The level-zero vector representation, its affinization and the level-one crystal energy

The derived operators on $V$ are well defined and give a $\mathbb{Q}(q)$-linear action of the generators. Each $E_i,F_i$ formula assigns to each $v_j$ either $0$ or a unique basis vector, and each $K_h$ multiplies $v_j$ by the nonzero scalar $q^{⟨{\lambda }_j,h⟩}$, so $E_i,F_i,K_h$ are well-defined linear endomorphisms of the finite-dimensional space $V$; $K_0=\mathrm{i}\mathrm{d}$ and $K_{h+h^{\mathrm{\prime }}}=K_h{K}_{h^{\mathrm{\prime }}}$ hold because $⟨{\lambda }_j,\cdot ⟩$ is additive, so the toral assignment is a homomorphism $P_{\mathrm{c}\mathrm{l}}^{\vee }\to \mathrm{G}\mathrm{L}(V)$.

The derived defining relations hold on $V$. For $h\in P_{\mathrm{c}\mathrm{l}}^{\vee }$ and $j\equiv i+1$, conjugation gives $K_h{E}_i{K}_h^{-1}{v}_j=q^{⟨{\lambda }_{j-1}-{\lambda }_j,h⟩}{v}_{j-1}$ with cyclic representatives; the weight difference is ${\alpha }_i$ for $j\ge 2$ and ${\alpha }_0-\delta$ for $j=1$, and $⟨\delta ,h⟩=0$, so the coefficient is $q^{⟨{\alpha }_i,h⟩}$ in both cases. Reversing the arrow gives $K_h{F}_i{K}_h^{-1}=q^{-⟨{\alpha }_i,h⟩}{F}_i$. For distinct colors $i,t$, $E_i{F}_t{v}_j$ can be nonzero only if $j\equiv t$ and $j+1\equiv i+1$, which forces $t=i$; $F_t{E}_i{v}_j$ can be nonzero only if $j\equiv i+1$ and $j-1\equiv t$, again forcing $t=i$, so both vanish when $t\mathrm{\ne }i$. For equal colors, $E_i{F}_i{v}_j={\delta }_{j\equiv i}{v}_j$ and $F_i{E}_i{v}_j={\delta }_{j\equiv i+1}{v}_j$, and $K_i{v}_j=q^{{\delta }_{i,j}-{\delta }_{i,j-1}}{v}_j$; thus $(K_i-K_i^{-1})\mathrm{/}(q-q^{-1})$ acts by $1,-1,0$ respectively when $j\equiv i$, $j\equiv i+1$, or neither. Finally $E_i^2=F_i^2=0$ because two successive steps of one color would require $i\equiv i+1$. For $e\ge 3$ and $a_{it}=0$, $E_i{E}_t$ requires $t=i+1$ and $E_t{E}_i$ requires $t=i-1$, both excluded by nonadjacency; the corresponding $F$ paths require $t=i-1$ or $t=i+1$ and also vanish. For $a_{it}=-1$, the outside terms of $E_i^2{E}_t-(q+q^{-1})E_i{E}_t{E}_i+E_t{E}_i^2$ vanish by $E_i^2=0$; its middle path would require, from right to left, residues $i+1\to i\to i-1$ and then $i-1\equiv i+1$, impossible for $e\ge 3$. The middle $F_i{F}_t{F}_i$ path requires $i\to i+1\to i+2$ and then $i+2\equiv i$, also impossible. For $e=2$, $a_{it}=-2$ and every cubic Serre term $E_i^{3-r}{E}_t{E}_i^r$ ($0\le r\le 3$) has at least one side exponent at least $2$, so it vanishes; the same holds for $F$. These arguments check all derived relations, including the double edge.

