arXiv research

Surveying matrix group actions on manifolds.

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

The study restricts matrix group actions on CAT(0) spaces and uniquely arcwise connected spaces, proving fixed points are inevitable.

Equivalence proven between equivariant K-theory and K-homology for certain matrix group actions.

Innovates rotation index for matrix pairs, solving group action problems.

Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the c…

Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a compact Riemann surface. This generating set and basis reflect the action of G in…

We show that for k at least 3, given any matrix in GL(k,Z), there is a hyperbolic fully irreducible automorphism of the free group of rank k whose induced action on Z^k is the given matrix.

Constructs CAT(0) actions for certain groups without unipotent elements.

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $Σ$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …

Let SL(n,Z) be the special linear group over integers and $M =S^r_1 \times S^r_2,T^r_1 \times S^r_2$ , or $T^r_0 \times S^r_1 \times S^r_2$, products of spheres and tori. We prove that any group action of SL(n,Z) on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1$. This confirms a conjec…

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).

Attention tokens are group elements, negated algebra norms.

Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and $M^{r}$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism $\mathrm{SL}_{n}(% \mathbb{Z})\righta…

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group $G$, we find some quite simple $R-$matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "exten…

Paper proposes a new approach to optimal transport for vector and matrix densities.

The paper uncovers symmetries in large language models through layer-peeled optimization.

The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N…

In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with n orb…

New model predicts drug effects across various cell types using causal imputation.

We define a family of representations $\{ρ_n\}_{n\geq 0}$ of a pure braid group $P_{2k}$. These representations are obtained from an action of $P_{2k}$ on a certain type of $A_2$ web space with color $n$. The $A_2$ web space is a generalization of the Kauffman bracket skein module of a disk with marked points on its bo…

Proves curvature positivity of invariant direct images in complex geometry.

Linear representations help embed manifolds into matrix spaces.

PSI-LinUCB improves scalability for large recommender systems.

The paper simplifies conditions for optimal paths on manifolds avoiding obstacles.

Identifies latent actions and dynamics from offline data with diverse demonstrators.

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph automorphisms, with finite quotient) the Mahler measure of its Lap…

Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…

In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge…

We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms, connection 1-forms and equivariant $3\times3$ positive-definite symmetric matrix-va…

Researchers use shape analysis to recover protein structures from Cryo-EM data.

The paper studies acylindrical actions on trees and proves acylindrical hyperbolicity of Baumslag-Solitar groups.

For an oriented surface of genus g with b boundary components, we construct a rational map from a subset of C^{6g-6+3b} onto an open algebraic subset of the PSL(2,C)-character variety as an analogue of the Fenchel-Nielsen coordinates. After taking the quotient by an action of a finite group, we obtain a parametrization…

Formula for sectional curvatures on matrix groups.

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a large…

New criteria for non-isometric group actions in metric spaces.

Non-proper surface group action on product of trees found.

Conditions for reducing quasi-actions to tree actions and group properties.

Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.

Classifies cobounded hyperbolic actions of metabelian groups.

Unified treatment of eigenvalue processes using Riemannian geometry.

A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to in…

Algorithm designs neural group actions for symmetric transformations.

The study examines how perturbations of lattice actions on group boundaries behave.

We introduce and systematically study the concept of a growth tight action. This generalizes growth tightness for word metrics as initiated by Grigorchuk and de la Harpe. Given a finitely generated, non-elementary group $G$ acting on a $G$--space $\mathcal{X}$, we prove that if $G$ contains a strongly contracting eleme…