Cyclotomic diagram duality and graded module duals

Let $B=R^{\mathrm{\Lambda }}(\beta )$ be a standard graded cyclotomic block (The cyclotomic KLR quotient at a dominant integral weight) and let $\psi$ be the descended anti-involution fixing $e(\nu ),x_a,{\tau }_a$; it is well defined because its individual first-strand polynomial generators are fixed, and it is the generator-fixing member of the two $\psi$'s in play (The KLR Cartan (graded Hom) pairing on the projective Grothendieck group). Write shifts internally as $(M\{r\}{)}_m=M_{m-r}$ (Associative graded algebras, bimodules, and internal shifts).

Vector duality. For a finite graded $B$-module $M$ define $D_{\psi }M:={\mathrm{HOM}}_k(M,k{)}^{\psi },$ the finite $k$-linear dual with $B$-action $(af)(m)=f(\psi (a)m)$. It is a finite graded $B$-module, reverses internal shifts, and $D_{\psi }^{2}M\cong M$.

Projective Hom duality. For a finite graded PROJECTIVE $P$ define instead $P^{\mathrm{\#}}:={\mathrm{HOM}}_B(P,B{)}^{\psi },$ with the $\psi$-twisted $B$-action. This is again a finite graded projective, it reverses internal shifts, and $(P^{\mathrm{\#}}{)}^{\mathrm{\#}}\cong P$ on finite graded projectives (Finite graded projective modules).

They are different dualities. Hom duality is not applied to a nonprojective simple to produce its cover; $D_{\psi }$ and $\mathrm{\#}$ differ already in rank one, where $B=k[x]\mathrm{/}{x}^{\lambda }$ has $B^{\mathrm{\#}}=B$ while $D_{\psi }B\cong B\{-2d_i(\lambda -1)\}$.

Normalizations used in the based statements. By Cyclotomic projective covers and dualities descend through the quotient, each simple shift orbit has a unique $D_{\psi }$-self-dual representative $S$, and its unique normalized cover $P$ with ${\mathrm{Hom}}^0(P,S)=k$ is $\mathrm{\#}$-self-dual; these normalized representatives are the ones used in the based statements below. The symmetric theorem Cyclotomic KLR blocks are finite-dimensional graded symmetric algebras additionally gives $D_{\psi }P\cong P^{\mathrm{\#}}\{-2{\mathrm{def}}_{\mathrm{\Lambda }}(\beta )\}$; it does not change the canonical projective normalization. No self-dual shift is asserted for every indecomposable finite module, and no choice is used.