论文标题
在移动的伯恩赛德比塞特函数的基本代数上
On the essential algebra of the shifted Burnside biset functor
论文作者
论文摘要
我们描述了Burnside Biset函数的基本代数，$ \ widehat {kb_t}（g）$，在两种情况下，$ t $ t $ the oft $ t $移动。首先，当$ g $和$ t $都是有限的阿贝尔集团时，$ k $是一个特征$ 0 $的领域。在这种情况下，$ \ wideHat {kb_t}（g）$是对移位星代数的商的同构，该代数是根据$ g \ times g \ times g \ times t $的子组定义的。第二种情况是$ g $和$ t $是满足$（| g |，| t |）= 1 $而$ k $的任何有限组是通勤的统一戒指。在这种情况下，$ \ wideHat {kb_t}（g）$与$ out（g）$和$ kb^{z（g）}（t）$的半领产品同构，是$ t $的$ t $的单个burnside ring，带有$ z（g）$的系数。 本文的目的是考虑$ \ wideHat {kb_t}（g）$的自然发电机集，来自$ kb_t（g \ times g）$中的瞬态元素，并探索某些情况，在此组合中，可以为$ \ wideHat {kb_t}（g）提供基础。
We describe the essential algebra, $\widehat{kB_T}(G)$, of the Burnside biset functor shifted by a group $T$, at a group $G$, in two cases. First, when $G$ and $T$ are both finite abelian groups and $k$ is a field of characteristic $0$. In this case, $\widehat{kB_T}(G)$ is isomorphic to a quotient of the shifted star algebra, which is defined in terms of the subgroups of $G\times G\times T$. The second case is when $G$ and $T$ are any finite groups satisfying $(|G|, |T|)=1$ and $k$ is a commutative unitary ring. In this case, $\widehat{kB_T}(G)$ is isomorphic to a semidirect product of $Out(G)$ and $kB^{Z(G)}(T)$, the monomial Burnside ring of $T$ with coefficients in $Z(G)$. The aim of the article is to consider the natural set of generators of $\widehat{kB_T}(G)$ coming from the transitive elements in $kB_T(G\times G)$ and explore some cases in which it is possible to give a basis for $\widehat{kB_T}(G)$ in this set.
