Orbit Closures Research Articles

The Cramer varieties [formula omitted

In this paper we study quasi-homogeneous affine algebraic varieties, that is, varieties obtained as closures of orbits of suitable group representations. We also discuss one interesting case that has links with the Orthogonal Grassmannian OGr(5,10). The main aim is to write the tangent bundle and the canonical class of quasi-homogeneous affine algebraic varieties in terms of group representations.

Extremal words in morphic subshifts

Given an infinite word x over an alphabet A, a letter b occurring in x, and a total order σ on A, we call the smallest word with respect to σ starting with b in the shift orbit closure of x an extremal word of x. In this paper we consider the extremal words of morphic words. If x=g(fω(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantee that all extremal words are morphic. This happens, in particular, when x is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the period-doubling word and the Chacon word and a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin–Shapiro word.

Homological approach to the Hernandez-Leclerc construction and quiver varieties

In a previous paper the authors have attached to each Dynkin quiver an associative algebra. The definition is categorical and the algebra is used to construct desingularizations of arbitrary quiver Grassmannians. In the present paper we prove that this algebra is isomorphic to an algebra constructed by Hernandez-Leclerc defined combinatorially and used to describe certain graded Nakajima quiver varieties. This approach is used to get an explicit realization of the orbit closures of representations of Dynkin quivers as affine quotients.

Combinatorial Algorithms for Computing Degenerations of Modules of Finite Dimension

We present combinatorial algorithms for solving three problems that appear in the study of the degeneration order ≤deg for the variety of finite-dimensional modules over a k-algebra Λ, where M ≤deg N means that a module N belongs to an orbit closure $\overline{\cal{O}(M)}$ of a module M in the variety of Λ-modules. In particular, we introduce algorithmic techniques for deciding whether or not the relation M ≤deg N holds and for determining all predecessors (resp. succesors) of a given module M with respect to ≤deg. The order ≤deg plays an important role in modern algebraic geometry and module theory. Applications of our technique and experimental tests for particular classes of algebras are presented. The results show that a computer algebra technique and algorithmic computer calculations provide important tools in solving theoretical mathematics problems of high computational complexity. The algorithms are implemented and published as a part of an open source GAP package called QPA.

Modules and quiver representations whose orbit closures are hypersurfaces

Modules and quiver representations whose orbit closures are hypersurfaces

Degenerate group of type A: Representations and flag varieties

We consider the degeneration of a simple Lie group which is a semidirect product of its Borel subgroup and a normal Abelian unipotent subgroup. We introduce a class of highest weight representations of the degenerate group of type A, generalizing the construction of PBW-graded representations of the classical group (PBW is an abbreviation for “Poincare-Birkhoff-Witt”). Following the classical construction of flag varieties, we consider the closures of orbits of the Abelian unipotent subgroup in projectivizations of the representations. We show that the degenerate flag varieties Fna and their desingularizations Rn can be obtained via this construction. We prove that the coordinate ring of Rn is isomorphic as a vector space to the direct sum of the duals of the highest weight representations of the degenerate group. At the end we state several conjectures on the structure of the highest weight representations of the degenerate group of type A.

Orbit Closure Hierarchies of Skew-symmetric Matrix Pencils

We study how small perturbations of a skew-symmetric matrix pencil may change its canonical form under congruence. This problem is also known as the stratification problem of skew-symmetric matrix ...

On Kapranov's description of $\ov{M}_{0,n}$ as a Chow quotient

We provide a direct proof, valid in arbitrary characteristic, of the result originally proven by Kapranov over C that the Hilbert quotient (P^1)^n//_HPGL_2 and Chow quotient (P^1)^n//_{Ch}PGL_2 are isomorphic to \overline{M}_{0,n}. In both cases this is done by explicitly constructing the universal family of orbit closures and then showing that the induced morphism is an isomorphism onto its image. The proofs of these results in many ways reduce to the case n = 4; in an appendix we outline a formalism of this phenomenon relating to certain operads.

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent $\alpha>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.

Shadowing Property, Weak Mixing and Regular Recurrence

We show that a non-wandering dynamical system with the shadowing property is either equicontinuous or has positive entropy and that in this context uniformly positive entropy is equivalent to weak mixing. We also show that weak mixing together with the shadowing property imply the specification property with a special kind of regularity in tracing (a weaker version of periodic specification property). This in turn implies that the set of ergodic measures supported on the closures of orbits of regularly recurrent points is dense in the space of all invariant measures (in particular, invariant measures in such a system form the Poulsen simplex, up to an affine homeomorphism).

Stationary measures and invariant subsets of homogeneous spaces (III)

Let G be a real Lie group, ? be a lattice in G and G be a compactly generated closed subgroup of G . If the Zariski closure of the group Ad(G) is semisimple with no compact factor, we prove that every G -orbit closure in G/? is a finite volume homogeneous space. We also establish related equidistribution properties.

