The categorical ${\mathfrak{s}\mathfrak{l}}_2$ commutator for cyclotomic KLR functors

The comparison square. Write $n=\mathrm{\mid }\beta \mathrm{\mid }$, $s=({\alpha }_i\mid 2\mathrm{\Lambda }-\beta )$, $\kappa =-\tau$, and use $K_0,K_1,P,Q$ from the proved comparison in [F1]. Put $C_0=e(\beta ,i)K_0$ and $C_1=e(\beta ,i)K_1$. If $\delta =\beta -{\alpha }_i\in Q_{+}$ put $D_0=R(\beta )e(\delta ,i){\otimes }_{R(\delta )}e(\delta ,i)A$ and $D_1=R(\beta )e(i,\delta ){\otimes }_{R(\delta )}e(\delta ,i)A$; otherwise both are zero. Restricting the comparison at $\beta$ and applying the comparison at $\delta$ to $E_{i}A$ give exact rows $0\to q^s{D}_1\stackrel{P^{\mathrm{\prime }}}{\to }{q}_i^{-2}{D}_0\to q_i^{-2}{F}_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}A\to 0$ and $0\to q^s{C}_1\stackrel{P}{\to }{C}_0\to E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}A\to 0$. The vertical maps are $U:D_1\to C_1$, $x\otimes y\mapsto x{\xi }_n(y)$, and $T:D_0\to C_0$, $x\otimes y\mapsto x{\kappa }_{n}y$, with the indicated shifts. Here $P^{\mathrm{\prime }}$ multiplies the first factor by $a(x_1){\kappa }_1\cdots {\kappa }_{n-1}$; right bilinearity of $P$ gives $PU=T{P}^{\mathrm{\prime }}$ on elementary tensors, both represented by $xa(x_1){\kappa }_1\cdots {\kappa }_{n}y$. The generic PBW maps of [F1] make $U,T$ injective, with cokernels $q^{-({\alpha }_i\mid \beta )}A[t]$ and $A[t]$ respectively, retaining the transferred color $i$. The induced coefficient map $\mathcal{A}:q_i^{2h}A[t]\to A[t]$ is therefore defined by $\mathcal{A}B=CP$, where $B$ takes the full-transfer coefficient in $C_1$ and $C$ projects $C_0=T(D_0)\oplus A[t]$ onto its polynomial summand. The square consists of $(R(\beta ),A)$-bimodule maps; its coefficient map and snake sequence are $A$-bimodule maps. Only $B,P$ commute with their endpoint-dot actions, and no $A[t]$-linearity of $\mathcal{A}$ is asserted. The snake lemma gives $0\to \ker \mathcal{A}\to q_i^{-2}{F}_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}A\to E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}A\to \mathrm{coker}\mathcal{A}\to 0$.

The two comparison products. The projectivity proof gives $QP$ as right multiplication on $K_1$ by $a_i(t){\prod }_{{\nu }_a\mathrm{\ne }i}{Q}_{i{\nu }_a}(t,x_a)$ on each old word $\nu$. Call this central polynomial $Z(t)$. The same element acts on $K_0$ with $t=x_{n+1}$, and $P$ intertwines the actions. Thus $(PQ-Z)P=0$, so $PQ-Z$ factors through $Be$. That quotient is killed by a scalar monic polynomial by [F2], whereas $K_0$ is projective over $A[t]$ and has no such polynomial torsion. Consequently $PQ=Z$ on $K_0$ as well. For $p={\beta }_i$, set $S(t)={\sum }_{\nu }{\prod }_{{\nu }_a=i}(t-x_a{)}^{2}e(\nu )$ and choose $\gamma \in k^{\times }$ so ${\gamma }^{-1}=(-1{)}^p{\prod }_{j\mathrm{\ne }i}{t}_{ij}^{{\beta }_j}$, where $t_{ij}$ is the leading $u^{-a_{ij}}$ coefficient of $Q_{ij}$. Then $W(t)=\gamma (-1{)}^p{a}_i(t){\sum }_{\nu }{\prod }_{{\nu }_a\mathrm{\ne }i}{Q}_{i{\nu }_a}(t,x_a)e(\nu )$ is central monic of degree $h+2p$, and $S$ is central monic of degree $2p$. In $Q$ the leading full-transfer coefficient is $(-1{)}^{p}S(t)$: each equal-color $g$ contributes $-(t-x_a{)}^2{\kappa }_a$ and each unequal-color $g$ contributes ${\kappa }_a$; every omitted crossing has a shorter permutation and no full-transfer coset. Applying $\mathcal{A}B=CP$ to $Q(t^k)$ and using $PQ=Z$ proves $\mathcal{A}(t^{k}S)={\gamma }^{-1}{t}^{k}W$ for every $k\ge 0$.

