The goal of our work is to study the spaces of primitive elements of some combinatorial Hopf algebras, whose underlying vector spaces admit linear basis labelled by subsets of the set of maps between finite sets. In order to deal with these objects we introduce the notion of shuffle algebras, which are coloured algebras where composition is not always defined. We define bialgebras in this framework and compute the subpaces of primitive elements associated to them. These spaces of primitive elements...

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $𝔽S_n$ of the symmetric group $S_n$ over a field $𝔽$ of characteristic 0 (or $p>n$). The goal is to obtain a constructive version of the isomorphism $\psi :{⨁}_{\lambda }{M}_{d_{\lambda }}\left(𝔽\right)⟶𝔽S_n$ where $\lambda$ is a partition of $n$ and $d_{\lambda }$ counts the standard tableaux of shape $\lambda$....

On introduit une opérade anticyclique $\mathbf{V}$ définie par une présentation ternaire quadratique. On montre qu’elle admet une base indexée par les arbres binaires planaires. On relie cette construction à la famille des treillis de Tamari ${\left({\mathsf{Y}}_n\right)}_{n\ge 0}$ en construisant un isomorphisme entre $\mathbf{V}(2n+1)$ et le groupe de Grothendieck de la catégorie $mod{\mathsf{Y}}_n$ qui envoie la base de $\mathbf{V}(2n+1)$ sur les classes des modules projectifs et qui transforme la structure anticyclique de $\mathbf{V}$ en la transformation de Coxeter de la catégorie dérivée de $mod{\mathsf{Y}}_n$. La dualité...