Frobenius forms and semisimplicity of cyclotomic quiver Hecke superalgebras

We determine the Nakayama automorphism of the canonical homogeneous Frobenius form on cyclotomic quiver Hecke superalgebras. We give explicit necessary and sufficient conditions on the dominant weight and the rank for semisimplicity of cyclotomic quiver Hecke superalgebras of types $A_e^{(1)}$, $C_e^{(1)}$, $A_{2e}^{(2)}$, $D_{e+1}^{(2)}$, $A_\infty$, $B_\infty$, and $C_\infty$. As an application, we prove that a cyclotomic Hecke–Clifford superalgebra is semisimple if and only if the corresponding Poincaré polynomial is nonzero.