The vector representation has a perfect crystal of level one and a lower global basis

Perfectness of $B$ except connectedness. By [F1] every $\epsilon (b_j)={\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$ and $\phi (b_j)={\mathrm{\Lambda }}_j^{\mathrm{c}\mathrm{l}}$ has level one, and the two maps are bijections $\{1,\dots ,e\}\to \{{\mathrm{\Lambda }}_0^{\mathrm{c}\mathrm{l}},\dots ,{\mathrm{\Lambda }}_{e-1}^{\mathrm{c}\mathrm{l}}\}=(P_{\mathrm{c}\mathrm{l}}^{+}{)}_1$ because both $j\mapsto {\mathrm{\Lambda }}_j^{\mathrm{c}\mathrm{l}}$ and $j\mapsto {\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$ run through the $e$ fundamental classes; hence $B_{\min }=\{b:⟨c,\epsilon (b)⟩=1\}=B$ and axiom (P3) holds. For (P1): the weights ${\lambda }_j$ of $V$ are pairwise distinct with multiplicity one by [F1], and the finite Weyl group of the subdiagram $I\setminus \{0\}$ acts transitively on them: its reflection $s_i$ exchanges ${\lambda }_i^{\mathrm{c}\mathrm{l}}$ and ${\lambda }_{i+1}^{\mathrm{c}\mathrm{l}}$ and fixes the others, so adjacent exchanges reach every weight, so they form the finite Weyl orbit of ${\lambda }^{\circ }$ with one-dimensional weight spaces. Moreover ${\lambda }_j^{\mathrm{c}\mathrm{l}}={\lambda }_1^{\mathrm{c}\mathrm{l}}-{\sum }_{i=1}^{j-1}{\alpha }_i^{\mathrm{c}\mathrm{l}}$, by telescoping the vector-weight differences, so the required finite negative-cone bound holds and the top weight space is one-dimensional.

Connectedness of $B\otimes B$ for the fixed coproduct. Use the fixed rank-one tensor rule of Rank-one tensor strings and their crystal limit rule, applied to the coefficient-one strings established in the displayed vector formulas: $f_i$ selects the first factor if ${\phi }_i(b_a)>{\epsilon }_i(b_b)$, and $e_i$ does so if ${\phi }_i(b_a)\ge {\epsilon }_i(b_b)$. For an off-diagonal pair $(b_a,b_b)$ take $i\equiv a-1$. Then ${\phi }_i(b_a)=0$ and ${\epsilon }_i(b_b)=0$, because $b\mathrm{\ne }a$, so the non-strict tie makes $e_i$ act on the first factor and sends $b_a$ to $b_{a-1}$ with cyclic representatives. Repeat until the first factor equals the second; at most $e-1$ steps are needed. From $(b_j,b_j)$, $f_j$ acts first on the first factor since $1>0$, producing $(b_{j+1},b_j)$, then on the second factor since $0\le 0$, producing $(b_{j+1},b_{j+1})$. Thus every pair is connected to a diagonal and all diagonals are mutually connected, proving (P2). No attempt to apply $e_b$ to an arbitrary $b_a$ is made.

Lower basis and separated charts of $V$. Define coefficient bar by $q\mapsto q^{-1}$ and ${\bar{v}}_j=v_j$. The matrices of $E_i,F_i$ have entries $0,1$, so $\overline{E_{i}v}=E_i\bar{v}$ and $\overline{F_{i}v}=F_i\bar{v}$; derived torals satisfy the corresponding bar-inverted toral identity. Put $V_A={⨁}_{j}A{v}_j$, $V_0={⨁}_j{A}_0{v}_j$ and $V_{\mathrm{\infty }}={\bar{V}}_0$. Their intersection is ${⨁}_j\mathbb{Z}{v}_j$: coefficientwise an integral Laurent polynomial regular at both zero and infinity is an integer constant. Its residue map at zero is the basis map $v_j\mapsto v_j+q{V}_0$, giving the balanced integral lifts $G(b_j)=v_j$. For a fixed color the nontrivial string is $v_i,v_{i+1}$, with $F_i{v}_i=v_{i+1}$ and $E_i{v}_{i+1}=v_i$ and coefficients $1$. Every other vector is a singleton killed by both operators. Therefore the normalized Kashiwara operators have the same $0,1$ matrices on this basis and preserve the genuine local chart; their residue arrows are $e_i,f_i$ from [F1]. Also $E_i^2=F_i^2=0$, hence the divided operators of order at least two vanish. The positive integral chart ${⨁}_j\mathbb{Z}[q]v_j$ is retained separately and is stable here because the matrices are $0,1$; it is not the local coefficient ring.

The affinized chart. Put $L_{\mathrm{a}\mathrm{f}\mathrm{f}}:={⨁}_m{A}_0{u}_m$ and $G(z^a{b}_j)=z^a{v}_j=u_{j-ae}$; extend the bar involution by $\bar{z}=z$. Then $\bar{\cdot }$ is semilinear with coefficient bar $q\mapsto q^{-1}$, fixes every $G(z^a{b}_j)$, and satisfies $\overline{E_{i}w}=E_i\bar{w}$, $\overline{F_{i}w}=F_i\bar{w}$ for $w\in V_{\mathrm{a}\mathrm{f}\mathrm{f}}$: the only loop shift occurs for $i=0$, where $E_0(z^a{v}_1)=z^{a+1}{v}_e$ and $\overline{z^{a+1}{v}_e}=z^{a+1}{v}_e=E_0\overline{z^a{v}_1}$, and the remaining cases have coefficients in $\{0,1\}$. The weight of $G(z^a{b}_j)$ is ${\mathrm{\Lambda }}_j-{\mathrm{\Lambda }}_{j-1}+a\delta$, and the string bound of step 1.3 is unchanged because each $i$-orbit is either the same nontrivial two-vertex string shifted by loops or a frozen singleton. The finite module $V$ carries only the $P_{\mathrm{c}\mathrm{l}}$-weights ${\lambda }_j$ and no dominant level-one highest weight vector, so it is a level-$0$ $U_q^{\mathrm{\prime }}(\mathfrak{g})$-module and not a level-one $P$-graded highest weight module; the $A=\mathbb{Z}[q,q^{-1}]$, $A_0=\mathbb{Q}[q{]}_{(q)}$ and positive $\mathbb{Z}[q]$ charts are used separately and are not identified.

Conclusion. The level-$0$ vector module has the level-one perfect crystal $B=\{b_1,\dots ,b_e\}$ with the explicit $\epsilon ,\phi$, connected tensor square and $B_{\min }=B$; the basis $G(b_j)=v_j$ is a lower global basis of $V$ whose nontrivial $i$-strings have two vertices with coefficients $1$ and whose remaining color strings are frozen singletons and whose divided operators of order $\ge 2$ vanish; and the affinized chart $G(z^a{b}_j)=z^a{v}_j$ has the bar, lower-basis and rank-one string properties of (ii) with the full $P$-weight. The affinized crystal is not perfect: $u_m$ and $u_{m-e}$ have the same $\epsilon$ and $\phi$, so these maps are not injective.