Indecomposable solutions with permutation brace of size $p^3$ and permutation braces of small sizes

Andrew Darlington; Magdalena Wiertel

Home/Archive/arXiv

arXiv · 2607.09898

Indecomposable solutions with permutation brace of size $p^3$ and permutation braces of small sizes

Andrew Darlington·Magdalena Wiertel

View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the construction of Bachiller, Ced\'o and Jespers, we give a complete classification of the indecomposable involutive set-theoretic solutions to the Yang-Baxter equation whose permutation brace has size $p^3$, where p is an odd prime. We also give an algorithm for systematically producing all indecomposable involutive solutions with a given permutation brace, and use this to enumerate these with the permutation brace of size up to 107.

Explore related subjects

math.GR

Keep this discovery

BibTeX RIS

Andrew Darlington, Magdalena Wiertel. 2026-07-10. Indecomposable solutions with permutation brace of size $p^3$ and permutation braces of small sizes. https://arxiv.org/abs/2607.09898

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

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗

arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗

arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