Generic KLR induction and restriction: mixed relations and the opposite-end exact sequence

Conventions and PBW coordinates. Put ${\kappa }_a=-{\tau }_a$ for every color pair, so ${\kappa }_w=(-1{)}^{\mathrm{\ell }(w)}{\tau }_w$ has the same degree and is a PBW basis with the source dot and braid signs. For $n\ge 1$, embed $R(n)$ on positions $1,\dots ,n$ and $R_1(n)={\xi }_{n}R(n)$ on positions $2,\dots ,n+1$; their overlap is $R_1(n-1)$ on positions $2,\dots ,n$. Write $k^I={⨁}_{j\in I}k{ϵ}_j$ for the color algebra. Choosing reduced words adapted to the cosets gives the left PBW decomposition $R(n+1)={⨁}_{a=1}^{n+1}R(n,1){\kappa }_n\cdots {\kappa }_a$ and the right PBW decomposition $R(n+1)={⨁}_{a=1}^{n+1}{\kappa }_a\cdots {\kappa }_n(k^I[x_{n+1}]{\otimes }_{k}R(n))$, where the product for $a=n+1$ is $1$. These are labelled decompositions: polynomial and idempotent coefficients keep the transferred color, and crossing degrees are summed on the actual word. PBW independence follows by using those reduced coset words followed by a reduced word in $S_n$; the resulting products have additive lengths and are precisely the chosen PBW basis.

The empty-word case. At $\beta =0$ the KLR presentation gives $R(0)=k$ and $R({\alpha }_j)=k[x_1]e(j)$ with $\mathrm{deg}{x}_1=2d_j$. The empty word has no last strand, so its $E_i$-restriction is zero on the weight-graded category, without introducing a negative-height algebra. Both append inductions of a graded $k$-module $M$ are $k[x_1]e(j){\otimes }_{k}M$. Orthogonality of $e(i),e(j)$ gives $E_i{F}_{j}M=E_i{\bar{F}}_{j}M=0$ for $i\mathrm{\ne }j$, whereas $E_i{F}_{i}M=E_i{\bar{F}}_{i}M\cong M{\otimes }_{k}k[t_i]$ by $x_1^a\otimes m\mapsto m\otimes t_i^a$, naturally and in degree zero. Thus (a) is zero-to-zero, (b) has precisely the polynomial summand with coefficient $1$, and (c) is $0\to 0\to M{\otimes }_{k}k[t_i]\stackrel{\mathrm{i}\mathrm{d}}{\to }M{\otimes }_{k}k[t_i]\to 0$, since $({\alpha }_i\mid 0)=0$. The uncolored overlap formula involving $R_1(n-1)$ below is used only for $n\ge 1$.

The same-end double cosets. For $n\ge 1$ define $T:R(n){\otimes }_{R(n-1)}R(n)\to R(n+1)$ by $x\otimes y\mapsto x{\kappa }_{n}y$, with both outer algebras embedded on the first $n$ positions. It is balanced because ${\kappa }_n$ commutes with the generators on positions $1,\dots ,n-1$. Decompose the first factor as the right $R(n-1)$-module ${⨁}_{a=1}^n{\kappa }_a\cdots {\kappa }_{n-1}(k^I[x_n]{\otimes }_{k}R(n-1))$. For a polynomial/color coefficient $f(x_n)$, the dot move gives $f(x_n){\kappa }_n={\kappa }_{n}f(x_{n+1})+c$, where $c$ lies in the polynomial/idempotent algebra on the first $n+1$ positions and its last-color projector equals its adjacent-color projector; consequently ${\kappa }_a\cdots {\kappa }_{n-1}cy$ lies in $R(n,1)$. Modulo $R(n,1)$ the image is therefore exactly ${\kappa }_a\cdots {\kappa }_{n}f(x_{n+1})y$. These are the right PBW cosets $a=1,\dots ,n$, bijectively, with coefficient $1$, so the induced quotient map is an isomorphism. Inclusion of $R(n,1)$ then gives the actual bimodule isomorphism $T\oplus \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{l}:R(n){\otimes }_{R(n-1)}R(n)\oplus R(n,1)\to R(n+1)$: injectivity and surjectivity follow by first passing to the quotient and then to the included summand. This retains all nonidentity cosets, rather than discarding cosets $a>1$.

The opposite-end overlap map. For $n\ge 1$, multiplication defines a balanced $(R(n),R_1(n))$-bimodule map $U:R(n){\otimes }_{R_1(n-1)}{R}_1(n)\to R(n+1)$, $x\otimes y\mapsto xy$. The PBW decomposition in the last $n$ positions is $R_1(n)={⨁}_{a=2}^{n+1}(R_1(n-1){\otimes }_k{k}^I[x_{n+1}]){\kappa }_n\cdots {\kappa }_a$. Tensoring this displayed free decomposition identifies the domain of $U$ with ${⨁}_{a=2}^{n+1}R(n,1){\kappa }_n\cdots {\kappa }_a$. Multiplication maps each basis element to the identically named PBW element, so $U$ is injective and its image $J$ is exactly those cosets. Using the opposite PBW decomposition gives also $J={⨁}_{a=0}^{n-1}{\kappa }_a\cdots {\kappa }_{1}R(1,n)$, where $a=0$ is the empty product. Thus $J$ is the overlap bimodule, not a two-sided ideal, and the quotient has the sole remaining transfer coset $R(n,1){\kappa }_n\cdots {\kappa }_1$, equivalently ${\kappa }_n\cdots {\kappa }_{1}R(1,n)$.