CLASSIFICATION OF ORBIT CLOSURES IN THE VARIETY OF THREE-DIMENSIONAL NOVIKOV ALGEBRAS

We classify the orbit closures in the variety [Formula: see text] of complex, three-dimensional Novikov algebras and obtain the Hasse diagrams for the closure ordering of the orbits. We provide invariants which are easy to compute and which enable us to decide whether or not one Novikov algebra degenerates to another Novikov algebra.

Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations

In this paper, we obtain a characterization of the recurrence of a continuous vector field w of a closed connected surface M as follows. The following are equivalent: (1) w is pointwise recurrent. (2) w is pointwise almost periodic. (3) w is minimal or pointwise periodic. Moreover, if w is regular, then the following are equivalent: (1) w is pointwise recurrent. (2) w is minimal or the orbit space M/w is either [0,1], or S1. (3) R is closed (where R:={(x,y)∈M×M|y∈O(x)¯} is the orbit closure relation). On the other hand, we show that the following are equivalent for a codimension one foliation F on a compact connected manifold: (1) F is pointwise almost periodic. (2) F is minimal or compact. (3) F is R-closed. Also we show that if a foliated space on a compact metrizable space is either minimal or both compact and without infinite holonomy, then it is R-closed.

Lyndon words and Fibonacci numbers

It is a fundamental property of non-letter Lyndon words that they can be expressed as a concatenation of two shorter Lyndon words. This leads to a naive lower bound ⌈log2(n)⌉+1 for the number of distinct Lyndon factors that a Lyndon word of length n must have, but this bound is not optimal. In this paper we show that a much more accurate lower bound is ⌈logϕ(n)⌉+1, where ϕ denotes the golden ratio (1+5)/2. We show that this bound is optimal in that it is attained by the Fibonacci Lyndon words. We then introduce a mapping Lx that counts the number of Lyndon factors of length at most n in an infinite word x. We show that a recurrent infinite word x is aperiodic if and only if Lx⩾Lf, where f is the Fibonacci infinite word, with equality if and only if x is in the shift orbit closure of f.

Nakajima Varieties and Repetitive Algebras

We realize certain graded Nakajima varieties of finite Dynkin type as orbit closures of repetitive algebras of Dynkin quivers. As an application, we show that the perverse sheaves introduced by Nakajima to describe irreducible characters of quantum loop algebras are isomorphic to the intersection cohomology sheaves of these orbit closures.

Resolutions of defining ideals of orbit closures for quivers of type $A_3$

We construct explicitly a minimal free resolution of the defining ideal of an orbit closure arising from a representation of the non-equioriented $A_3$ quiver. The resolution is a generalization of Lascoux's resolution for determinantal ideals. The case of non-equioriented $A_3$ quiver is made special by the fact that, in this case, every orbit closure admits a so-called $1$-step desingularization. Using the resolution we give a description of the minimal set of generators of the defining ideal. The resolution also allows us to read off some geometric properties of the orbit closure, like normality and Cohen-Macaulay. In addition, we give a characterization for the orbit closure to be Gorenstein.

Geometry of orbits of permanents and determinants

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the padded permanent (i.e., the permanent multiplied by a power of a linear form).

Log minimal model program for the moduli space of stable curves: the first flip

We give a geometric invariant theory (GIT) construction of the log canonical model M ¯ g (a) of the pairs (M ¯ g ,ad) for a?(7/10�?,7/10] for small ??Q + . We show that M ¯ g (7/10) is isomorphic to the GIT quotient of the Chow variety of bicanonical curves; M ¯ g (7/10-?) is isomorphic to the GIT quotient of the asymptotically-linearized Hilbert scheme of bicanonical curves. In each case, we completely classify the (semi)stable curves and their orbit closures. Chow semistable curves have ordinary cusps and tacnodes as singularities but do not admit elliptic tails. Hilbert semistable curves satisfy further conditions; e.g., they do not contain elliptic chains. We show that there is a small contraction ?:M ¯ g (7/10+?)?M ¯ g (7/10) that contracts the locus of elliptic bridges. Moreover, by using the GIT interpretation of the log canonical models, we construct a small contraction ? + :M ¯ g (7/10-?)?M ¯ g (7/10) that is the Mori flip of ? .

Norm closures of orbits of bounded operators

To every bounded linear operator $A$ between Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$ three cardinals $\iota_r(A)$, $\iota_i(A)$ and $\iota_f(A)$ and a binary number $\iota_b(A)$ are assigned in terms of which the descriptions of the norm closures of the orbits $\{G A L^{-1}:\ L \in \mathcal{G}_1,\ G \in \mathcal{G}_2\}$ are given for $\mathcal{G}_1$ and $\mathcal{G}_2$ (chosen independently) being the trivial group, the unitary group or the group of all invertible operators on $\mathcal{H}$ and $\mathcal{K}$, respectively.

Schubert calculus of Richardson varieties stable under spherical Levi subgroups

We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for similar rules in types $B$ and $D$, associated to Richardson varieties stable under spherical Levi subgroups of $SO(2n+1,\C)$ and $SO(2n,\C)$, respectively.