$\tau_{X,Y} {\colon}X \otimes Y\to Y \otimes X$ in ${\mathcal{S}}$ is depicted as $$\psfrag{Z}[Bl][Bl]{\scalebox{.8}{$X$}}
\psfrag{X}[Br][Br]{\scalebox{.8}{$X$}}
\psfrag{A}[Bl][Bl]{\scalebox{.8}{$Y$}}
\psfrag{Y}[Br][Br]{\scalebox{.8}{$Y$}}
\tau_{X,Y}=\,\; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{symmetry.eps}}}} \,.$$ The axioms of a symmetry are depicted as follows: for all $X,Y,Z \in {\mathrm{Ob}}({\mathcal{S}})$,

[cc]{}

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       \[Bl\]\[Bl\] \[Br\]\[Br\] \[Bl\]\[Bl\] \[Br\]\[Br\] = \[Bl\]\[Bl\] \[Bl\]\[Bl\] \[Br\]\[Br\] \[Br\]\[Br\] \[Bl\]\[Bl\] \[Br\]\[Br\] ,
   \[2em\] \[Bl\]\[Bl\] \[Br\]\[Br\] \[Bl\]\[Bl\] \[Br\]\[Br\] = \[Bl\]\[Bl\] \[Bl\]\[Bl\] \[Br\]\[Br\] \[Br\]\[Br\] \[Bl\]\[Bl\] \[Br\]\[Br\] ,
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& \[Bl\]\[Bl\] \[Br\]\[Br\] \[Bl\]\[Bl\] \[Br\]\[Br\] = \[Bl\]\[Bl\] \[Br\]\[Br\] .

The naturality of $\tau$ ensures that we can push boxes across a strand without changing the morphism represented by the diagram: for any morphism $f$ in ${\mathcal{S}}$, $$\psfrag{f}[Bc][Bc]{\scalebox{.9}{$f$}}
{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{symmetry-bl.eps}}}} \; = \; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{symmetry-ur.eps}}}} \quad \text{and} \quad
{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{symmetry-br.eps}}}} \; = \; {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{symmetry-ul.eps}}}} \;.$$ For a detailed treatment of the Penrose graphical calculus, we refer to [@JS] or [@TV].

Representation of the symmetric group {#sect-repres-Sn}
-------------------------------------

Recall that a symmetric monoidal category provides representations of the symmetric groups. More precisely, the axioms of a symmetry imply that to any $A \in {\mathrm{Ob}}({\mathcal{S}})$ and $n \in {\mathbb{N}}$ is associated a unique group morphism $$\left \{ \begin{array}{ccc}\mathfrak{S}_n & \to & {\mathrm{End}}_{\mathcal{S}}(A^{\otimes n}) \\ \sigma & \mapsto & P_\sigma \end{array} \right.$$ such that for all $1 \leq i \leq n-1$, the transposition $(i,i+1)$ is mapped 