The KLR Cartan (graded Hom) pairing on the projective Grothendieck group

Let $Q$ be a KLR datum as in The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum and let $\psi$ be the horizontal-reflection anti-involution of $R(\beta )$ (reflection of diagrams about a horizontal axis, which reverses the order of crossings); for a finitely generated graded left module $P$ let $P^{\psi }$ be the right module with $p\cdot r:=\psi (r)p$. For $\beta \in Q_{+}$ define the Cartan pairing $([P],[Q]{)}_{\beta }:={\mathrm{g}\mathrm{d}\mathrm{i}\mathrm{m}}_k(P^{\psi }{\otimes }_{R(\beta )}Q)\in \mathbb{Z}[q,q^{-1}]\cdot (\beta {)}_q,(\beta {)}_q:={\prod }_i{\prod }_{a=1}^{{\beta }_i}\frac{1}{1-q^{2d_{i}a}},$ for finitely generated graded projective modules $P,Q$, where ${\mathrm{g}\mathrm{d}\mathrm{i}\mathrm{m}}_{k}M={\sum }_{m\in \mathbb{Z}}{q}^m{\dim }_k{M}_m$; the tensor is over the graded algebra with the total-degree grading (Graded balanced tensor product and homogeneous Hom). Equivalently define the hom pairing $([P],[Q]{)}_{\beta }^{\mathrm{\prime }}:={\mathrm{g}\mathrm{d}\mathrm{i}\mathrm{m}}_k{\mathrm{HOM}}_{R(\beta )}(P,Q)$ (Projective–module Hom pairing on class generators, Projective Hom pairing descends and is graded sesquilinear); then $([P],[Q]{)}_{\beta }^{\mathrm{\prime }}=([P^{\mathrm{\#}}],[Q]{)}_{\beta }$ for the graded left projective $P^{\mathrm{\#}}:={\mathrm{HOM}}_{R(\beta )}(P,R(\beta ){)}^{\psi }$. Thus the tensor form is symmetric and bilinear; the Hom form is conjugate-linear in its first variable and linear in its second. For the tensor form $([P],q[Q])=q([P],[Q])$ and $(q[P],[Q])=q([P],[Q])$, while the Hom form obeys $h(q^r[P],q^s[Q])=q^{s-r}h([P],[Q])$. Both are additive, vanish on different-weight components, and descend to $K_0(R)={⨁}_{\beta }{K}_0(R(\beta ))$: the tensor form is $A$-bilinear and the Hom form is $A$-sesquilinear (first variable conjugate-linear), with values in $\mathbb{Q}(q)$, more precisely in the localization adjoining all $(1-q_i^{2a}{)}^{-1}$; the values lie in $\mathbb{Z}[q,q^{-1}]\cdot (\beta {)}_q$ because $P^{\psi }{\otimes }_{R(\beta )}Q$ is a finite free graded module over the central color-symmetric polynomial subalgebra $C_{\beta }$ of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules. Normalization: $([P_i],[P_i])=(1-q^{2d_i}{)}^{-1}=(1-q_i^2{)}^{-1}$ and $([P_i],[P_j])=0$ for $i\mathrm{\ne }j$, where $P_i=R({\alpha }_i)$.

Well-definedness. $P^{\psi }{\otimes }_{R(\beta )}Q$ is a graded $k$-vector space of finite dimension in each degree and bounded below, so its graded dimension is a well-defined element of $\mathbb{Z}((q))$; the monic-root tower and PBW calculation in step 1.1 of Graded Krull-Schmidt and shift-orbit bases for finitely generated KLR modules explicitly prove that $R(\beta )$ is finite graded free over $C_{\beta }$, although freeness is not a separate assertion of that lemma's Statement. Write $V=P^{\psi }{\otimes }_{R}Q$. Taking the split inclusions of $P$ and $Q$ into finite shifted-free $R$-modules and tensoring their splitting maps exhibits $V$ as a direct summand of a finite sum of shifts of $R$. Thus $V$ is finite graded projective over $C_{\beta }$. This upgrades the value to $\mathbb{Z}[q,q^{-1}]\cdot (\beta {)}_q$: lifting a homogeneous $k$-basis of $V\mathrm{/}{C}_{\beta }^{+}V$ gives a surjection from a finite shifted free $C_{\beta }$-module with vanishing cokernel, whose projective splitting has kernel $K$ with $K=C_{\beta }^{+}K$, also vanishing by the least-degree argument; hence the Hilbert series is a finite Laurent polynomial times $\mathrm{g}\mathrm{d}\mathrm{i}\mathrm{m}{C}_{\beta }=(\beta {)}_q$. Symmetry of the tensor form is commutativity of the balanced tensor $P^{\psi }{\otimes }_{R}Q\cong Q^{\psi }{\otimes }_{R}P$ under the flip; the comparison form is the evaluation isomorphism ${\mathrm{HOM}}_R(P,R){\otimes }_{R}Q\to {\mathrm{HOM}}_R(P,Q)$, $f\otimes v\mapsto (p\mapsto f(p)v)$, an isomorphism for shifted free $P$ and hence for all finite graded projective summands, twisted by $\psi$ on the first factor.

Shifts and the two normalizations. The total Hom-degree convention is that of Projective Hom pairing descends and is graded sesquilinear, so $h(P\{r\},Q\{s\})=q^{s-r}h(P,Q)$; no scalar conjugation of the tensor value is valid, and the rank-one values $(R({\alpha }_i),R({\alpha }_i))=1\mathrm{/}(1-q_i^2)$ already show that the Hom form is not the plain scalar extension of the tensor form. The generator values follow from $P_i=R({\alpha }_i)=k[x_1]$, $P_i^{\psi }{\otimes }_{R({\alpha }_i)}{P}_i\cong k[x_1]$ with $\mathrm{deg}{x}_1=2d_i$, and $([P_i],[P_j])=0$ for $i\mathrm{\ne }j$ because the two weight components are orthogonal. The cited projective/module Hom suppliers concern finite-dimensional algebras. Here the same shift and additivity formulas follow directly from homogeneous Hom and evaluation on finite shifted-free modules and summands; their finite-dimensional hypotheses are not applied to $R(\beta )$. The anti-involution $\psi$ is recorded explicitly because it fixes the sign of the shift in the Hom form. No choice principle is used.