that this would be the last example of exceptional triangle and quadrilateral unfoldings.

Reconstructing orbit closures from their boundaries

We introduce and study diamonds of G L + ( 2 , R ) \mathrm {GL}^+(2, \mathbb {R}) -invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We classify a surprisingly rich collection of diamonds where the two degenerations are contained in “trivial” invariant subvarieties. Our main results have been applied to classify large collections of invariant subvarieties; the statement of those results do not involve diamonds, but their proofs rely on them.

On abelian closures of infinite non-binary words

Two finite words u and v are called abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). In this paper, we investigate how this property extends to non-binary words. We consider the abelian closures of most natural generalizations of Sturmian words to non-binary alphabets, such as balanced words and minimal complexity words. We characterize the abelian closures of words in these families and show that in both families, there exist both words which satisfy the property A(x)=Ω(x) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion Ω(x)⊂A(x). We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

AbstractWe study the joint distribution of values of a pair consisting of a quadratic form ${\mathbf q}$ and a linear form ${\mathbf l}$ over the set of integral vectors, a problem initiated by Dani and Margulis [Orbit closures of generic unipotent flows on homogeneous spaces of $\mathrm{SL}_3(\mathbb{R})$ . Math. Ann.286 (1990), 101–128]. In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ , then under the assumptions that for every $(\alpha , \beta ) \in {\mathbb {R}}^2 \setminus \{ (0,0) \}$ , the form $\alpha {\mathbf q} + \beta {\mathbf l}^2$ is irrational and that the signature of the restriction of ${\mathbf q}$ to the kernel of ${\mathbf l}$ is $(p, n-1-p)$ , where ${3\le p\le n-2}$ , the number of vectors $v \in {\mathbb {Z}}^n$ for which $\|v\| < T$ , $a < {\mathbf q}(v) < b$ and $c< {\mathbf l}(v) < d$ is asymptotically $ C({\mathbf q}, {\mathbf l})(d-c)(b-a)T^{n-3}$ as $T \to \infty $ , where $C({\mathbf q}, {\mathbf l})$ only depends on ${\mathbf q}$ and ${\mathbf l}$ . The density of the set of joint values of $({\mathbf q}, {\mathbf l})$ under the same assumptions is shown by Gorodnik [Oppenheim conjecture for pairs consisting of a linear form and a quadratic form. Trans. Amer. Math. Soc.356(11) (2004), 4447–4463].

Invariant measures for -free systems revisited

Abstract For $\mathscr {B} \subseteq \mathbb {N} $ , the $ \mathscr {B} $ -free subshift $ X_{\eta } $ is the orbit closure of the characteristic function of the set of $ \mathscr {B} $ -free integers. We show that many results about invariant measures and entropy, previously only known for the hereditary closure of $ X_{\eta } $ , have their analogues for $ X_{\eta } $ as well. In particular, we settle in the affirmative a conjecture of Keller about a description of such measures [G. Keller. Generalized heredity in $\mathcal B$ -free systems. Stoch. Dyn.21(3) (2021), Paper No. 2140008]. A central assumption in our work is that $\eta ^{*} $ (the Toeplitz sequence that generates the unique minimal component of $ X_{\eta } $ ) is regular. From this, we obtain natural periodic approximations that we frequently use in our proofs to bound the elements in $ X_{\eta } $ from above and below.

Classification of Orbit Closures in the Variety of 4-Dimensional Symplectic Lie Algebras

Classification of Orbit Closures in the Variety of 4-Dimensional Symplectic Lie Algebras

Open homomorphisms of minimal flows

The consideration of homomorphisms (‘‘extensions’’) of minimal flows is central to topological dynamics. Building upon two earlier papers of the first author, with two different collaborators, we study open extensions of minimal flows. These are classified using maximally highly proximal generators, certain closed subsets of the universal minimal flow. The orbit closures of such in the space of closed subsets are the maximally highly proximal minimal flows. Given a maximally highly proximal generator C, the action of the universal minimal flow on a space gives rise to a “C extension” (the inverse images are determined by C). These are open and every open homomorphism is in fact a C extension for some MHP generator C. This generalizes the classical special case of RIC extensions. Moreover, the open extensions can be represented as quasi-factors (minimal sets in the space of closed subsets). We also consider a category whose objects are extensions of minimal flows and define the morphisms of these.

Minimal set of generators of ideals defining nilpotent orbit closures

Over a field of characteristic 0 0 , we construct a minimal set of generators of the defining ideals of closures of nilpotent conjugacy class in the set of n × n n \times n matrices. This modifies a conjecture of Weyman and provides a complete answer to it.

Exploring projective equivalences between closures of orbits

Motivated by results in the literature that use representations and group actions to produce nice geometric results about algebraic varieties, this article studies projective equivalence relations between closures of orbits for several complex algebraic group actions on , where is a complex representation of . In particular, we study the cases when is one of the following:, , , and . On the way, we also obtain some interesting geometric results from studying these orbits.

One-skeleton posets of Bruhat interval polytopes

Introduced by Kodama and Williams, Bruhat interval polytopes are generalized permutohedra closely connected to the study of torus orbit closures and total positivity in Schubert varieties. We show that the 1-skeleton posets of these polytopes are lattices and classify when the polytopes are simple, thereby resolving open problems and conjectures of Fraser, of Lee–Masuda, and of Lee–Masuda–Park. In particular, we classify when generic torus orbit closures in Schubert varieties are smooth.

Equivariant degenerations of plane curve orbits

In a series of papers, Aluffi and Faber computed the degree of the G L 3 GL_3 -orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how orbits degenerate under some natural specializations, yielding a fairly complete picture in the case of plane quartics.

Saturated Orbit Closures in the Hodge Bundle

Abstract We give a new proof of the classification of $\operatorname {GL}^{+}(2,{\mathbb {R}})$-orbit closures that are saturated for the absolute period foliation of the Hodge bundle. As a consequence, we obtain a short proof of the classification of closures of leaves of the absolute period foliation of the Hodge bundle. Our approach is based on a method for classifying $\operatorname {GL}^{+}(2,{\mathbb {R}})$-orbit closures using deformations of flat pairs of pants.

Around Exponential-Algebraic Closedness

AbstractWe present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: francesco.gallinaro@mathematik.uni-freiburg.deURL: https://etheses.whiterose.ac.uk/31077/