The affinized space is well defined and carries the displayed operators. By [F5] each $u_m$ is $z^{a(m)}{v}_{j(m)}$ for a unique pair, so $\{u_m\}$ is a $\mathbb{Q}(q)$-basis of $V_{\mathrm{a}\mathrm{f}\mathrm{f}}$. The formulas $E_i{u}_m={\delta }_{m\equiv i+1}{u}_{m-1}$ and $F_i{u}_m={\delta }_{m\equiv i}{u}_{m+1}$ define linear endomorphisms, and they agree with the $z$-picture: if $m=j-ae$ with $j\equiv i+1$ and $j\mathrm{\ne }1$, then $m-1=(j-1)-ae$ and $u_{m-1}=z^a{v}_{j-1}$, while for $j=1$ one has $m=1-ae$, $m-1=e-(a+1)e$ and $u_{m-1}=z^{a+1}{v}_e$, matching $E_i(z^a{v}_1)=z^{a+{\delta }_{i0}}{v}_e$ because $j=1\equiv i+1$ forces $i=0$; the $F_i$ computation is the mirror image.

$V_{\mathrm{a}\mathrm{f}\mathrm{f}}$ is a full $U_q(\mathfrak{g})$-module with the stated weights and integrable strings. If $E_i{u}_m\mathrm{\ne }0$, then $\mathrm{w}\mathrm{t}(u_{m-1})-\mathrm{w}\mathrm{t}(u_m)={\alpha }_i$: for $j(m)\ge 2$ this is ${\lambda }_{j(m)-1}-{\lambda }_{j(m)}$; for $j(m)=1$ the new pair is $(e,a(m)+1)$, giving ${\lambda }_e+\delta -{\lambda }_1={\alpha }_0$ by [F3]. Reversing these arrows gives weight difference $-{\alpha }_i$ for $F_i$. Hence the full toral conjugation relations hold for every $h\in P^{\vee }$, and the additive weight pairing gives $K_0=1$ and $K_h{K}_{h^{\mathrm{\prime }}}=K_{h+h^{\mathrm{\prime }}}$. The mixed and Serre computations of step 1.2 apply to the same integer residue paths on $u_m$; when a same-color mixed composition is nonzero, its two opposite steps return to $u_m$, so their coefficients and the $K_i$ commutator are unchanged. Thus [F1] gives the full module structure, and $K_d{u}_m=q^{a(m)}{u}_m$ by [F6]. Each color acts on disjoint pairs $\{u_{i+ke},u_{i+ke+1}\}$ with $K_i$ eigenvalues $q,q^{-1}$ and one-step raising and lowering maps, and on singleton vectors of other residues with $K_i$ eigenvalue $1$ and both maps zero. Every basis vector therefore lies in a finite-dimensional rank-one stable space, so the module is integrable and its displayed weight decomposition has level $0$. Finally multiplication by $z$ commutes with $E_i,F_i$ and every derived $K_h$, since $⟨\delta ,h⟩=0$ there. The subspace $(z-1)V_{\mathrm{a}\mathrm{f}\mathrm{f}}$ is consequently stable under the genuine subalgebra $U_q^{\mathrm{\prime }}(\mathfrak{g})$, and its quotient, identified with $V$ by $z^a{v}_j\mapsto v_j$, carries exactly the finite operators of step 1.2. This proves the claimed derived module action without assigning $K_d$ to finite $V$.