Correct same-end colored kernels. For an input weight $\beta$ set $\delta =\beta -{\alpha }_i$ and $\gamma =\delta +{\alpha }_j$. When $\delta \in Q_{+}$, the kernel of $E_i{F}_j$ is $e(\gamma ,i)R(\beta +{\alpha }_j)e(\beta ,j)$, an $(R(\gamma ),R(\beta ))$-bimodule. Cutting the isomorphism of step 2.1 by these idempotents gives the image under $T$ of $R(\gamma )e(\delta ,j){\otimes }_{R(\delta )}e(\delta ,i)R(\beta )$, the kernel of $F_j{E}_i$. The factors project onto the two colors interchanged by ${\kappa }_n$, so its degree is exactly $-b_{ij}$, and the image is the domain shifted by $-b_{ij}$. If $\delta \mathrm{\notin }{Q}_{+}$, $E_i$ has zero restriction on the input weight because no word of weight $\beta$ ends in $i$; the transfer summand is zero, without writing any algebra at a negative weight. For $i\mathrm{\ne }j$ the one-strand-enlarged word then also has no $i$ endpoint, so $E_i{F}_j$ is zero. The other summand $R(n,1)$ preserves the last color, so it vanishes exactly for $i\mathrm{\ne }j$; for $i=j$ its corner is $R(\beta ){\otimes }_{k}k[t_i]$, with $\mathrm{deg}{t}_i=2d_i$. Hence the bimodule maps give (a)'s same-end isomorphism and (b), with coefficient $1$ on the polynomial summand. Tensoring an actual bimodule isomorphism with any graded input module preserves it; no exactness or flatness assumption is needed for this passage.

The quotient and its right splitting. Put $w={\kappa }_n\cdots {\kappa }_1$. Move a dot or crossing along $w$ by the local relations. Modulo $J$ these moves give $x_{a}w=w{x}_{a+1}$ for $1\le a\le n$, $x_{n+1}w=w{x}_1$, and ${\kappa }_{a}w=w{\kappa }_{a+1}$ for $1\le a<n$. Dot corrections remove one crossing and braid corrections remove the three-crossing braid block; in each case the surviving permutation fails to transfer the first endpoint all the way to the last endpoint, so its PBW coset is among $a=2,\dots ,n+1$ and lies in $J$. Idempotents simply follow the same cyclic permutation. These generator identities prove $af(x_{n+1})w=w{\xi }_n(a)f(x_1)$ in the quotient. Taking the coefficient of $w$ therefore identifies the underlying bimodule quotient with $R(n){\otimes }_k{k}^I[t]$, the right $R(n)$ action being through ${\xi }_n$, and both indicated endpoint dots becoming $t$. The color algebra is retained because the transferred endpoint can have any color. Its right PBW expression $w(k^I[x_1]{\otimes }_k{R}_1(n))$ provides a right $R_1(n)$-linear section of the quotient map by choosing those same PBW representatives. It need not be a bimodule section. Hence the sequence defined by $U$ and this coefficient map is right-split before any tensoring.

Opposite-end colored kernels and degrees. With $\delta ,\gamma$ as in step 3.1 and $\delta \in Q_{+}$, the kernel of $E_i{\bar{F}}_j$ is $e(\gamma ,i)R(\beta +{\alpha }_j)e(j,\beta )$, with right $R(\beta )$ action through ${\xi }_n$. Cutting $U$ identifies its overlap submodule with $R(\gamma )e(j,\delta ){\otimes }_{R(\delta )}e(\delta ,i)R(\beta )$, the kernel of ${\bar{F}}_j{E}_i$, by the plain multiplication map; this map adds no crossing and has degree zero. The quotient transfer word $w$ takes bottom $(j,\nu )$ to top $(\nu ,j)$, so this quotient corner vanishes for $i\mathrm{\ne }j$ and gives the unshifted opposite-end isomorphism in (a). For $i=j$ and $\nu$ of weight $\beta$, $we(i,\nu )$ has degree $-{\sum }_a{b}_{i{\nu }_a}=-({\alpha }_i\mid \beta )$. Thus the quotient corner is $q^{-({\alpha }_i\mid \beta )}R(\beta ){\otimes }_{k}k[t_i]$, with the label $i$ fixed and $\mathrm{deg}{t}_i=2d_i$. The right section in step 3.2 respects these color projectors and this degree shift, so the colored sequence is split as right $R(\beta )$-modules. Tensoring with any graded $M$ retains its injective first map and exact quotient and gives precisely (c), functorially, with the stated shift. If $\delta \mathrm{\notin }{Q}_{+}$ the overlap corner is zero; at $\beta =0$ this agrees with step 1.2.

Conclusion. Steps 1.2 and 3.1 establish the same-end mixed isomorphisms, and step 4.1 establishes the opposite-end isomorphism and exact sequence for every weight. All maps are the explicit homogeneous PBW corner maps above; the opposite-end sequence is right-split before tensoring, and the uncolored expression retains its transferred-color labels and uses $n\ge 1$. The dots, crossing signs and shifts are those of the fixed native grading, with the source normalization $\kappa =-\tau$ affecting only unit PBW coefficients. No quotient flatness, uniform shift for all labelled cosets, or negative-height algebra is assumed.