Orbit Closures Research Articles

Type A quiver loci and Schubert varieties

We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.

Existence of Affine Pavings for Varieties of Partial Flags Associated to Nilpotent Elements

The flag variety of a complex reductive linear algebraic group |$G$| is by definition the quotient |$G/B$| by a Borel subgroup. It can be regarded as the set of Borel subalgebras of |${\rm Lie}(G)$|⁠. Given a nilpotent element |$e$| in |${\rm Lie}(G)$|⁠, one calls Springer fiber the subvariety formed by the Borel subalgebras which |$e$| belongs to. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by De Concini, Lusztig, and Procesi asserts that when |$G$| is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In this paper, we study varieties generalizing the Springer fibers in two ways: being contained in a partial flag variety |$G/P$| (the quotient by a parabolic subgroup, instead of a Borel subgroup) and defined by a more general belonging condition (in terms of an ideal of |${\rm Lie}(P))$|⁠. These varieties arise, for instance, as fibers of resolutions of nilpotent orbit closures. The main result of the paper is a generalization of De Concini, Lusztig, and Procesi's theorem to this context.Communicated by Corrado de Concini

Distance between unitary orbits of normal elements in simple [formula omitted]-algebras of real rank zero

Let x,y be two normal elements in a unital simple C⁎-algebra A. We introduce a function Dc(x,y) and show that in a unital simple AF-algebra there is a constant 1>C>0 such thatC⋅Dc(x,y)≤dist(U(x),U(y))≤Dc(x,y), where U(x) and U(y) are the closures of the unitary orbits of x and of y, respectively. We also generalize this to unital simple C⁎-algebras with real rank zero, stable rank one and weakly unperforated K0-group. More complicated estimates are given in the presence of non-trivial K1-information.

HIGHEST-WEIGHT VECTORS FOR THE ADJOINT ACTION OF GL n ON POLYNOMIALS, II

Let G = GL n be the general linear group over an algebraically closed field k and let \( \mathfrak{g}=\mathfrak{g}{\mathfrak{l}}_n \) be its Lie algebra. Let U be the subgroup of G which consists of the upper unitriangular matrices. Let k[\( \mathfrak{g} \)] be the algebra of polynomial functions on g and let k[\( \mathfrak{g} \)] G be the algebra of invariants under the conjugation action of G. For all weights χ ∈ ℤ n with χ2 ≤ 0 or χ n − 1 ≥ 0 we give explicit bases for the k[\( \mathfrak{g} \)] G -module k \( {\left[\mathfrak{g}\right]}_{\chi}^U \) of highest weight vectors of weight χ. We also give bases for the vector spaces k[C] χ U where C is a nilpotent orbit closure. This extends earlier results to a much bigger class of weights. To express our semi-invariants in terms of matrix powers we prove certain Cayley-Hamilton type identities.

On Shuffling of Infinite Square-Free Words

In this paper we answer two recent questions from Charlier et al. (2014) and Harju (2013) about self-shuffling words. An infinite word $w$ is called self-shuffling, if $w=\prod_{i=0}^\infty U_iV_i=\prod_{i=0}^\infty U_i=\prod_{i=0}^\infty V_i$ for some finite words $U_i$, $V_i$. Harju recently asked whether square-free self-shuffling words exist. We answer this question affirmatively. Besides that, we build an infinite word such that no word in its shift orbit closure is self-shuffling, answering positively a question of E. Charlier et al.

Cylinder deformations in orbit closures of translation surfaces

Let M be a translation surface. We show that certain deformations of M supported on the set of all cylinders in a given direction remain in the GL(2,R)-orbit closure of M. Applications are given concerning complete periodicity, field of definition, and the number of of parallel cylinders which may be found on a translation surface in a given orbit closure. The proof uses Eskin-Mirzakhani-Mohammadi's recent theorem on orbit closures of translation surfaces, as well as results of Minsky-Weiss and Smillie-Weiss on the dynamics of horocycle flow.

