removing one box from $\widetilde{\lambda}$. Let $\{Q_s\}$ be the set of tableaux obtained from $P$ by replacing ${\mbox{\boldmath $\alpha$}}^{(i-1/2)}$ with $\widetilde{\lambda}^{-}_{(s)}$.

Then we define $(E_i)_{QP}$ to be $$(E_i)_{Q_sP}
    = \frac{h(\widetilde{\lambda})}{h(\widetilde{\lambda}^{-}_{(s)})}.$$ Here $h(\lambda)$ is the product of hook lengths defined by $$h(\lambda)  = \prod_{x\in\lambda} h_{\lambda}(x)$$ and $h_{\lambda}(x)$ is the [*hook-length*]{} at $x\in\lambda$.

Note that the matrix $E_i$ is determined by the label $\widetilde{\lambda}$ itself not by the vertex at which the tableau $P$ goes through. In other words, if another vertex in different level, say $i'$, has the same label $\widetilde{\lambda}$, then $E_{i'}$ becomes the same matrix.

Let $v(\lambda^-_{(s)}, \lambda)$ be the standard vector which corresponds to a tableau whose $(i-1)$-st, $(i-1/2)$-th and $i$-th coordinate $({\mbox{\boldmath $\alpha$}}^{(i-1)}, {\mbox{\boldmath $\alpha$}}^{(i-1/2)}, {\mbox{\boldmath $\alpha$}}^{(i)})$ are labeled by $(\lambda, \lambda^-_{(s)}, \lambda)$. Then for a tableau $P$ which goes through $\lambda$ at the $(i-1)$-st and the $i$-th coordinate of $P$, we have $$\rho(e_i)(v_P)
=
\sum_{s'} \frac{h(\lambda)}{h(\lambda^{-}_{(s')})}v(\lambda^-_{(s')}, \lambda).$$ Here $\lambda^{-}_{(s')}$ runs through Young diagrams obtained from $\lambda$ by removing one box.

![Representation spaces for $\rho(e_i)$[]{data-label="fig:repE4a"}](19.eps)

![Representation spaces for $\rho(e_i)$[]{data-label="fig:repE4b"}](20.eps)

Suppose that tableaux $\{p_r\}$ goes through paths in pictures illustrated in Figure \[fig:repE4a\] or \[fig:repE4b\]. Then we have $$\begin{aligned}
\rho(e_i)(v_0) &=&
\frac{h(\widetilde{\emptyset})}{h(\widehat{\emptyset})}v_0 = Qv_0,\\
\rho(e_i)(v_1\ v_2) &=& (v_1\ v_2)
\begin{pmatrix}
    h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{\emptyset})
    &h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{\emptyset})\\
    h(\widetilde{{\mbox{\tiny\yng(1)}}})/h(\widehat{{\mbox{\tiny\yng(1)}}