arXiv research

The paper connects GL-racks to knot coloring invariants.

Extends gl(m|k) construction using Hilbert scheme of points.

Study of generalized Legendrian racks and their GL-structures.

We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

We study $GL(2)$-structures on differential manifolds. The structures play a fundamental role in the geometric theory of ordinary differential equations. We prove that any $GL(2)$-structure on an even dimensional manifold give rise to a certain almost-complex structure on a bundle over the original manifold. Further, w…

Researchers compute $\mathfrak{gl}_2$-skein modules for lens spaces.

The study classifies and normalizes 3D gl-regular Nijenhuis operators.

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

We construct a canonical frame for an arbitrary Gl(2)-structure thus solving the equivalence problem for Gl(2)-structures. Our treatment includes also a problem of contact equivalence of ordinary differential equations and applies to certain classes of vector distributions. Additionally we characterise Gl(2)-structures…

We construct a right-invariant differential calculus on the quantum supergroup GL$_h(1| 1)$ and obtain the $h$-deformed superalgebra of GL$_h(1| 1)$.

Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). Class III-1 CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_5(C). Class III-2 CR manifolds have initial G-…

New basis for quantum gl_N invariants derived from Macdonald polynomials.

New homology for links in annulus discovered.

Simplified computation of symmetric gl_1 homology for links.

Study GLS estimator properties in multivariate regression with heteroskedastic and autocorrelated errors.

Extended quantum state result for gl_n weight systems.

In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a $\mathfrak{gl}_n$-web version of Hu--Mathas' graded cellular basis and has two major application…

The two-parametric quantum deformation of the algebra of coordinate functions on the supergroup GL$(1| 1)$ via a contraction of GL$_{p,q}(1| 1)$ is presented. Related differential calculus on the quantum superplane is introduced.

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-stru…

Random Forests adapted for dependent data using GLS.

In this paper, we consider the Graphical Lasso (GL), a popular optimization problem for learning the sparse representations of high-dimensional datasets, which is well-known to be computationally expensive for large-scale problems. Recently, we have shown that the sparsity pattern of the optimal solution of GL is equiv…

Paper defines and computes a new weight system for gl_N Lie algebra.

It is known that the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$ has a finite generating set. In this paper, we give a finite presentation of the level $2$ principal congruence subgroup of $GL(n;\mathbb{Z})$.

Study flat GL(1|1) connections using fatgraphs and coordinates.

This paper studies gl-regular Nijenhuis operators and their properties.

The purpose of this paper is to study the Plancherel formula for the spaces of $L^2$-sections of the line bundles over the pseudo-Riemannian space $G/H$, where $G={\rm SL}(n+1, {\mathbb R})$ and $H={\rm S}({\rm GL}(1, {\mathbb R})\times {\rm GL}(n, {\mathbb R}))$. The formula is given in an explicit form by means of sp…

Paper proves Legendrian knots with same GL-rack have similar invariants.

Researchers describe local properties of Haantjes operators.

The paper constructs compatible Poisson brackets on gl(N).

Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a ca…

We use the technique of quantum skew Howe duality to investigate the monoidal category of exterior powers of the standard representation of $U_q(\mathfrak{gl}(1|1))$. This produces a complete diagrammatic description of the category in terms of trivalent graphs, with the usual MOY relations plus one additional family o…

New structure for quantum algebra representations.

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of…

We introduce a construction of the differential calculus on the quantum supergroup GL$_{p,q}(1| 1)$. We obtain two differential calculi, respectively, associated with the left and right Cartan-Maurer one-forms. We also obtain the quantum superalgebra of GL$_{p,q}(1| 1)$. Although all of the structures we obtain are der…

New invariant from Viro's gl(1|1) polynomial distinguishes lens spaces.

The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exp…

Innovative rack theory applied to Legendrian links.

We define parameter dependent $\mathfrak{gl}_2$-foams and their associated web and arc algebras, and verify that they specialize to several known $\mathfrak{sl}_2$ or $\mathfrak{gl}_2$ constructions related to higher link and tangle invariants. Moreover, we show that all these specializations are equivalent, and we ded…

GL-LowPopArt improves minimax-optimal estimation for trace regression.

We find generators for the full rational loop group of GL(n,C) as well as for the subgroup consisting of loops that satisfy the reality condition with respect to the noncompact real form GL(n,R). We calculate the dressing action of some of those generators on the positive loop group, and apply this to the ZS-AKNS flows…

Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).

Enhances stability ranges for Torelli and congruence subgroup homologies.

We consider rank 3 distributions with growth vector (3,5,6). The class of such distributions splits into three subclasses: parabolic, hyperbolic and elliptic. In the present paper, we deal with the parabolic case. We provide a classification of such distributions and exhibit connections between them and Gl(2)-structure…

In this note we consider some properties of $GL_n(\mathbb{R})$ with the Semi-Riemannian structure induced by the trace metric $g$. In particular we study geodesics and curvature tensors. Moreover we prove that $GL_n$ has a suitable foliation, whose leaves are isometric to $(SL_n(\mathbb{R}), g)$, while its component of…

The differential calculus on the quantum supergroup GL$_q(1| 1)$ was introduced by Schmidke {\it et al}. (1990 {\it Z. Phys. C} {\bf 48} 249). We construct a differential calculus on the quantum supergroup GL$_q(1| 1)$ in a different way and we obtain its quantum superalgebra. The main structures are derived without an…

A complete classification of all continuous GL(n) contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.