Combinatorial Results on (1, 2, 1, 2)-Avoiding GL(p, ℂ) × GL(q, ℂ)-Orbit Closures on GL(p+q, ℂ)/B

Combinatorial Results on (1, 2, 1, 2)-Avoiding GL(<i>p</i>, ℂ) × GL(<i>q</i>, ℂ)-Orbit Closures on GL(<i>p</i>+<i>q</i>, ℂ)/<i>B</i>

A study of the representations supported by the orbit closure of the determinant

A study of the representations supported by the orbit closure of the determinant

Characteristic Classes of Singular Toric Varieties

AbstractIn this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch‐Riemann‐Roch theorem. As important special cases, we obtain new (or recover well‐known) formulae for the Baum‐Fulton‐MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom‐Milnor L‐classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus‐orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus‐invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice‐point counting in lattice polytopes and their subcomplexes, yielding generalized Pick‐type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz‐Riemann‐Roch theorem of Edidin‐Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L‐classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Bruhat Order on Partial Fixed Point Free Involutions

The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex triangulates a ball. The rank-generating function of $PF_n$ is computed and the resulting polynomial is contrasted with the Hasse-Weil zeta function of the variety of skew-symmetric matrices over finite fields.

A study of the representations supported by the orbit closure of the determinant

AbstractWe show the existence of a large family of representations supported by the orbit closure of the determinant. However, the validity of our result is based on the validity of the celebrated ‘Latin square conjecture’ due to Alon and Tarsi or, more precisely, on the validity of an equivalent ‘column Latin square conjecture’ due to Huang and Rota.

On a Morelli type expression of cohomology classes of torus orbifolds

Let X be a complete toric variety of dimension n and \del the fan in a lattice N associated to X. For each cone \sigma of \del there corresponds an orbit closure V(\sigma) of the action of complex torus on X. The homology classes {[V(\sigma)]| \dim \sigma=k} form a set of specified generators of H_{n-k}(X,Q). Then any x\in H_{n-k}(X,Q) can be written in the form \[ x=\sum_{\sigma\in\del_X, \dim\sigma=k}\mu(x,\sigma)[V(\sigma)]. \] A question occurs whether there is some canonical way to express \mu(x,\sigma). Morelli gave an answer when X is non-singular and at least for x= \T_{n-k}(X) the Todd class of X. However his answer takes coefficients in the field of rational functions of degree 0 on the Grassmann manifold G_{n-k+1}(N_Q) of (n-k+1)-planes in N_Q. His proof uses Baum-Bott's residue formula for holomorphic foliations applied to the action of complex torus on X. On the other hand there appeared several attempts for generalizing non-singular toric varieties in topological contexts. Such generalized manifolds of dimension 2n acted on by a compact n dimensional torus T are called by the names quasi-toric manifolds, torus manifolds, toric manifolds, toric origami manifolds, topological toric manifolds and so on. Similarly torus orbifold can be considered. To a torus orbifold $X$ a simplicial set \del_X called multi-fan of X is associated. A question occurs whether a similar expression to Morelli's formula holds for torus orbifolds. It will be shown the answer is yes in this case too at least when the rational cohomology ring H^*(X)_Q is generated by H^2(X)_Q. Under this assumption the equivariant cohomology ring with rational coefficients H^*_T(X,Q) is isomorphic to H^*_T(\del_X,Q), the face ring of the multi-fan \del_X, and the proof is carried out on H^*_T(\del_X,Q) by using completely combinatorial terms.

Infinite self-shuffling words

In this paper we introduce and study a new property of infinite words: An infinite word x∈AN, with values in a finite set A, is said to be k-self-shuffling(k≥2) if x admits factorizations: x=∏i=0∞Ui(1)⋯Ui(k)=∏i=0∞Ui(1)=⋯=∏i=0∞Ui(k). In other words, there exists a shuffle of k-copies of x which produces x. We are particularly interested in the case k=2, in which case we say x is self-shuffling. This property of infinite words is shown to be independent of the complexity of the word as measured by the number of distinct factors of each length. Examples exist from bounded to full complexity. It is also an intrinsic property of the word and not of its language (set of factors). For instance, every aperiodic word contains a non-self-shuffling word in its shift orbit closure. While the property of being self-shuffling is a relatively strong condition, many important words arising in the area of symbolic dynamics are verified to be self-shuffling. They include for instance the Thue–Morse word fixed by the morphism 0↦01, 1↦10. As another example we show that all Sturmian words of slope α∈R∖Q and intercept 0<ρ<1 are self-shuffling (while those of intercept ρ=0 are not). Our characterization of self-shuffling Sturmian words can be interpreted arithmetically in terms of a dynamical embedding and defines an arithmetic process we call the stepping stone model. One important feature of self-shuffling words stems from their morphic invariance: The morphic image of a self-shuffling word is self-shuffling. This provides a useful tool for showing that one word is not the morphic image of another. In addition to its morphic invariance, this new notion has other unexpected applications particularly in the area of substitutive dynamical systems. For example, as a consequence of our characterization of self-shuffling Sturmian words, we recover a number theoretic result, originally due to Yasutomi, on a classification of pure morphic Sturmian words in the orbit of the characteristic.

