Orbit Closures Research Articles

Orbital Degeneracy Loci II: Gorenstein Orbits

AbstractIn [3] we introduced orbital degeneracy loci as generalizations of degeneracy loci of morphisms between vector bundles. Orbital degeneracy loci can be constructed from any stable subvariety of a representation of an algebraic group. In this paper we show that their canonical bundles can be conveniently controlled in the case where the affine coordinate ring of the subvariety is Gorenstein. We then study in a systematic way the subvarieties obtained as orbit closures in representations with finitely many orbits, and we determine the canonical bundles of the corresponding orbital degeneracy loci in the Gorenstein cases. Applications are given to the construction of low-dimensional varieties with negative or trivial canonical bundle.

No occurrence obstructions in geometric complexity theory

The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classesVPws\mathrm {VP}_{\mathrm {ws}}andVNP\mathrm {VNP}. In 2001 Mulmuley and Sohoni suggested studying a strengthened version of this conjecture over complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibitingoccurrence obstructions, which are irreducible representations ofGLn2(C)\mathrm {GL}_{n^2}(\mathbb {C}), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem viamultiplicity obstructionsas proposed by Mulmuley and Sohoni in 2001.

The cross-section of a spherical double cone

We show that the poset of SL(n)-orbit closures in the product of two partial flag varieties is a lattice if the action of SL(n) is spherical.

Resolutions of nilpotent orbit closures via Coulomb branches of 3-dimensional mathcal{N}=4 theories

The Coulomb branches of certain 3-dimensional mathcal{N}=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional Lie algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic resolution, (ii) the form of the symplectic resolution, and (iii) the Mukai flops in the case of multiple resolutions. Moreover, the (characteristic) height two nilpotent orbit closures are resolved by cotangent bundles of Hermitian symmetric spaces and the unitary Coulomb branch quiver realisations exhaust all the possibilities.

Quasi-uniform convergence in dynamical systems generated by an amenable group action

We study properties of the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz sequences for amenable residually finite groups. We introduce the notion of a regular Toeplitz sequence for an arbitrary amenable residually finite group and demonstrate that all regular Toeplitz sequences define minimal and uniquely ergodic systems. We prove path-connectivity of this family with respect to the Weyl pseudometric. This leads to a new proof of a theorem of Fabrice Krieger, which says that the values of entropy of Toeplitz sequences can have arbitrary finite entropy.

Involution words: Counting problems and connections to Schubert calculus for symmetric orbit closures

Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of “admissible sequences” by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of “involution Schubert polynomials” can be viewed as a Billey–Jockusch–Stanley type formula for cohomology class representatives of On- and Sp2n-orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type B.

Quantizations of Regular Functions on Nilpotent Orbits

We study the quantizations of the algebras of regular functions on nilpo- tent orbits. We show that such a quantization always exists and is unique if the orbit is birationally rigid. Further we show that, for special birationally rigid orbits, the quan- tization has integral central character in all cases but four (one orbit in E7 and three orbits in E8). We use this to complete the computation of Goldie ranks for primitive ideals with integral central character for all special nilpotent orbits but one (in E8). Our main ingredient are results on the geometry of normalizations of the closures of nilpotent orbits by Fu and Namikawa.

NILPOTENT SUBSPACES AND NILPOTENT ORBITS

Let $G$ be a semisimple complex algebraic group with Lie algebra $\mathfrak{g}$. For a nilpotent $G$-orbit ${\mathcal{O}}\subset \mathfrak{g}$, let $d_{{\mathcal{O}}}$ denote the maximal dimension of a subspace $V\subset \mathfrak{g}$ that is contained in the closure of ${\mathcal{O}}$. In this note, we prove that $d_{{\mathcal{O}}}\leq {\textstyle \frac{1}{2}}\dim {\mathcal{O}}$ and this upper bound is attained if and only if ${\mathcal{O}}$ is a Richardson orbit. Furthermore, if $V$ is $B$-stable and $\dim V={\textstyle \frac{1}{2}}\dim {\mathcal{O}}$, then $V$ is the nilradical of a polarisation of ${\mathcal{O}}$. Every nilpotent orbit closure has a distinguished $B$-stable subspace constructed via an $\mathfrak{sl}_{2}$-triple, which is called the Dynkin ideal. We then characterise the nilpotent orbits ${\mathcal{O}}$ such that the Dynkin ideal (1) has the minimal dimension among all $B$-stable subspaces $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$, or (2) is the only $B$-stable subspace $\mathfrak{c}$ such that $\mathfrak{c}\cap {\mathcal{O}}$ is dense in $\mathfrak{c}$.