The coefficient recurrence and division. Let $u_k=e(\beta ,i){\kappa }_n\cdots {\kappa }_1{x}_1^{k}e(i,\beta )\in C_1$. It satisfies $B(u_k)=t^k$. In the same-end direct sum write $P(u_k)=T({\psi }_k)+{\varphi }_k$, with ${\psi }_k\in D_0$ and ${\varphi }_k=\mathcal{A}(t^k)\in A[t]$. Since $P(u_{k+1})=P(u_k)x_{n+1}$ and $T(z)x_{n+1}=T(z(x_n\otimes 1))+E(z)$, where $E(x\otimes y)=xy\in A$, comparison of the summands gives ${\psi }_{k+1}={\psi }_k(x_n\otimes 1)$ and ${\varphi }_{k+1}=t{\varphi }_k+E({\psi }_k)$. These identities use the source $\kappa$ mixed relation and the actual multiplication counit. In particular $\mathcal{A}(at)-\mathcal{A}(a)t\in A$ for all $a\in A[t]$. Induction on the degree of $f$, using $\mathcal{A}(atg)-\mathcal{A}(a)tg=[\mathcal{A}(atg)-\mathcal{A}(at)g]+[\mathcal{A}(at)-\mathcal{A}(a)t]g$, gives ${\mathrm{deg}}_t(\mathcal{A}(af)-\mathcal{A}(a)f)<{\mathrm{deg}}_{t}f$. With $f=S$ and step 2.1, this says $t^{k}W=(\gamma {\varphi }_k)S+r_k$, ${\mathrm{deg}}_t{r}_k<2p$. Thus $\gamma {\varphi }_k$ is exactly the monic-division quotient, zero when $h+k<0$ and monic of degree $h+k$ otherwise.

The positive branch. If $h\ge 0$, the values $\gamma {\varphi }_k$ have distinct unit leading degrees $h+k$. Finite triangular elimination over $A$ makes $\mathcal{A}$ injective and identifies its cokernel with ${⨁}_{k=0}^{h-1}A{t}^k$. The inclusion of these polynomial classes in $C_0$, followed by $C_0\to E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}A$, is a bimodule section of the last snake map. It is the ordinary adjunction unit followed by $k$ endpoint dots, of degree $2d_{i}k$. Hence $E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}A\cong q_i^{-2}{F}_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}A\oplus {⨁}_{k=0}^{h-1}A\{2d_{i}k\}$. At $h=0$ the sum is empty.

The negative branch. If $h=-r\le 0$, division makes $\mathcal{A}$ surjective with kernel $q_i^{-2r}{⨁}_{j=0}^{r-1}A{t}^j$. Its snake inclusion sends $t^j$ to the image of ${\psi }_j$ in $q_i^{-2}{F}_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}A$. The multiplication counit and its dotted versions descend from $D_0$ to this quotient. On the included kernel, their coefficient matrix is $E({\psi }_{j+k})$ for $0\le j,k<r$. Step 3.1 gives $E({\psi }_m)=0$ for $m<r-1$ and $E({\psi }_{r-1})={\gamma }^{-1}$, since ${\varphi }_m=0$ for $m<r$ and ${\varphi }_r={\gamma }^{-1}$. Reversing column order therefore makes this matrix triangular with invertible scalar diagonal. Its finite recursive inverse, consisting of bimodule maps, supplies a retraction of the snake inclusion and splits the sequence. The dotted counit at index $k$ has target shift $-2d_i(k+1)$ after the $q_i^{-2}$ shift on $D_0$, so $q_i^{-2}{F}_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}A\cong E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}A\oplus {⨁}_{k=0}^{r-1}A\{-2d_i(k+1)\}$. At $r=0$ this agrees with step 4.1. At the vacuum the direct calculation is $E_i^{\mathrm{\Lambda }}{F}_i^{\mathrm{\Lambda }}k=k[t]\mathrm{/}{t}^{{\lambda }_i}$ and $F_i^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}k=0$; it gives the positive branch without any negative-height algebra.

Unequal-color descent and functoriality. For $i\mathrm{\ne }j$ let $S_n=R(n+1)$, $L=R(n)$ on the first $n$ positions, $L[t]=R(n,1)$, $w={\kappa }_1\cdots {\kappa }_n$ and $w^{\mathrm{\prime }}={\kappa }_n\cdots {\kappa }_1$. Expanding the first-strand cyclotomic ideal by the right cosets and then the left cosets of [F1] gives $S_{n}a(x_1)S_n=S_{n}a(x_1)L+La(x_1)S_n+L[t]w^{\mathrm{\prime }}a(x_1)wL[t]$: all partial transfer words lie in the first two summands, and only the full transfer remains. In its $(i,j)$ last-color corner the last summand is zero, because $w^{\mathrm{\prime }}$ transfers its top color to the first position and $w$ transfers its bottom color there, giving $e(i,n)a(x_1)e(j,n)=0$. Thus the corner ideal is exactly the sum of the source and target induced cyclotomic ideals. Quotienting the generic same-end corner isomorphism gives $E_i^{\mathrm{\Lambda }}{F}_j^{\mathrm{\Lambda }}\cong q^{-b_{ij}}{F}_j^{\mathrm{\Lambda }}{E}_i^{\mathrm{\Lambda }}$. For equal colors, tensor the split bimodule isomorphisms of steps 4.1 and 5.1 with any graded $A$-module: split bimodule maps remain inverse under tensoring. They give precisely (a) and (b), naturally, with the stated bimodule form on $A$. No flatness of a quotient or separability assumption is used.