Mould calculus -- on the secondary symmetries (Q321481)

Ecalle's moulds and their symmetries are studied. After an introduction of these objects, formal moulds are defined. To any mould \(M\) is attached two formal moulds, \(Mog\) (ordinary generating formal mould) and \(Meg\) (exponential generating formal mould). A Hopf-algebraic proof of the following result, due to \textit{J. Ecalle} [J. Théor. Nombres Bordx. 15, No. 2, 411--478 (2003; Zbl 1094.11032)], is announced: {\parindent=0.6cm\begin{itemize}\item[--] \(M\) is symmetral (respectively symmetrel) if, and only if, \(Mog\) is symmetral (respectively symmetril). \item[--] \(M\) is alternal (respectively alternel) if, and only if, \(Mog\) is alternal (respectively alternil). NEWLINENEWLINE\end{itemize}} The following result is announced: {\parindent=0.6cm\begin{itemize}\item[--] \(M\) is symmetral (respectively symmetrel) if, and only if, \(Meg\) is symmetral (respectively symmetrel). \item[--] \(M\) is alternal (respectively alternel) if, and only if, \(Meg\) is alternal (respectively alternel). NEWLINENEWLINE\end{itemize}}