The crystal data of $u_m$ and $b_j$ are as displayed. By step 2.1, $F_i{u}_m\mathrm{\ne }0$ exactly when $m\equiv i$, and then the image is $u_{m+1}$, so the $i$-string through $u_m$ has length ${\phi }_i(u_m)={\delta }_{m\equiv i}$; dually the $e_i$-string has length ${\epsilon }_i(u_m)={\delta }_{m\equiv i+1}$. Hence $\epsilon (u_m)={\sum }_i{\delta }_{m\equiv i+1}{\mathrm{\Lambda }}_i^{\mathrm{c}\mathrm{l}}={\mathrm{\Lambda }}_{j(m)-1}^{\mathrm{c}\mathrm{l}}$ and $\phi (u_m)={\mathrm{\Lambda }}_{j(m)}^{\mathrm{c}\mathrm{l}}$, which lie in $P_{\mathrm{c}\mathrm{l}}$ and have level $1$ by [F4]; the consistency identity ${\phi }_i(u_m)={\epsilon }_i(u_m)+⟨h_i,\mathrm{w}\mathrm{t}(u_m)⟩$ holds because $⟨h_i,\mathrm{w}\mathrm{t}(u_m)⟩={\delta }_{i,j(m)}-{\delta }_{i,j(m)-1}$ by [F6]. On $B$ the same computation with $\mathrm{w}\mathrm{t}(b_j)={\lambda }_j^{\mathrm{c}\mathrm{l}}$ gives $f_i{b}_j=b_{j+1}$ for $j\equiv i$, $e_i{b}_j=b_{j-1}$ for $j\equiv i+1$, $\epsilon (b_j)={\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$, $\phi (b_j)={\mathrm{\Lambda }}_j^{\mathrm{c}\mathrm{l}}$, and the two maps $j\mapsto {\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$ and $j\mapsto {\mathrm{\Lambda }}_j^{\mathrm{c}\mathrm{l}}$ are bijections from $\{1,\dots ,e\}$ onto the $e$ level-one fundamental classes $\{{\mathrm{\Lambda }}_0^{\mathrm{c}\mathrm{l}},\dots ,{\mathrm{\Lambda }}_{e-1}^{\mathrm{c}\mathrm{l}}\}$.

The opposite-channel tensor rule is a crystal rule on $B_{\mathrm{a}\mathrm{f}\mathrm{f}}\otimes B_{\mathrm{a}\mathrm{f}\mathrm{f}}$. For one fixed color write $+$ for a top vector with $({\epsilon }_i,{\phi }_i)=(0,1)$, $-$ for its bottom vector with $(1,0)$, and $o$ for a singleton with $(0,0)$. The displayed rule gives exactly the strings $++\stackrel{f_i}{\to }+-\stackrel{f_i}{\to }--$, $+o\stackrel{f_i}{\to }-o$, and $o+\stackrel{f_i}{\to }o-$, with $-+$ and $oo$ isolated; the displayed $e_i$ reverses each arrow, including the ties. Thus the string lengths $({\epsilon }_i,{\phi }_i)$ are respectively $(0,2),(1,1),(2,0)$ on the first string, $(0,1),(1,0)$ on the next two strings, and $(0,0)$ on the isolated pairs. These nine cases give ${\epsilon }_i(x\otimes y)=\max ({\epsilon }_i(x)-⟨h_i,\mathrm{w}\mathrm{t}(y)⟩,{\epsilon }_i(y))$ and ${\phi }_i(x\otimes y)=\max ({\phi }_i(x),{\phi }_i(y)+⟨h_i,\mathrm{w}\mathrm{t}(x)⟩)$, using step 3.1 on the factors. They imply ${\phi }_i-{\epsilon }_i=⟨h_i,\mathrm{w}\mathrm{t}(x)+\mathrm{w}\mathrm{t}(y)⟩$; each arrow changes the additive weight by $-{\alpha }_i$ and its inverse by ${\alpha }_i$. This verifies the claimed rule, which is the opposite-coproduct ${\otimes }_{+}$ rule in the cited KMPY section.

