Cyclotomic KLR blocks are finite-dimensional graded symmetric algebras

Finite dimensionality and the regular module. By [F1] every block $B$ is finite dimensional (ungraded in the monic branch), every dot is nilpotent in the standard case, and $R^{\mathrm{\Lambda }}(\beta +n{\alpha }_i)=0$ for large $n$. Sum the vacuum word projectives $F_{\nu }k=Be(\nu )$ over the words $\nu$ of weight $\beta$ to obtain the regular module $B={⨁}_{\nu }Be(\nu )$.

The actual perfect word form. Compose the actual second caps along $\nu$ to obtain ${ϵ}_{\nu }$, and put $\tau (a)={\sum }_{\nu }{ϵ}_{\nu }(e(\nu )ae(\nu ))$. By [F2], the composed shift is $t_{\beta }=(\beta ,\beta )-2(\mathrm{\Lambda },\beta )=-2{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )$. The actual word adjunctions identify ${\mathrm{HOM}}_B(M,B)$ with $DM\{-t_{\beta }\}$; at $M=B$ the isomorphism is $a\mapsto [b\mapsto \tau (ba)]$. It proves perfectness of every opposite-corner pairing, including off-diagonal corners, rather than assuming that a nonzero diagram closure detects every ideal.

Symmetry and its exact weights. Choose a total order on the finite color support and set $c_{\nu }={\prod }_{a<b,{\nu }_a>{\nu }_b}{t}_{{\nu }_a{\nu }_b}\mathrm{/}{t}_{{\nu }_b{\nu }_a}$, $c=\sum c_{\nu }e(\nu )$. The inversion product is intrinsic to the word and gives $c_{s_r\nu }\mathrm{/}{c}_{\nu }=t_{ji}\mathrm{/}{t}_{ij}$ for a bottom pair $(i,j)$. By [F2], $\tau (pa)=(t_{ij}\mathrm{/}{t}_{ji})\tau (ap)$; hence $\mathrm{tr}(a)=\tau (ca)$ is cyclic on unequal crossings. It is cyclic on dots, idempotents and equal-color crossings by the actual native mate identities. Moving generators successively through the functional proves symmetry on every product. All $c_{\nu }$ are nonzero degree-zero scalars, so multiplication by $c$ is invertible, preserving perfectness and degree. The symmetric pairing therefore identifies $DB$ with $B\{t_{\beta }\}$ as graded bimodules.

Finite projectives, boundaries and monic specialization. Evaluating the symmetric bimodule isomorphism on shifted free modules and cutting by an idempotent gives $D_{\psi }P\cong {\mathrm{HOM}}_B(P,DB{)}^{\psi }\cong P^{\mathrm{\#}}\{t_{\beta }\}$ for every finite projective $P$. Thus vector duality preserves projectives, without asserting $P^{\mathrm{\#}}=P$ for every shift or self-duality of every finite module. At a zero block the form is zero and its pairing is zero-to-zero; at rank zero it is the identity of $k$. At one strand it is the monic top-coefficient form, with degree $-2d_i({\lambda }_i-1)$ in the standard homogeneous case. The actual cap/terminal identities of [F2] are monic identities descended before taking the quotient. They therefore construct the same perfect symmetric form in every monic field deformation with shifts forgotten. Finite dimension and finite root strings in that branch come independently from [F1]; neither nilpotence nor homogeneous trace degree is claimed there.