Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d N = 2

In 3d $\N=2$ gauge theories, mirror symmetry as Fourier transformation of sphere partition functions can be used to check and derive dualities. In this notes, we study Abelian 3d $N=2$ theories with mixed Chern-Simons levels, in particular $(U(1)-[1])^{N}_{k_{ij}}$, which is $N$ copies of $U(1)-[1]$ coupled together by mixed Chern-Simons levels $k_{ij}$. It turns out that mirror symmetry gives rise to many mirror dual theories with different mixed CS levels and FI parameters, and mirror transformations form a finite group with nice mathematical structure. We also discuss the mirror transformations of $U(1)_k- [N] $ theories with brane constructions. The results obtained match with vortex partition functions calculated from open topological strings.