Classify Frobenius automorphisms in extended structures on kC_n

Prove that for the cyclic group C_n with generator g, the only Frobenius algebra automorphisms φ that can occur in an extended Frobenius structure on the group algebra kC_n are: (a) φ(g)=±g or φ(g)=ω_n g^{-1} when n is even; (b) φ(g)=g or φ(g)=ω_n g^{-1} when n is odd, where ω_n is any n-th root of unity in k.

Background

The paper classifies extended Frobenius structures for the group algebras kC2, kC3, and kC4, identifying all possible pairs (φ,θ) that extend the standard Frobenius algebra structure. These concrete cases suggest a pattern for the allowable Frobenius automorphisms φ in the extended structures.

The conjecture generalizes these specific classifications to all cyclic orders n, predicting exactly which algebra automorphisms φ can appear in an extended Frobenius structure on kC_n, distinguishing the even and odd cases and allowing a twist by an arbitrary n-th root of unity ω_n.

References

Conjecture Let $g$ be a generator of $C_n$. The following are the only possibilities for the Frobenius automorphism $\phi$ for an extended structure on $\Bbbk C_n$: \begin{enumerate}[\upshape (a)] \item $\phi(g)=\pm g$ or $\phi(g)=\omega_n g{-1}$ when $n$ is even, \smallskip

\item $\phi(g)=g$ or $\phi(g)=\omega_n g{-1}$ when $n$ is odd, \end{enumerate} where $\omega_n \in \Bbbk$ is any $n$-th root of unity.

— On extended Frobenius structures (2410.18232 - Czenky et al., 23 Oct 2024) in Section 2.2 (Classification results)