A study for the categorical representation theory for finite groups and algebra

(1)对于有限群和群的某个子群族，导出一个范畴和一个交换环。这个环称为广义伯恩赛德环。到目前为止，只有少数几个例子，这样的家庭的子群已被审查。在系数环是有理数在任意素数处的局部化的情况下，我们引进了一个新的族。该族由某些p-根式子群的正规化子组成。研究了环的同态核和单位及本原幂等元.证明了环的乘法单位所给出的广义特征标的次数与群的子群族的阶复形的Euler特征标一致。(2)利用Bouc和Thevenaz关于p-偏集函子的一个定理，建立了p-群的Dade群、交叉伯恩赛德环和Drinfeld双的有理表示环之间的关系. (3)通过对伯恩赛德Tambara函子的Dress构造，给出了有限群的交叉伯恩赛德环的构造.

(1) For a finite group and a certain family of subgroups of the group induce a categoryand a commutative ring. The ring is called generalized Burnside ring. So far only afew examples of such families of subgroups have been examined. We introduced anew family in the case when the coefficient ring is the localization of the rationalintegers at any prime. The family consists of normalizers of certain p-radical subgroups. We investigate the kernel of certain homomorphism and the units andprimitive idempotents of the ring. It is shown that the degree of the generalizedcharacter afforded by the multiplicative identity (unit) of the ring coincides withthe Euler characteristic of the order complex of the family of subgroups of the group.(2) By using a theorem of p-biset functor obtained by Bouc and Thevenaz, we have arelationship between the Dade group, crossed Burnside ring and rational representation ring of Drinfeld double for a p-group.(3) We described the construction of the crossed Burnside ring for a finite group via theDress construction applied to the Burnside Tambara functor.