The energy function is constant on every connected component of the crystal graph $B_{\mathrm{a}\mathrm{f}\mathrm{f}}\otimes B_{\mathrm{a}\mathrm{f}\mathrm{f}}$ for the rule of step 4.1. Write $x=z^a{b}_j$ with $m=j-ae$ and $y=z^b{b}_k$ with $n=k-be$, so $H_{\mathrm{o}\mathrm{p}}(x\otimes y)=\chi (j>k)+b-a$. By step 4.1 the arrow $f_i$ acts on the first factor precisely when ${\epsilon }_i(x)\ge {\phi }_i(y)$ and on the second factor when ${\epsilon }_i(x)<{\phi }_i(y)$, and an arrow is nonzero only when the selected factor lies in the domain of $f_i$; explicitly, the first-factor case is nonzero exactly for $j\equiv i$ (whence ${\epsilon }_i(x)=0$) and $k\equiv ̸i$ (whence ${\phi }_i(y)=0$), and the second-factor case is nonzero exactly for $k\equiv i$ and $j\equiv ̸i+1$. In the first case the new pair is $(z^{a^{\mathrm{\prime }}}{b}_{j^{\mathrm{\prime }}},z^b{b}_k)$ with $j^{\mathrm{\prime }}=j+1,a^{\mathrm{\prime }}=a$ if $j\mathrm{\ne }e$, and with $j^{\mathrm{\prime }}=1,a^{\mathrm{\prime }}=a-1$ if $j=e$ (then $i=0$): for $j\mathrm{\ne }e$ one has $k\mathrm{\ne }j$, so $\chi (j^{\mathrm{\prime }}>k)=\chi (j>k)$, and for $j=e$ one has $k\le e-1$, so $\chi (j^{\mathrm{\prime }}>k)-\chi (j>k)=[1>k]-1=-1$ while $a^{\mathrm{\prime }}-a=-1$, giving $\mathrm{\Delta }H=(b-a^{\mathrm{\prime }})-(b-a)+(-1)=1-1=0$. In the second case the new pair is $(z^a{b}_j,z^{b^{\mathrm{\prime }}}{b}_{k^{\mathrm{\prime }}})$ with $k^{\mathrm{\prime }}=k+1,b^{\mathrm{\prime }}=b$ if $k\mathrm{\ne }e$, and $k^{\mathrm{\prime }}=1,b^{\mathrm{\prime }}=b-1$ if $k=e$ (then $i=0$): for $k\mathrm{\ne }e$ one has $j\mathrm{\ne }k+1$, so $\chi (j>k^{\mathrm{\prime }})=\chi (j>k)$, and for $k=e$ one has $j\mathrm{\ne }1$, so $\chi (j>1)-\chi (j>e)=1-0=1$ while $b^{\mathrm{\prime }}-b=-1$, giving $\mathrm{\Delta }H=1-1=0$. The defining rule is reversible by $e_i$, so every edge of the graph is covered by these two cases and $H_{\mathrm{o}\mathrm{p}}$ is constant on components.

The energy function has the two loop-shift axioms, vanishes on extremal diagonals and has the stated positivity criterion. For $x=z^a{b}_j$ and $y=z^b{b}_k$: $H_{\mathrm{o}\mathrm{p}}(zx\otimes y)=\chi (j>k)+b-(a+1)=H_{\mathrm{o}\mathrm{p}}(x\otimes y)-1$ and $H_{\mathrm{o}\mathrm{p}}(x\otimes zy)=\chi (j>k)+(b+1)-a=H_{\mathrm{o}\mathrm{p}}(x\otimes y)+1$, which are the two axioms (E1) and (E2); $H_{\mathrm{o}\mathrm{p}}(z^a{b}_j\otimes z^a{b}_j)=0$ for every $j,a$, which includes every extremal element $z^a{b}_j$ because every color string through any $b_j$ is a singleton or a one-arrow string: at positive label the raising map vanishes and the Weyl reflection follows the lowering arrow, at negative label the lowering map vanishes and the reflection follows the raising arrow, and at zero label both vanish; these reflections stay in $B$, and the same argument with the loop shifts stays in $B_{\mathrm{a}\mathrm{f}\mathrm{f}}$, so the normalization (E4) holds. For $m_1=j-ae$ and $m_2=k-be$: if $j>k$ then $m_1-m_2=(j-k)+(b-a)e$ and $H_{\mathrm{o}\mathrm{p}}=\chi +b-a$; the case $b\ge a$ gives $H_{\mathrm{o}\mathrm{p}}\ge 1$ and $m_1-m_2\ge 1$; the case $b<a$ gives $H_{\mathrm{o}\mathrm{p}}\le 0$ and $m_1-m_2\le (j-k)-e\le e-1-e<0$. If $j\le k$ then $H_{\mathrm{o}\mathrm{p}}=b-a$, and $m_1>m_2$ forces $(b-a)e>k-j\ge 0$, so $b>a$ and $H_{\mathrm{o}\mathrm{p}}>0$; conversely $b>a$ gives $H_{\mathrm{o}\mathrm{p}}>0$ and $m_1-m_2=(j-k)+(b-a)e\ge e-(k-j)\ge 1$. Hence in all cases $H_{\mathrm{o}\mathrm{p}}(x\otimes y)>0⟺m_1>m_2$, and this criterion includes $e=2$.