Lexicographic shellability of partial involutions

In this manuscript we study inclusion posets of Borel orbit closures on (symmetric) matrices. In particular, we show that the Bruhat poset of partial involutions is a lexicographically shellable poset. We determine which subintervals of the Bruhat posets are Eulerian, and moreover, by studying certain embeddings of the symmetric groups and their involutions into rook matrices and partial involutions, respectively, we obtain new shelling orders on the corresponding order complexes.

Non-Veech surfaces in $${\mathcal{H}^{\rm hyp}(4)}$$ H hyp ( 4 ) are generic

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\), that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\)-orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\). The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmuller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\).

The field of definition of affine invariant submanifolds of the moduli space of abelian differentials

The field of definition of an affine invariant submanifold M is the smallest subfield of the reals such that M can be defined in local period coordinates by linear equations with coefficients in this field. We show that the field of definition is equal to the intersection of the holonomy fields of translation surfaces in M, and is a real number field of degree at most the genus. We show that the projection of the tangent bundle of M to absolute cohomology H^1 is simple, and give a direct sum decomposition of H^1. Applications include explicit full measure sets of translation surfaces whose orbit closures are as large as possible, and evidence for finiteness of algebraically primitive Teichm\"uller curves. The proofs use recent results of Artur Avila, Alex Eskin, Maryam Mirzakhani, Amir Mohammadi, and Martin M\"oller.

Computational Methods for Orbit Closures in a Representation with Finitely Many Orbits

The action of on has finitely many orbits. Kraśkiewicz and Weyman give a conjectural description of the minimal free resolution of the coordinate ring for certain orbit closures in . We verify their conjecture by constructing their complexes explicitly in Macaulay2 and checking that they are exact. We also construct equations for all the orbit closures and verify that they generate radical ideals.

Density of values of linear maps on quadratic surfaces

In this paper we investigate the density properties of the set of values of a linear map at integer points on a quadratic surface. In particular we show that this set is dense in the range of the linear map subject to certain algebraic conditions on the linear map and the quadratic form that defines the surface. The proof uses Ratner's theorem on orbit closures of unipotent subgroups acting on homogeneous spaces.

Suffix conjugates for a class of morphic subshifts

Let $A$ be a finite alphabet and $f:~A^{\ast }\rightarrow A^{\ast }$ be a morphism with an iterative fixed point $f^{{\it\omega}}({\it\alpha})$, where ${\it\alpha}\in A$. Consider the subshift $({\mathcal{X}},T)$, where ${\mathcal{X}}$ is the shift orbit closure of $f^{{\it\omega}}({\it\alpha})$ and $T:~{\mathcal{X}}\rightarrow {\mathcal{X}}$ is the shift map. Let $S$ be a finite alphabet that is in bijective correspondence via a mapping $c$ with the set of non-empty suffixes of the images $f(a)$ for $a\in A$. Let ${\mathcal{S}}\subset S^{\mathbb{N}}$ be the set of infinite words $\mathbf{s}=(s_{n})_{n\geq 0}$ such that ${\it\pi}(\mathbf{s}):=c(s_{0})f(c(s_{1}))f^{2}(c(s_{2}))\cdots \in {\mathcal{X}}$. We show that if $f$ is primitive, $f^{{\it\omega}}({\it\alpha})$ is aperiodic, and $f(A)$ is a suffix code, then there exists a mapping $H:~{\mathcal{S}}\rightarrow {\mathcal{S}}$ such that $({\mathcal{S}},H)$ is a topological dynamical system and ${\it\pi}:~({\mathcal{S}},H)\rightarrow ({\mathcal{X}},T)$ is a conjugacy; we call $({\mathcal{S}},H)$ the suffix conjugate of $({\mathcal{X}},T)$. In the special case where $f$ is the Fibonacci or Thue–Morse morphism, we show that the subshift $({\mathcal{S}},T)$ is sofic, that is, the language of ${\mathcal{S}}$ is regular.

ORBIT CLOSURES IN THE WITT ALGEBRA AND ITS DUAL SPACE

Working over an algebraically closed field of characteristic p &gt; 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p - 1 (in which case we only obtain a conditional statement) in the dual space W* under the induced action of G. As a corollary we prove that the algebra of invariants k[W*]G is trivial.