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

(1)对于有限群和群的某一族子群，诱导出范畴和交换环。这种环称为广义Burnside环。到目前为止，只研究了几个这样的子群族的例子。当系数环是有理整数在任意素数上的局部化时，我们引入了一个新的族。这个族由某些p-根子群的正规化子组成。我们研究了某些同态的核以及环的单位和本原幂等元。(2)利用Bouc和Theveraz得到的p-Biset函子定理，得到了p-群的Dade群、交叉Burnside环和Drinfeld二重环的有理表示环之间的关系.(3)利用Burnside-Tambara函子的Dress构造刻画了有限群的交叉Burnside环.

(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.