The minimal-parabolic KLR diagram proof category

Fix a symmetrizable Cartan datum with real simple roots (so $a_{ii}=2$ for every color $i$), a field $k$ placed in degree zero, and weighted-homogeneous KLR polynomials $Q_{jk}$ satisfying (Q1)-(Q4) as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum; write $b_{jk}=d_j{a}_{jk}=b_{kj}$. Fix a color $i$. The minimal-parabolic KLR diagram proof category ${\mathcal{P}}_i$ is the additive graded $k$-linear 2-category of weighted oriented diagrams described by the generators, objects and equations below.

Generators and composition. The objects are the weights $\mu \in P$. The generating $1$-morphisms are the downward generators $G_j:\mu \to \mu -{\alpha }_j$, one for every color $j$, and the single upward generator $H_i:\mu \to \mu +{\alpha }_i$. Horizontal composition reads chronological negative words from the vacuum outward. Take additive sums of objects and grading shifts, and use idempotent completion when speaking of word projectives. The generator $H_i$ denotes normalized projective restriction: in the polynomial flag realization at input weight $h=⟨h_i,\mu ⟩$ it is basic reversed flag restriction shifted by $-d_i(1+h)$; it is distinct from unshifted basic restriction.

The graded 2-morphism spaces. The weights are 0-objects. A 1-morphism is a finite composable word in the $G_j,H_i$, with its specified source and target weights; identities are empty words. For parallel words $u,v$, define ${\mathrm{Hom}}^n(u,v)$ as the degree-$n$ part of the $k$-span of formal oriented string diagrams generated by the dots, downward crossings and four units/counits below, modulo the displayed local relations. Every strand is labeled by its color, and its direction is downward for $G_j$ and upward for $H_i$. The top boundary is the source word and the bottom boundary the target word. Regions carry the unique weights obtained from the outside source weight by the indicated simple-root changes. Diagrams are generic: transverse double crossings, isolated dots and nondegenerate cups/caps, with no triple crossing or dot at a crossing/extremum. Oppositely oriented crossings denote the specified adjunction mates of a downward crossing, not new generators. Use planar isotopies preserving the ordered oriented local events; moving a crossing past an extremum uses the mate relations below. Vertical composition is stacking and horizontal composition is juxtaposition at a common weight; degrees add under either operation. The interchange relation and bilinearity are imposed. A degree-$n$ map from $u$ to $v$ is a degree-zero map $u\{n\}\to v$ with the internal-shift convention of the grading supplier. Additive sums, shifts and idempotent completion have their usual graded 2-category meanings applied to these Hom spaces.

Degrees and the native crossing. Dots have degree $2d_j$ and downward crossings have degree $-b_{jk}$, matching the grading of The KLR grading and the diagrammatic multiplication. The native crossing is $\tau$, with native equal-color divided difference $(f-sf)\mathrm{/}(x_1-x_2)$; for expressing the source diagram equations put $\kappa =-\tau$ for every color pair, retaining dots, idempotents and $Q$. Thus the native cubic braid numerator is globally opposite to the source numerator, while the quadratic $Q$ is unchanged. Impose the full downward KLR relations of the cited Definition.

Adjunctions and the rank-one matrix. At input $i$-weight $h$, the first unit $1\to H_i{G}_i$ and the second counit $H_i{G}_i\to 1$ have degree $d_i(1-h)$; the first counit $G_i{H}_i\to 1$ and the second unit $1\to G_i{H}_i$ have degree $d_i(1+h)$. Impose both adjunction triangles with their adjacent input-weight shifts and cyclicity of the dot. For $h\ge 0$ prescribe the degree-zero rank-one matrix $H_i{G}_i\cong G_i{H}_i\oplus {⨁}_{r=0}^{h-1}1\{d_i(1-h+2r)\};$ for $h<0$ prescribe $G_i{H}_i\cong H_i{G}_i\oplus {⨁}_{r=0}^{-h-1}1\{d_i(1+h+2r)\}\mathrm{.}$ The off-diagonal crossing component is the cup--source-crossing--cap transfer, and the identity columns and rows are the indicated dotted units/counits. Adjoin its actual inverse blocks, requiring both matrix inverse equations; the dotted dual rows and columns are the finite triangular coefficient convolution determined by bubble inversion, and the crossing block satisfies the local cross-and-cap equations. In particular, at $h<0$, if $r_h:G_i{H}_i\to H_i{G}_i\oplus D_{-h}$ is this matrix, then $r_h^{-1}{j}_H$ and $p_H{r}_h$ are the specified natural inclusion and projection with product $1_{H_i{G}_i}$.