Matrix orbit closures

Let G be the group $$\mathrm {GL}_r(\mathbf {C}) \times (\mathbf {C}^\times )^n.$$ We conjecture that the finely-graded Hilbert series of a G orbit closure in the space of r-by-n matrices is wholly determined by the associated matroid. In support of this, we prove that the coefficients of this Hilbert series corresponding to certain hook-shaped Schur functions in the $$\mathrm {GL}_r(\mathbf {C})$$ variables are determined by the matroid, and that the orbit closure has a set-theoretic system of ideal generators whose combinatorics are also so determined. We also discuss relations between these Hilbert series for related matrices, including their stabilizing behaviour as r increases.

Branes and the Kraft-Procesi transition: classical case

Moduli spaces of a large set of 3d mathcal{N}=4 effective gauge theories are known to be closures of nilpotent orbits. This set of theories has recently acquired a special status, due to Namikawa’s theorem. As a consequence of this theorem, closures of nilpotent orbits are the simplest non-trivial moduli spaces that can be found in three dimensional theories with eight supercharges. In the early 80’s mathematicians Hanspeter Kraft and Claudio Procesi characterized an inclusion relation between nilpotent orbit closures of the same classical Lie algebra. We recently [1] showed a physical realization of their work in terms of the motion of D3-branes on the Type IIB superstring embedding of the effective gauge theories. This analysis is restricted to A-type Lie algebras. The present note expands our previous discussion to the remaining classical cases: orthogonal and symplectic algebras. In order to do so we introduce O3-planes in the superstring description. We also find a brane realization for the mathematical map between two partitions of the same integer number known as collapse. Another result is that basic Kraft-Procesi transitions turn out to be described by the moduli space of orthosymplectic quivers with varying boundary conditions.

Governing singularities of symmetric orbit closures

We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL_p x GL_q acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.

Full-rank affine invariant submanifolds

We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense GL(2,R) orbit.

GL2R orbit closures in hyperelliptic components of strata

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of abelian differentials. This result implies finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2, R) orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.

Geometric representation of the infimax S-adic family

We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of substitutions from the family has an asymptotic fixed sequence whose shift orbit closure is an infimax minimal set $\Delta^+$. The subcollection of substitutions also generates an infinite Bratteli-Vershik diagram with prefix-suffix labeled edges. Paths in the diagram give the Dumont-Thomas expansion of sequences in $\Delta^+$ which in turn gives a projection onto the asymptotic stable direction of the infinite product of the Abelianization matrices. The projections of all sequences from $\Delta^+$ is the generalized Rauzy fractal which has subpieces corresponding to the images of symbolic cylinder sets. The intervals containing these subpieces are shown to be disjoint except at endpoints, and thus the induced map derived from the symbolic shift translates them. Therefore the process yields an Interval Translation Map, and the Rauzy fractal is proved to be its attractor.

The method of shifted partial derivatives cannot separate the permanent from the determinant

The method of shifted partial derivatives introduced A. Gupta et al. [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] and N. Kayal [An exponential lower bound for the sum of powers of bounded degree polynomials, ECCC 19, 2010, p. 81], was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n − m p e r m m \ell ^{n-m}\mathrm {perm}_m cannot be realized inside the G L n 2 GL_{n^2} -orbit closure of the determinant d e t n \mathrm {det}_n when n > 2 m 2 + 2 m n>2m^2+2m . Our proof relies on several simple degenerations of the determinant polynomial, Macaulay’s theorem, which gives a lower bound on the growth of an ideal, and a lower bound estimate from [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] regarding the shifted partial derivatives of the determinant.

