Categorification of based modules over the complex representation ring of S₄

The complex representation rings of finite groups are the fundamental class of fusion rings, categorified by the corresponding fusion categories of complex representations. The category of Z₊-modules of finite rank over such a representation ring is also semisimple. In this paper, we classify the irreducible based modules of rank up to 5 over the complex representation ring r(S₄) of the symmetric group S₄. Totally 16 inequivalent irreducible based modules are obtained. Based on such a classification result, we further discuss the categorification of based modules over r(S₄) by module categories over the complex representation category Rep(S₄) of S₄ arisen from projective representations of certain subgroups of S₄.