Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1301.2001

CSLs of the root lattice $\mathbf{A_4}$

Manuela Heuer·Peter Zeiner
View original source ↗Read source document ↗Export metadata ↓

Abstract

Recently, the group of coincidence isometries of the root lattice $A_4$ has been determined providing a classification of these isometries with respect to their coincidence indices. A more difficult task is the classification of all CSLs, since different coincidence isometries may generate the same CSL. In contrast to the typical examples in dimensions $d \leq 3$, where coincidence isometries generating the same CSL can only differ by a symmetry operation, the situation is more involved in 4 dimensions. Here, we find coincidence isometries that are not related by a symmetry operation but nevertheless give rise to the same CSL. We indicate how the classification of CSLs can be obtained by making use of the icosian ring and provide explicit criteria for two isometries to generate the same CSL. Moreover, we determine the number of CSLs of a given index and encapsulate the result in a Dirichlet series generating function.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Manuela Heuer, Peter Zeiner. 2013-01-09. CSLs of the root lattice $\mathbf{A_4}$. https://doi.org/10.1088/1742-6596%2F226%2F1%2F012024

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Equilibrium Laws for Julia's Zero and the Hyperbolic Zero of Binary Forms

For a real binary form with no real roots, Julia's zero and the hyperbolic zero are two SL(2,R)-equivariant points in the upper half-plane. We show that they admit closely related equilibrium characterizations, with radial weights given respectively by the hyperbolic tangent and hyperbolic sine of the distances to the roots. This common framework gives geometric criteria for coincidence of the two zero maps. They always agree for binary quartics; for binary sextics they agree exactly when the three upper-half-plane roots form an equilateral hyperbolic triangle, or are collinear with one root the hyperbolic midpoint of the other two. We also obtain a corresponding result for collinear binary octics. The two equilibrium laws further reveal a sharp difference in the influence of distant roots. A strict majority of roots confined to a compact set keeps Julia's zero in a compact set, and the threshold one-half is optimal. In contrast, an escaping minority can force the hyperbolic zero to escape at linear scale. We also illustrate the computational advantages of the explicit formula for the hyperbolic zero.

math.MG↗
arXiv2026

Hopf quotients of the infinite-dimensional Gaussian pyramid

We study the infinite-dimensional Gaussian pyramid and its quotients by the global sign flip and the $U(1)$-Hopf action. We resolve affirmatively a long-standing problem posed by Tomohiro Fukaya around 2014: these three limiting geometries are pairwise non-similar, meaning that no positive rescaling makes any two of them coincide.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov--Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.

math.MG↗