The word-mirror involution of a KLR algebra and the quantum star

Relation check. Assign to each generator its mirror image $\rho$; the idempotents map to idempotents, dots to dots with reversed positions, and crossings to minus crossings with reversed positions, all of the same degrees because dot colors are mirrored and $b_{ij}=b_{ji}$. The quadratic relation uses (Q2): $Q_{ij}(u,v)=Q_{ji}(v,u)$, and the two crossing signs cancel in the square, so $\rho ({\tau }_a^2-Q_{{\nu }_a{\nu }_{a+1}})=0$. The two mixed equal-color constants exchange under reversal and the crossing minus corrects their sign; the three-crossing braid words exchange positions, and their odd crossing sign together with the reversal of numerator and denominator gives the actual native divided-braid correction. Far commutativity, dot commutativity and the dot slides map to the same relations with mirrored indices, and applying $\rho$ twice fixes every generator, so $\rho$ is a degree-preserving algebra automorphism with ${\rho }^2=1$; since the generator-fixing anti-involution $\psi$ fixes all generators and reverses multiplication while $\rho$ is a homomorphism, $\rho \psi =\psi \rho$.

Module twist and dualities. Twisting left modules by $a\cdot m=\rho (a)m$ is an equivalence of graded module categories, since $\rho$ is an automorphism. Finite vector duality $M\mapsto M^{\ast }$ satisfies $(M^{\ast }{)}^{\ast }=M$ and the twist commutes with it because the dual action is built from $\rho (a)$ applied to the same finite-dimensional spaces; projective Hom duality $P\mapsto {\mathrm{HOM}}_B(P,B)$ likewise commutes, as the evaluation pairings are preserved. The separated-block formula is ${\rho }^{\ast }\mathrm{Ind}(M,N)\cong \mathrm{Ind}({\rho }^{\ast }N,{\rho }^{\ast }M)$ without a shift: reflecting the block embedding exchanges the $\beta$ and $\gamma$ blocks and the tensor-factor flip gives the reflected induced module, with the same degrees because the reflection uses the equal-degree assignment.

The induced quantum star. Under the categorification $\gamma$ of [F2] the word projective of $\nu$ represents $F_{{\nu }_1}\cdots F_{{\nu }_n}$, so the reflection equivalence acts on $K_0(R)$ by reversing the order of multiplication of the divided generators and fixing each $F_i$ and $q$; this is exactly the quantum multiplication-reversing star anti-automorphism $\ast$.

The cyclotomic generator. Applying $\rho$ to the first-strand cyclotomic generator $a_{{\nu }_1}(x_1)e(\nu )$ gives $a_{{\nu }_1}(x_n)e(\mathrm{rev}\nu )$, with the SAME original first color, so $\rho$ identifies the first-strand and last-strand cyclotomic quotients; preservation of the first ideal itself is not asserted in general, since a monic polynomial in the first dot is moved to the last dot unless the polynomial is constant. All computations are finite and choice-free.