t{\mu}\in{\mbox{\boldmath $\Lambda$}}_{i+\frac{1}{2}}$ if $\widehat{\mu}$ is obtained from $\widetilde{\lambda}$ by removing a box ($i = 0, 1, 2, \ldots n-1$) \[resp. ($i=0, 1, 2, \dots, n$)\],

-   join $\widehat{\mu}\in{\mbox{\boldmath $\Lambda$}}_{i-\frac{1}{2}}$ and $\widetilde{\lambda}\in{\mbox{\boldmath $\Lambda$}}_i$ if $\widetilde{\lambda}$ is obtained from $\widehat{\mu}$ by adding a box ($i = 1, 2, \ldots n$).

For a pair of Young diagrams $({\mbox{\boldmath $\alpha$}}, {\mbox{\boldmath $\beta$}})$, if ${\mbox{\boldmath $\beta$}}$ is obtained from ${\mbox{\boldmath $\alpha$}}$ by one of the method above, we write this as ${\mbox{\boldmath $\alpha$}}\smile{\mbox{\boldmath $\beta$}}$.

Finally, we define the sets of the tableaux. For a half integer $n\in\frac{1}{2}\mathbb{Z}$ and ${\mbox{\boldmath $\alpha$}}\in{\mbox{\boldmath $\Lambda$}}_n$, we define ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, [*tableaux of shape ${\mbox{\boldmath $\alpha$}}$*]{}, to be $$\begin{aligned}
{\mathbb T}({\mbox{\boldmath $\alpha$}})&=&
    \{P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})\ |
    \ {\mbox{\boldmath $\alpha$}}^{(j)} \in {\mbox{\boldmath $\Lambda$}}_j\ (j = 0, 1/2, \ldots, n),\\
& & \quad{\mbox{\boldmath $\alpha$}}^{(n)} = {\mbox{\boldmath $\alpha$}}, 
    {\mbox{\boldmath $\alpha$}}^{(j)}\smile{\mbox{\boldmath $\alpha$}}^{(j+1/2)}
    \ (j = 0, 1/2, \ldots, n-1/2)\}.\end{aligned}$$

Construction of representation {#sec:rep}
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Now we have defined the sets of tableaux, we define linear transformations among the tableaux.

Let ${\mathbb Q}$ be the field of rational numbers and $K_0 = {\mathbb Q}(Q)$ its extension. In the following, the linear transformations are defined over $K_0$. If they preserve the relations defined in the previous sections, they define representations of ${A}_n = {A}_n(Q)\otimes K_0$. Similar methods are used for example in the references [@AK; @GHJ; @Mu; @W1; @W2; @Ko2].

Let ${\mathbb V}(