TY - JOUR T1 - Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring AU - Chen , Zhichao AU - Cai , Jiayi AU - Meng , Lingchao AU - Li , Libin JO - Journal of Mathematical Study VL - 4 SP - 357 EP - 370 PY - 2021 DA - 2021/06 SN - 54 DO - http://doi.org/10.4208/jms.v54n4.21.02 UR - https://global-sci.org/intro/article_detail/jms/19288.html KW - Non-negative integer, matrix representation, irreducible $\mathbb{Z}_{+}$-module, $\mathbb{Z}_{+}$-ring. AB -

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.