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 $ \mathbb{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(S_4) $ of the symmetric group $ S_4 $. In total, 16 inequivalent irreducible based modules were obtained. In this process, the MATLAB program was used in order to obtain some representation matrices. Based on such a classification result, we further discuss the categorification of based modules over $ r(S_4) $ by module categories over the complex representation category $ {\rm Rep}(S_4) $ of $ S_4 $ arisen from projective representations of certain subgroups of $ S_4 $.
Citation: Wenxia Wu, Yunnan Li. Classification of irreducible based modules over the complex representation ring of $ S_4 $[J]. AIMS Mathematics, 2024, 9(7): 19859-19887. doi: 10.3934/math.2024970
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 $ \mathbb{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(S_4) $ of the symmetric group $ S_4 $. In total, 16 inequivalent irreducible based modules were obtained. In this process, the MATLAB program was used in order to obtain some representation matrices. Based on such a classification result, we further discuss the categorification of based modules over $ r(S_4) $ by module categories over the complex representation category $ {\rm Rep}(S_4) $ of $ S_4 $ arisen from projective representations of certain subgroups of $ S_4 $.
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Wenxia Wu, Yunnan Li. Classification of irreducible based modules over the complex representation ring of $ S_4 $[J]. AIMS Mathematics, 2024, 9(7): 19859-19887. doi: 10.3934/math.2024970
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