{\mbox{\boldmath $\alpha$}})
= \oplus_{P \in {\mathbb T}({\mbox{\boldmath $\alpha$}}) }K_0 v_P$ be a vector space over $K_0$ with the standard basis $\{v_P|P\in {\mathbb T}({\mbox{\boldmath $\alpha$}})\}$.

For generators $e_i$, $f_i$ and $s_i$ of ${A}_n$, we define linear maps $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$, $\rho_{{\mbox{\boldmath $\alpha$}}}(f_i)$ and $\rho_{{\mbox{\boldmath $\alpha$}}}(s_i)$ on ${\mathbb V}({\mbox{\boldmath $\alpha$}})$ giving the matrices $E_i$ $F_i$ and $M_i$ respectively with respect to the basis $\{ v_P | P\in {\mathbb T}({\mbox{\boldmath $\alpha$}}) \}$.

Definition of $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)$
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Firstly, we define a linear map for $e_i$.

For a tableaux $P = ({\mbox{\boldmath $\alpha$}}^{(0)}, {\mbox{\boldmath $\alpha$}}^{(1/2)}, \ldots, {\mbox{\boldmath $\alpha$}}^{(n)})$ of ${\mathbb T}({\mbox{\boldmath $\alpha$}})$, we define $\rho_{{\mbox{\boldmath $\alpha$}}}(e_i)(v_P)
= \sum_{Q \in {\mathbb T}({\mbox{\boldmath $\alpha$}})}(E_i)_{QP}v_Q$. Let $Q = ({\mbox{\boldmath $\alpha$}}^{\prime(0)}, {\mbox{\boldmath $\alpha$}}^{\prime(1/2)},
\ldots, {\mbox{\boldmath $\alpha$}}^{\prime(n)})$.

If there is an $i_0 \in \{1/2, 1, \ldots, n-1/2 \} \setminus \{i-1/2\}$ such that ${\mbox{\boldmath $\alpha$}}^{(i_0)}\neq {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$, then we put $$(E_i)_{QP} = 0.$$ In the following, we consider the case that ${\mbox{\boldmath $\alpha$}}^{(i_0)} = {\mbox{\boldmath $\alpha$}}^{\prime(i_0)}$ for $i_0\in\{0, 1/2, 1, \ldots, n-1/2\}\setminus\{i-1/2\}$.

If ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ are not labeled by the same Young diagram, then we put $$(E_i)_{QP} = 0.$$

We consider the case ${\mbox{\boldmath $\alpha$}}^{(i-1)}$ and ${\mbox{\boldmath $\alpha$}}^{(i)}$ have the same label $\widetilde{\lambda}$. In this case, the possible vertices as ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ have labels $\{\widetilde{\lambda}^{-}_{(s)}\}$, which are obtained by 