@article{Wu2024, 
author = {Wenxia Wu and Yunnan Li},
title = {Classification of irreducible based modules over the complex representation ring of        S    4},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {7},
pages = {19859-19887},
keywords = {Z+-module, Z+-ring, representation ring, module category, symmetric group},
url = {https://www.sciopen.com/article/10.3934/math.2024970},
doi = {10.3934/math.2024970},
abstract = {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  (      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        R    e    p    (      S    4    ) of        S    4   arisen from projective representations of certain subgroups of        S    4  .}
}