Cayley unitary elements in group algebras under oriented involutions
John H. Castillo, Yzel Wlly Gómez-Espíndola, Alexander Holguín-Villa
TL;DR
This work extends the construction of Cayley unitary elements from group algebras to the setting of oriented involutions, defining and analyzing the skew-symmetric subalgebra $\mathbf{F}G^{-}$ and the unitary elements $u_{[\beta]}=(1-\beta)(1+\beta)^{-1}$ for $\beta\in\mathbf{F}G^{-}$. It establishes invertibility criteria and explicit coefficient formulas for inverses of expressions like $1+\beta$, revealing Fibonacci-like sequences that govern the coefficients (notably when $\beta=q(x-x^{-1})$ with $q$ real and $x$ of finite order). In the oriented classical case, it shows that for elements with $\sigma(x)=-1$, the invertibility of $1+(x+x^{-1})$ occurs precisely when the order satisfies $o(x)\equiv2$ or $4$ (mod $6$), and it provides explicit Cayley unitary elements built from these skew-symmetric components, with concrete examples in small groups. Overall, the paper broadens the repertoire of unitary elements in group algebras under oriented involutions and highlights order-based patterns with potential applications in cryptography and coding theory.
Abstract
Let $\mathbf{F}$ be a real extension of $\mathbb{Q}$, $G$ a finite group and $\mathbf{F}G$ its group algebra. Given both a group homomorphism $σ:G\rightarrow \{\pm1\}$ (called an orientation) and a group involution $^\ast:G \rightarrow G$ such that $gg^\ast\in N=ker(σ)$, an oriented group involution $\circledast$ of $\mathbf{F}G$ is defined by $α=\sum_{g\in G}α_{g}g \mapsto α^\circledast=\sum_{g\in G}α_{g}σ(g)g^{\ast}$. In this paper, in case the involution on $G$ is the classical one, $x\mapsto x^{-1}$, $β=x+x^{-1}$ is a skew-symmetric element in $\mathbf{F}G$ such that $1+β$ is invertible, for $x\in G$ with $σ(x)=-1$, we consider Cayley unitary elements built out of $β$. We prove that the coefficients of $(1+β)^{-1}$ involve an interesting sequence which is a Fibonacci-like sequence.