arXiv subjects

Charlotte Roelants

Publications and source records attributed to Charlotte Roelants.

3 recordsLinked to original sources

arXiv2026

Non-degeneracy of Killing forms on real conjugacy classes of finite groups

Killing forms on finite groups arise as special cases of braided Killing forms on braided Lie algebras. If $\mathcal{C}$ is a conjugation-stable subset of a finite group $G$, the Killing form on $\mathbb{C}\mathcal{C}$ is given by $K_\mathcal{C}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b \in \mathcal{C}$. It is conjectured in previous work by López Peña, Majid and Rietsch that $K_\mathcal{C}$ is non-degenerate for any real conjugacy class $\mathcal{C}$ in a finite simple group. In this article, we reformulate the conjecture and introduce combinatorial conditions - the $\textit{1-element condition}$ and the $\textit{2-element condition}$ - that are sufficient for non-degeneracy to hold. This allows us to prove the conjecture for simple groups of the form $\mathrm{PSL}_2(q)$ and certain conjugacy classes in the alternating and symmetric groups. Moreover, we verify computationally that every real conjugacy class in a simple group of order $\leq 10^9$ fulfills at least one of these two conditions, thereby significantly extending the computational evidence for the conjecture. This raises the question whether these conditions are satisfied by all conjugacy classes in finite simple groups.

math.GR↗

arXiv2025

On reducible Killing forms for groups of Lie type

Killing forms on finite groups arise as examples of braided Killing forms on braided Lie algebras. For a finite group $G$ and a $G$-stable subset $\mathcal{C}$, the Killing form associated with $\mathbb{C}[\mathcal{C}]$ is given by $K_{\mathcal{C}}(a,b) = |C_G(ab) \cap \mathcal{C}|$ for $a,b\in \mathcal{C}$. Motivated by Cartan's criterion for semisimplicity of Lie algebras, and previous work of López Peña, Majid, and Rietsch, we study the non-degeneracy and irreducibility of $K_{\mathcal{C}}$ when $\mathcal{C}$ is a conjugacy class of involutions or unipotent elements in a finite simple group of Lie type and Lie rank one. Our approach suggests interesting connections with character theory, related counting formulas, and the study of commuting graphs.

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↗