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[*[Key words: Finite group. Mackey functor. Symmetric Algebra. Symmetric monoidal category. Burnside Ring.]{}*]{}

[*[A.M.S. subject classification: 19A22, 20C05, 18D10,16W99.]{}*]{}

Trace maps for Mackey algebras.
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Introduction.
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Let $R$ be a unital commutative ring and $G$ be a finite group. The notion of Mackey functor was introduced by Green in $1971$. For him a Mackey functor is an axiomatisation of the comportment of the representations of a finite group. There are now several possible definitions of Mackey functors, in this paper we use the point of view of Dress who defined the Mackey functors as particular bivariant functors and we use the Mackey algebra introduced by Thévenaz and Webb. In [@tw] they proved that a Mackey functor is nothing but a module over the so-called Mackey algebra. Numerous properties of this algebra are known: this algebra shares a lot of properties with the group algebra. For example, the Mackey algebra is a free $R$-module, and its $R$-rank doesn’t depend on the ring $R$. If we work with a $p$-modular system which is “large enough”, there is a decomposition theory, in particular the Cartan matrix of this algebra is symmetric. However there are some differences, over a field of characteristic $p>0$, where $p\mid |G|$, the determinant of the Cartan matrix is not a power of the prime number $p$ in general, and as shown in $\cite{tw}$ the Mackey algebra is seldom a self-injective algebra. One may wonder about a stronger property for the Mackey algebra: when is the Mackey algebra a symmetric algebra? The answer to this question depends on the ring $R$. When $R$ is a field of characteristic $0$ or coprime to $|G|$, the Mackey algebra is semi-simple (see [@tw_simple]), so it is clearly a sy