Explicit rank-one inverse blocks. Put $P=G_i{H}_i$, $V=H_i{G}_i$ at the common outside weight $\mu$, and write ${\eta }_{\mu }:1_{\mu }\to H_i{G}_i$, ${\epsilon }_{\mu }:G_i{H}_i\to 1_{\mu }$ for the first adjunction and ${\eta }_{\mu }^{\mathrm{\prime }}:1_{\mu }\to G_i{H}_i$, ${\epsilon }_{\mu }^{\mathrm{\prime }}:H_i{G}_i\to 1_{\mu }$ for the second. Let ${\tau }^{GG}$ be the native equal-color downward crossing and ${\tau }^{HH}$ its twofold adjunction mate. Fix the native sideways mates by $T=(H_i{G}_i{\epsilon }_{\mu })(H_i{\tau }^{GG}{H}_i)({\eta }_{\mu }{G}_i{H}_i),S=(G_i{H}_i{\epsilon }_{\mu }^{\mathrm{\prime }})(G_i{\tau }^{HH}{G}_i)({\eta }_{\mu }^{\mathrm{\prime }}{H}_i{G}_i)\mathrm{.}$ Their middle words are respectively $H_i{G}_i{G}_i{H}_i$ and $G_i{H}_i{H}_i{G}_i$; the displayed final counits consume their rightmost $G_i{H}_i$ and $H_i{G}_i$. Thus $T:P\to V$ and $S:V\to P$ are typed, with degree $d_i(1-h)-2d_i+d_i(1+h)=0$. The source-crossing component in the preceding rank-one matrix is $-T$. For $h\ge 0$, its identity columns are $C_k=x^k{\eta }_{\mu }$, and its identity rows are $R_k={\sum }_{a+j=h-1-k}{A}_{\mu ,j}{\epsilon }_{\mu }^{\mathrm{\prime }}{x}^a,0\le k<h,$ where $a,j\ge 0$. The matrix is $\rho =(-T,(C_k{)}_k):P\oplus D_h\to V$ and its specified inverse is $\sigma =(S,(R_k{)}_k):V\to P\oplus D_h$. Impose both equations $\sigma \rho =1$ and $\rho \sigma =1$; in components these say $-ST=1_P$, $R_{k}T=0$, $S{C}_k=0$, $R_k{C}_{\mathrm{\ell }}={\delta }_{k\mathrm{\ell }}$ and $-TS+{\sum }_k{C}_k{R}_k=1_V$. For $h<0$, take $C_k^{-}=x^k{\eta }_{\mu }^{\mathrm{\prime }}$ and $R_k^{-}={\sum }_{a+j=-h-1-k}{\mathsf{H}}_{\mu ,j}{\epsilon }_{\mu }{x}^a,0\le k<-h,$ again with $a,j\ge 0$. Now $\rho =(-T,(R_k^{-}{)}_k):P\to V\oplus D_{-h}$ and $\sigma =(S,(C_k^{-}{)}_k):V\oplus D_{-h}\to P$; impose $\rho \sigma =1$, $\sigma \rho =1$. Equivalently $-TS=1_V$, $T{C}_k^{-}=0$, $R_k^{-}S=0$, $R_k^{-}{C}_{\mathrm{\ell }}^{-}={\delta }_{k\mathrm{\ell }}$ and $-ST+{\sum }_k{C}_k^{-}{R}_k^{-}=1_P$. The shift summands $D_h,D_{-h}$ are exactly those displayed above. The row degrees are respectively $d_i(h-1-2k)$ and $-d_i(1+h+2k)$, opposite the column degrees, so every matrix component is degree zero after the indicated shifts. Empty identity sums at $h=0$ give the same inverse equations. These are imposed relations in the presentation; a claim that the concrete flag bimodules satisfy them still requires their local realization proof.

Bubble series and slides. The normalized far-left bubble series $A_{\mu }(t),{\mathsf{H}}_{\mu }(t)$ have constant coefficient $1$ and satisfy $A_{\mu }(t){\mathsf{H}}_{\mu }(t)=1$. Genuine negative-degree bubbles vanish. The $A$ coefficient with index $m$ is the clockwise bubble with $-h-1+m$ dots when this count is nonnegative, extended by formal inverse coefficients otherwise; its degree is $2d_{i}m$. The other family has dot count $h-1+\mathrm{\ell }$ and degree $2d_i\mathrm{\ell }$. Impose the finite-coefficient slides verified on the monic flag model: adjoining an $i$ root $z$ divides $A$ by $(1-zt{)}^2$ and multiplies $\mathsf{H}$ by that factor; adjoining a spectator root $y$ of color $j\mathrm{\ne }i$ multiplies $A$ by $q_{ij}^{\mathrm{r}\mathrm{e}\mathrm{v}}(y,t)=t^{-a_{ij}}{Q}_{ij}(t^{-1},y)\mathrm{/}{t}_{ij}$, where $t_{ij}$ is the nonzero extremal coefficient, and divides $\mathsf{H}$ by it. Every reciprocal coefficient is defined by the finite recursion with constant coefficient $1$.

