The Classification Of The Irreducible Z₊- Modules Over Near-group Fusion Ring K（Q₈,n）

The near-group categories,which are an important class of fusion categories.The concept of a near-group fusion ring comes from the Grothendick rings of fusion categories.J.Siehler first introduced the concept of the near-group categories.Ostrik provided the number of an non-equivalence Z+-modules that are irreducible is limited over a Z+-ring A of finite rank.In this paper,we utilize the classification of Z+modules that are irreducible over a near-group fusion ring and give all irreducible Z+modules over the near-group fusion ring K（Q8,n）,Q8 is called dihedral group of order 8 which is composed of i and j.The paper is organized as follows:In the introduction part,we introduce the research background,research method,current status of research and the result of research.In Chapter 1,we state the exact definition of Z+-ring and Z+-module.The second part introduce decision theorem for a Z+-module over a Z+-ring of finite rank to be irreducible;The third part provide the range of the ranks of irreducible Z+-module over a near-group fusion ring,a decision theorem for judging a near-group fusion ring is integeral or non-integeral and a theorem of Frobenius-Perron for positive matrix.The forth part we introduce the number of direct summands in the decomposition of an irreducible Z+-module over a near-group fusion ring.The last part we provide the general method of the classification of irreducible Z+-module over near-group fusion ring.we not only provide a classification method for irreducibl Z+-module over an non-integeral fusion ring;but also give a classification method for irreducible Z+-module over an integeral fusion ring.In Chapter 2,we explicitly classify the irreducible Z+-modules over K（Q8,n）.Dicussing K（Q8,n）is integeral near-group fusion ring when n=2 or n=7,we find that n=2 the number of direct summands less than or equal to 3 and n=7 the number of direct summands less than or equal to 9.By calculation,we know K（Q8,2）has 87 irreducible Z+-modules and K（Q8,7）has 277 irreducible Z+-modules.In Chapter 3,according to classification theorem we know when n≠2,7 the number of direct summands equal to 2,then we give the construction method of irreducible Z+-modules over the near-group fusion rings K（Q8,n）when n≠2,7.By give n and substitute in Theorem we can obtain the specific irreducible Z+-modules;when n=0,1,by calculation,we get K（Q8,0）has 23 irreducible Z+-modules and K（Q8,1）has 16 irreducible Z+-modules.