arXiv research

Picard modular groups are shown to be generated by complex reflections.

Defines fundamental racks for braid spaces of complex reflection groups.

Let $V$ be a finite dimensional complex vector space and $W\subset \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. A classical conjecture predicts that $V^{\reg}$ is a $K(pi,1)$ space. When $W$ is a complexified real reflection group, the conjecture f…

New link groups are derived from torus necklaces, connecting braid groups to reflection groups.

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connectio…

This paper studies parabolic quasi-Coxeter elements in complex reflection groups and their combinatorial properties.

New reflection groups derived from torus knots with finite meridians.

Study of generalized J-groups and their presentations.

Ehrenborg and Jung recently related the order complex for the lattice of d-divisible partitions with the simplicial complex of pointed ordered set partitions via a homotopy equivalence. The latter has top homology naturally identified as a Specht module. Their work unifies that of Calderbank, Hanlon, Robinson, and Wach…

The braid group of a complex reflection group is shown to be an index d subgroup.

After establishing the uniqueness of the continuation of local Cauchy data for harmonic maps between two Riemannian manifolds M and N, we prove (i) a reflection principle for a smooth minimal submanifold Y of a Riemannian manifold M that contains a reflective submanifold of M as a hypersurface and (ii) the reflection p…

The paper defines parabolic subgroups for complex braid groups and proves they form a lattice.

Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.

New method trains reflected Schrödinger bridges without complex derivatives.

We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to ind…

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

Let $V$ be a finite dimensional complex vector space and $W\subseteq \GL(V)$ be a finite complex reflection group. Let $V^{\reg}$ be the complement in $V$ of the reflecting hyperplanes. We prove that $V^{\reg}$ is a $K(π,1)$ space. This was predicted by a classical conjecture, originally stated by Brieskorn for complex…

Defines invariants for reflection groups and connects them to Frobenius structures.

Researchers describe unitary representations of mixed braid groups.

Uniform diameter bound for reflection group disk patterns.

We review the complex differential geometry of the space of oriented affine lines in ${\Bbb{R}}^3$ and give a description of Hamilton's characteristic functions for reflection in an oriented C$^1$ surface in terms of this geometry.

The final result of this article gives the order of the extension $$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$)…

Self-reflective VAE improves inference and generative modeling without complex components.

New Garside structures found for torus knot groups and related braid groups.

Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to corres…

Study reflection symmetry and APS boundary conditions on a warped cylinder.

This paper compares self-reflection and budget tuning for LLMs, revealing domain-specific performance gains.

The study determines discreteness of complex hyperbolic triangle groups.

Constructs hyperbolic reflection groups with 3D limit sets.

Study on Morse homology for reflection actions on manifolds.

Study geometric properties of a complex hyperbolic group action.

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…

We present several formulas for the traces of elements in complex hyperbolic triangle groups generated by complex reflections. The space of such groups of fixed signature is of real dimension one. We parameterise this space by a real invariant alpha of triangles in the complex hyperbolic plane. The main result of the p…

Study complex reflections in 3D hyperbolic geometry, finding new representations.

We consider a cocompact discrete reflection group $W$ of a CAT(0) space $X$. Then $W$ becomes a Coxeter group. In this paper, we study an analogy between the Davis-Moussong complex $Σ(W,S)$ and the CAT(0) space $X$, and show several analogous results about the limit set of a parabolic subgroup of the Coxeter group $W$.

For a real oriented hyperplane arrangement, we show that the corresponding Salvetti complex is homotopy equivalent to the complement of the complexified arrangement. This result was originally proved by M. Salvetti. Our proof follows the framework of a proof given by L. Paris and relies heavily on the notation of orien…

Study complex reflections in infinite Coxeter tetrahedron moduli space.

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}_{\C}^n$ is an element in ${\rm U}(n,1)$ with negative type eigenvalue $λ$, $|λ|=1$, of multiplicity $k+1$ and positive type eigenvalue $1$ of multiplicity $n-k$. We prove that a holomorphic isometry of ${\rm H}_{\C}^n$ is a product of at most fou…

Researchers solved a complex problem for a specific type of 4-manifolds.

In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.

This paper develops the first method for the exact simulation of reflected Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The running time of generating exact samples of non-stationary RBM at any time $t$ is uniformly bounded by $\mathcal{O}(1/\barγ^2)$ where $\barγ$ is the average drift of…

Study local wild mapping class groups for irregular connections on complex curves.

Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and χbe a multiplicative character of G. Let Ω^χbe the R-module of χ-invariant differential forms. We define a multiplication in Ω^χand show that under this multiplication Ω^χha…

Reflected Diffusion Models improve on score-based models by incorporating data constraints.

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds. The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinite…

We study relations between reflections in (positive or negative) points in the complex hyperbolic plane. It is easy to see that the reflections in the points q_1,q_2 obtained from p_1,p_2 by moving p_1,p_2 along the geodesic generated by p_1,p_2 and keeping the (dis)tance between p_1,p_2 satisfy the bending relation R(…

Efficiently optimizes orthogonal and Stiefel matrices on parallel units.

Study of wild mapping class groups on complex reflection groups.