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

Given a classical semisimple complex algebraic group G and a symmetric pair (G,K) of non-Hermitian type, we study the closures of the spherical nilpotent K-orbits in the isotropy representation of K. For all such orbit closures, we study the normality, and we describe the K-module structure of the ring of regular functions of the normalizations.

On dynamics of graph maps with zero topological entropy

We explore the dynamics of graph maps with zero topological entropy. It is shown that a continuous map f on a topological graph G has zero topological entropy if and only if it is locally mean equicontinuous, that is the dynamics on each orbit closure is mean equicontinuous. As an application, we show that Sarnak’s Möbius disjointness conjecture is true for graph maps with zero topological entropy. We also extend several results known in interval dynamics to graph maps. We show that a graph map has zero topological entropy if and only if there is no 3-scrambled tuple if and only if the proximal relation is an equivalence relation; a graph map has no scrambled pairs if and only if it is null if and only if it is tame.

Nilpotent orbits and the Coulomb branch of T\u03c3(G) theories: special orthogonal vs orthogonal gauge group factors

Coulomb branches of a set of 3d mathcal{N} = 4 supersymmetric gauge theories are closures of nilpotent orbits of the algebra mathfrak{so}(n) . From the point of view of string theory, these quantum field theories can be understood as effective gauge theories describing the low energy dynamics of a brane configuration with the presence of orientifold planes [1]. The presence of the orientifold planes raises the question to whether the orthogonal factors of a the gauge group are indeed orthogonal O(N ) or special orthogonal SO(N ). In order to investigate this problem, we compute the Hilbert series for the Coulomb branch of Tσ(SO(n)∨) theories, utilizing the monopole formula. The results for all nilpotent orbits from mathfrak{so}(3) to mathfrak{so}(10) which are special and normal are presented. A new relationship between the choice of SO/O(N ) factors in the gauge group and the Lusztig’s Canonical Quotient overline{A}left({mathcal{O}}_{lambda}right) of the corresponding nilpotent orbit is observed. We also provide a new way of projecting several magnetic lattices of different SO(N ) gauge group factors by the diagonal action of a {mathbb{Z}}_2 group.

Quiver theories and formulae for nilpotent orbits of Exceptional algebras

We treat the topic of the closures of the nilpotent orbits of the Lie algebras of Exceptional groups through their descriptions as moduli spaces, in terms of Hilbert series and the highest weight generating functions for their representation content. We extend the set of known Coulomb branch quiver theory constructions for Exceptional group minimal nilpotent orbits, or reduced single instanton moduli spaces, to include all orbits of Characteristic Height 2, drawing on extended Dynkin diagrams and the unitary monopole formula. We also present a representation theoretic formula, based on localisation methods, for the normal nilpotent orbits of the Lie algebras of any Classical or Exceptional group. We analyse lower dimensioned Exceptional group nilpotent orbits in terms of Hilbert series and the Highest Weight Generating functions for their decompositions into characters of irreducible representations and/or Hall Littlewood polynomials. We investigate the relationships between the moduli spaces describing different nilpotent orbits and propose candidates for the constructions of some non-normal nilpotent orbits of Exceptional algebras.

Mapping class group dynamics and the holonomy of branched affine structures

We classify, up to few exceptions, the orbit closures of the {mathrm {Mod}}(Sigma )-action on the affine character variety chi ( mathrm {Aff}({mathbb {C}})). We obtain from this classification that the only obstruction for a non-abelian representation rho : pi _1 Sigma longrightarrow mathrm {Aff}({mathbb {C}}) to be the holonomy of a branched affine structure on Sigma is to be Euclidean and not to have positive volume, where Sigma is a closed oriented surface of genus g ge 2.