Four oriented same-color equations. The following notation specifies actual local diagrams, all with color $i$. The two-crossing equations have common outside weight $\mu$; the single curls have the adjacent region weights specified individually below. Read tangent directions in the page. Let $C_L,C_R$ be the cap arch (two bottom endpoints and no top endpoints), whose tangent at its peak points left or right, respectively. Their bottom arrows, from left to right, are $(\downarrow ,\uparrow )$ and $(\uparrow ,\downarrow )$. Let $U_L,U_R$ be the cup valley (two top endpoints and no bottom endpoints), whose tangent at its trough points left or right; their top arrows are $(\uparrow ,\downarrow )$ and $(\downarrow ,\uparrow )$. An exponent on $C$ or $U$ counts dots on its open arc. Let $B_{\mathrm{c}\mathrm{c}\mathrm{w}}^b$ and $B_{\mathrm{c}\mathrm{w}}^b$ be the counterclockwise and clockwise bubbles with literal dot count $b$, located to the right of the cap/cup in the outside region of weight $\mu$. Their negative dot counts are exactly the fake bubbles defined by the coefficient conventions above, not zero by convention. Write ${\eta }_{\mu }:1_{\mu }\to H_i{G}_i$ and ${\epsilon }_{\mu }^{\mathrm{\prime }}:H_i{G}_i\to 1_{\mu }$ for the first unit and second counit, and ${\eta }_{\mu }^{\mathrm{\prime }}:1_{\mu }\to G_i{H}_i$, ${\epsilon }_{\mu }:G_i{H}_i\to 1_{\mu }$ for the second unit and first counit. Let ${\tau }^{HH}$ be the twofold adjunction mate of the native equal-color downward crossing. Define $L_{\mu }=(H_i{\epsilon }_{\mu }^{\mathrm{\prime }})({\tau }^{HH}{G}_i)(H_i{\eta }_{\mu }),R_{\mu }=({\epsilon }_{\mu }{H}_i)(G_i{\tau }^{HH})({\eta }_{\mu }^{\mathrm{\prime }}{H}_i)\mathrm{.}$ The first formula starts and ends at $H_i$ with input $\mu$; the second starts and ends at $H_i$ with input $\mu -{\alpha }_i$. Thus both units and counits in $L_{\mu }$ have degree $d_i(1-h)$, while those in $R_{\mu }$ have degree $d_i(1+h)$. The crossing degree $-2d_i$ makes their degrees $-2d_{i}h$ and $2d_{i}h$ respectively. Products in these formulas apply the rightmost map first, with adjacent weights determined by composition. Geometrically $L_{\mu }$ is an upward-strand curl with its loop bulging LEFT and its left-adjacent region of weight $\mu$. This is an endomorphism of $H_i$ at input $\mu$, and its bubble on the right-hand side is in the left region $\mu$. Let $R_{\mu }$ be the upward-strand curl with its loop bulging RIGHT and its right-adjacent region of weight $\mu$. Its left region is $\mu -{\alpha }_i$, so it is an endomorphism of $H_i$ at input $\mu -{\alpha }_i$, and its bubble is in the right region $\mu$. Both have one native crossing and the two extremal turns of the loop. Let $D_{\downarrow \uparrow }$ and $D_{\uparrow \downarrow }$ be the two opposite crossing mates composed in the two orders on their indicated straight boundaries, with common outside region $\mu$. Impose $L_{\mu }={\sum }_{a+b=-1}{x}^a{B}_{\mathrm{c}\mathrm{c}\mathrm{w},\mu }^b,R_{\mu }=-{\sum }_{a+b=-1}{x}^a{B}_{\mathrm{c}\mathrm{w},\mu }^b,$ $D_{\downarrow \uparrow }=-1_{\downarrow \uparrow }+{\sum }_{a+b+c=-2}{C}_L^a{B}_{\mathrm{c}\mathrm{w},\mu }^b{U}_R^c,$ $D_{\uparrow \downarrow }=-1_{\uparrow \downarrow }+{\sum }_{a+b+c=-2}{C}_R^a{B}_{\mathrm{c}\mathrm{c}\mathrm{w},\mu }^b{U}_L^c\mathrm{.}$ Composition in the last two displays is top-to-bottom: $U$ first, then the outside bubble, then $C$. The single-curl equations are upward-strand endomorphism equations; their cup/cap forms are their specified adjunction mates, not cap drawings with unspecified endpoint arrows. In every sum, $a,c$ on open strands are nonnegative. The clockwise bubble with literal $b$ dots is the $A_{\mu }$ coefficient of index $b+h+1$, and the counterclockwise bubble is the ${\mathsf{H}}_{\mu }$ coefficient of index $b-h+1$, where $h=⟨h_i,\mu ⟩$. Coefficients of negative index are zero and the index-zero coefficient is one. The nonnegative coefficient and open-dot indices make each sum finite. The opposite-crossing equations have degree zero: their correction degrees are $2d_i(1-h)+2d_i(a+b+c+h+1)=0$ and $2d_i(1+h)+2d_i(a+b+c-h+1)=0$. The two curls have degrees $-2d_{i}h$ and $2d_{i}h$ at their respective, different input weights.

