arXiv subjects

Arpan Kanrar

Publications and source records attributed to Arpan Kanrar.

5 recordsLinked to original sources

arXiv2026

On indecomposable involutive solutions to the Yang-Baxter equation whose squaring map is a $p$-cycle

The pioneering work of Rump, which proved Gateva-Ivanova's conjecture concerning the decomposability of square-free solutions to the Yang-Baxter equation, significantly motivated further research into the associated squaring map $T$. This line of inquiry has yielded numerous decomposability theorems based on the underlying structure of $T$. Two seminal questions, posed by Ramírez and Vendramin, ask about the existence of certain indecomposable involutive solutions whose squaring maps are transpositions or $3$-cycles. In this paper, we explore these problems by examining the case where $T$ is a $p$-cycle, for an arbitrary prime number $p$. We provide negative answers to the aforementioned questions under the assumption that the solution has nilpotent permutation group or has prime-power cardinality, providing also some decomposition criteria in this setting. Moreover, we show that, in the particular case of latin solutions, the situation is more rigid.

math.QA↗

arXiv2025

On Modular maximal-cyclic braces

Inspired by a conjecture by Guarnieri and Vendramin concerning the number of braces with a generalized quaternion adjoint group, many researchers have studied braces whose adjoint group is a non-abelian $2$-group with a cyclic subgroup of index $2$. Following this direction, braces with generalized quaternion, dihedral, and semidihedral adjoint groups have been classified. It was found that the number of such braces stabilizes as the group order increases. In this paper, we consider the remaining open case of modular maximal-cyclic groups. We show that these braces possess only one non-cyclic additive group structure, and, in contrast to previous findings, the number of such braces increases with increasing order.

math.GR↗

arXiv2025

Central series' and ($n$)-isoclinism of skew left braces

The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central series' interact `nicely' with respect to commutation. That allows us to define a concept of $n$-isoclinism of skew left braces in $\mathcal{I}_n$, by using a concept of brace commutator words, which we have introduced. We prove results on $1$-isoclinism (isoclinism) of skew left braces analogous to important results in group theory. For any two symmetric $n$-isoclinic skew left braces $A$ and $B$, we prove that, there exist skew left braces $C$ and $R$ such that both $A$ and $B$ are $n$-isoclinic to both $C$ and $R$ and (i) $A$ and $B$ are quotient skew left braces of $C$; (ii) $A$ and $B$ are sub-skew left braces of $R$. Connections between a skew left brace and the group which occurs as a natural semi-direct product of additive and multiplicative groups of the skew left brace are investigated, and it is proved that $n$-isoclinism is preserved from braces to groups. We also show that various nilpotency concepts on skew left braces are invariant under $n$-isoclinism.

math.RA↗

arXiv2023

Involutive Yang-Baxter groups never act as Frobenius groups

A conjecture of S. Ram\'ırez states that every indecomposable non-degenerate involutive set-theoretic solution to the Yang-Baxter equation with dihedral permutation group of order $2n$ has cardinality $2n$. The conjecture is verified for odd $n$ and disproved for even $n$. The proof for odd $n$ is obtained from the more general result that the permutation group of a finite solution never acts as a Frobenius group.

math.GR↗

arXiv2023

Cycle matrices: A combinatorial approach to the set-theoretic solutions of the Quantum Yang-Baxter Equation

An $n\times n$ matrix $M=[m_{ij}]$ with $m_{ij}\in U_n=\{1,2,\ldots,n\}$ will be called a cycle matrix if $(U_n,\cdot)$ is a cycle set, where $i\cdot j=m_{ij}$. We study these matrices in this article. Using these matrices, we give some recipes to construct solutions, which include the multipermutation level $2$ solutions. As an application of these, we construct a multi-permutation solution of level $r$ for all $r\geq 1$. Our method gives alternate proof that the class of permutation groups of solutions contains all finite abelian groups.

math.GR↗