The ground sequence satisfies the two ground identities. Let $b_s^{\circ }=u_{M-s}$ and write $M-s=j-ae$ and $M-s-1=k-be$ with $j,k\in \{1,\dots ,e\}$. If $j\mathrm{\ne }1$ then $k=j-1$ and $b=a$; if $j=1$ then $k=e$ and $b=a+1$. In the first case $\chi (j>k)=1$ and $b-a=0$, in the second $\chi (1>e)=0$ and $b-a=1$; hence $H_{\mathrm{o}\mathrm{p}}(b_s^{\circ }\otimes b_{s+1}^{\circ })=1$ in both cases, for every $M$ and every $s\ge 0$. For the $\epsilon$-$\phi$ identity: $b_s^{\circ }=u_{M-s}$ has $\epsilon (b_s^{\circ })={\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$ and $b_{s+1}^{\circ }=u_{M-s-1}$ has $\phi (b_{s+1}^{\circ })={\mathrm{\Lambda }}_k^{\mathrm{c}\mathrm{l}}$, and $k\equiv M-s-1\equiv j-1$ in $\mathbb{Z}\mathrm{/}e\mathbb{Z}$ with $k=j-1$ when $j\mathrm{\ne }1$ and $k=e$ representing $0=j-1$ when $j=1$; hence $\epsilon (b_s^{\circ })=\phi (b_{s+1}^{\circ })$.

Conclusion. The derived module $V$ is a well-defined finite-dimensional $U_q^{\mathrm{\prime }}(\mathfrak{g})$-module of level $0$ with basis $v_j$ of classical weights $({\mathrm{\Lambda }}_j-{\mathrm{\Lambda }}_{j-1}{)}^{\mathrm{c}\mathrm{l}}$ and crystal operators $f_i{b}_j=b_{j+1}$ for $j\equiv i$, $e_i{b}_j=b_{j-1}$ for $j\equiv i+1$, with $\epsilon (b_j)={\mathrm{\Lambda }}_{j-1}^{\mathrm{c}\mathrm{l}}$, $\phi (b_j)={\mathrm{\Lambda }}_j^{\mathrm{c}\mathrm{l}}$, both of level $1$, by steps 1.1, 1.2 and 3.1; its affinization $V_{\mathrm{a}\mathrm{f}\mathrm{f}}$ has basis $u_m=z^a{v}_j$ with $m=j-ae$, weights ${\mathrm{\Lambda }}_j-{\mathrm{\Lambda }}_{j-1}+a\delta$ and $K_d$ eigenvalue $q^a$, carries the integrable action $E_i{u}_m={\delta }_{m\equiv i+1}{u}_{m-1}$, $F_i{u}_m={\delta }_{m\equiv i}{u}_{m+1}$ by step 2.1, and has ordinary order $u_m>u_n⟺m>n$; the opposite-channel tensor rule and energy $H_{\mathrm{o}\mathrm{p}}(z^a{b}_j\otimes z^b{b}_k)=\chi (j>k)+b-a$ with $l(z^a{b}_j)=ae-j$ satisfy the two loop-shift axioms, constancy on components, the extremal normalization and the positivity criterion by steps 5.1 and 6.1; and the charge-$M$ ground sequence $b_s^{\circ }=u_{M-s}$ satisfies $\epsilon (b_s^{\circ })=\phi (b_{s+1}^{\circ })$ and $H_{\mathrm{o}\mathrm{p}}(b_s^{\circ }\otimes b_{s+1}^{\circ })=1$ by step 7.1. All statements hold for every $e\ge 2$.