The native single-crossing signs follow from $\kappa =-\tau$; two-crossing signs are unchanged. The literal triple-dot sum is $-2$, not the compact $-1$ printed in Webster's Figure 2: in the full positive-weight equation, coefficient indices satisfy $f_1+f_2+f_3=h-1$ and the bubble has literal count $b=-h-1+f_2$, giving $a+b+c=-2$; the negative-weight equation uses $g_1+g_2+g_3=-h-1$ and $b=h-1+g_2$, giving the same value. Reflection exchanges the two bubble orientations and preserves region weights. The single-curl coefficient sums instead have total $h$ or $-h$, yielding literal $a+b=-1$. At $h=0$ the triple sums are empty and the single curls are respectively $+1$ and $-1$ on their specified shifted input weights. At the $h=1$, $p(u)=u$ vacuum, $H_i{G}_i=k$, $G_i{H}_i=0$, $A_{\mu }=1$ and the only surviving clockwise bubble has $b=-2$; thus the first cancellation is $0=-1_k+1_k$, with no degree or vacuum contradiction.

Distinct-color crossing mates. For $j\mathrm{\ne }i$, at input weight $\mu$ put $\xi =\mu +{\alpha }_i-{\alpha }_j$, so $X=G_j{H}_i$ and $Y=H_i{G}_j$ both go from $\mu$ to $\xi$. Let ${\tau }^{G_i{G}_j}:G_i{G}_j\to G_j{G}_i$ and ${\tau }^{G_j{G}_i}:G_j{G}_i\to G_i{G}_j$ denote the two native downward crossings. Specify the handed pair by the actual typed mate composites $a=(H_i{G}_j{\epsilon }_{\mu })(H_i{\tau }^{G_i{G}_j}{H}_i)({\eta }_{\xi }{G}_j{H}_i):X\to Y,$ $b=({\epsilon }_{\xi }^{\mathrm{\prime }}{G}_j{H}_i)(H_i{\tau }^{G_j{G}_i}{H}_i)(H_i{G}_j{\eta }_{\mu }^{\mathrm{\prime }}):Y\to X\mathrm{.}$ The intermediate words are respectively $H_i{G}_i{G}_j{H}_i$ and $H_i{G}_j{G}_i{H}_i$; in $a$ the final first counit consumes the rightmost $G_i{H}_i$, while in $b$ the final second counit consumes the leftmost $H_i{G}_i$. Impose $ba=t_{ij}{1}_X$ and $ab=t_{ij}{1}_Y$ for this same handed pair. Thus $a$ and $t_{ij}^{-1}b$ are mutually inverse and give $H_i{G}_j\cong G_j{H}_i$, commuting with the $j$ dot. If $h=⟨h_i,\mu ⟩$, both composites have degree $d_i(1+h)+d_i(-1-h+a_{ij})-b_{ij}=0$. Replacing both native crossings by the source crossings $\kappa =-\tau$ negates $a$ and $b$ separately and leaves both cancellation scalars unchanged.

There is no $H_j$ in this minimal category. The analogous calculation in the separately distinguished category ${\mathcal{P}}_j$ has scalar $t_{ji}$ in both orders for its own handed pair. That does not assign $t_{ji}$ to the other composition of the fixed pair $a,b$ above. Comparing the two partial actions, or asserting a full double-right-mate ratio, requires the separate coherent assembly proof; it is not an additional relation imposed here.

Status of the relations. These equations are relations of this proof-tool category. Their consistency and homogeneous realization are established by the polynomial flag model (The minimal-parabolic polynomial flag model realizes the graded diagram relations ↗), rather than assumed for the actual cyclotomic induction/restriction action; no claim about the latter is made here. This definition uses no choice: it is a presentation by generators and homogeneous equations, and every sl(2) infinite family appearing in the matrix and bubble clauses is a finite coefficient recursion from a constant coefficient $1$.