\in I \mid {\langle \mu,\,\alpha_{i}^{\vee} \rangle}=0 \bigr\}.$$

Standard paths. {#subsec:SP}
---------------

We consider the following condition on $\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$: $$\label{eq:SM}
\begin{cases}
\text{there exist $x,\,y \in W_{{\mathrm{af}}}$ such that 
$x {\succeq}y$ in $W_{{\mathrm{af}}}$}, \\[1mm]
\text{and such that ${\Pi^{J}}(x)=\kappa(\pi)$, ${\Pi^{K}}(y)=\iota(\eta)$};
\end{cases} \tag{\sf SP}$$ we set $${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu):=
  \bigl\{\pi \otimes \eta \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu) \mid 
  \text{$\pi \otimes \eta$ satisfies condition \eqref{eq:SM}}
  \bigr\}.$$

\[thm:SMT\] The set ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu) \sqcup \{{\bm{0}}\}$ is stable under the action of the Kashiwara [(]{}or, root[)]{} operators $e_{i}$, $f_{i}$, $i \in I_{{\mathrm{af}}}$, on ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda) \otimes {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$[;]{} in particular, ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ is a crystal with weights in $P_{{\mathrm{af}}}$. Moreover, ${\mathbb{S}^{\frac{\infty}{2}}}(\lambda+\mu)$ is isomorphic as a crystal to ${\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda+\mu)$.

We will give a proof of Theorem \[thm:SMT\] in Section \[sec:prf-SMT\].

Defining chains. {#subsec:DC}
----------------

\[dfn:DC\] Let $\pi=(x_{1},\,\dots,\,x_{s}\,;\,{\mathbf{a}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\lambda)$ and $\eta=(y_{1},\,\dots,\,y_{p}\,;\,{\mathbf{b}}) \in {\mathbb{B}^{{\frac{\infty}{2}}}}(\mu)$. A defining chain for $\pi \otimes \eta$ is a sequence $x_{1}',\,\dots,\,x_{s}',\,y_{1}',\,\dots,\,y_{p}'$ of elements in $W_{{\mathrm{af}}}$ satisfying the condition: $$\label{eq:DC}
\begin{cases}
x_{1}' {\succeq}\cdots {\succeq}x_{s}' {\succeq}y_{1}' {\succeq}\cdots {\succeq}y_{p}' 
 \quad \text{\rm in $W_{{\mathrm{af}}}$}; \\[1.5mm]
{\Pi^{J}}(x_{u}')=x_{u} 
 \quad \text{\rm for $1 \le u \le s$}; \\[1.5mm]
{\Pi^{K}}(y_{q}')=y_{q}
 \quad \text{\rm for $1 \le q \le p$};
\end{cases}
\tag{\sf DC}
$$ we call $x_{1}'$ (resp., $y_{p}'$) the initial element (resp., the final element) of 