Dress induction and the Burnside quotient Green ring

Prime ideals of the Burnside ring of a saturated fusion system

In a 1969 article, A. Dress described the prime ideals of the Burnside ring of a finite group G and the inclusion relations between them. One may ask whether similar results exist for the Burnside ring of a saturated fusion system F on a finite p-group S. The present paper answers positively, providing a typology for the prime ideals of A(F) and for those of A(F) (p) as a consequence.

支架与拟群的Burnside环

Burnside rings for racks and quandles

本文利用Burnside环重构有限支架与拟群的分类理论，建立加法基、乘法生成元及标记理论，为分类提供新框架并加强与代数的联系。

关于Burnside环的全幂运算的像

On the image of the total power operation for Burnside rings

该研究证明Burnside环全幂运算的像落在组合子环内，构造了交换分次环并给出特征公式，推广了Frobenius–Wielandt同态和Burnside轨道计数引理。

Monomial $G$-posets and their Lefschetz invariants

Let $G$ be a finite group, and $C$ be an abelian group. We introduce the notions of $C$-monomial $G$-sets and $C$-monomial $G$-posets, and state some of their categorical properties. This gives in particular a new description of the $C$-monomial Burnside ring $B_C(G)$. We also introduce Lefschetz invariants of $C$-monomial $G$-posets, which are elements of $B_C(G)$. These invariants allow for a definition of a generalized tensor induction multiplicative map $\mathcal{T}_{U,λ}: B_C(G)\to B_C(H)$ associated to any $C$-monomial $(G,H)$-biset $(U,λ)$, which in turn gives a group homomorphism $B_C(…

Rational $p$-biset functors

In this paper, I give several characterizations of {\em rational biset functors over $p$-groups}, which are independent of the knowledge of genetic bases for $p$-groups. I also introduce a construction of new biset functors from known ones, which is similar to the Yoneda construction for representable functors, and to the Dress construction for Mackey functors, and I show that this construction preserves the class of rational $p$-biset functors.\par This leads to a characterization of rational $p$-biset functors as additive functors from a specific quotient category of the biset category to ab…