Cayley unitary elements in group algebras under oriented involutions

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 $\ddot{I}ƒ:G\to \{\pm 1\}$ (called an orientation) and a group involution ${}^{\ast }:G\to G$ such that $g{g}^{\ast }\in N=ker(\ddot{I}ƒ)$, an oriented group involution $⊛$ of $\mathbf{F}G$ is defined by $\hat{I}\pm ={\sum }_{g\in G}\hat{I}{\pm }_{g}g\mapsto \hat{I}{\pm }^{⊛}={\sum }_{g\in G}\hat{I}{\pm }_g\ddot{I}ƒ(g)g^{\ast }$. In this paper, in case the involution on $G$ is the classical one, $x\mapsto x^{-1}$, ${\hat{I}}^2=x+x^{-1}$ is a skew-symmetric element in $\mathbf{F}G$ such that $1+{\hat{I}}^2$ is invertible, for $x\in G$ with $\ddot{I}ƒ(x)=-1$, we consider Cayley unitary elements built out of ${\hat{I}}^2$. We prove that the coefficients of $(1+{\hat{I}}^2{)}^{-1}$ involve an interesting sequence which is a Fibonacci-like sequence.

10 pages