Green fields

The center of a Green biset functor

For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of $A$-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group.

关于某些Green双集函子的分离性

On the separability of some Green biset functors

研究证明复字符的Green双集函子在Z上不分离，而移位后的Burnside双集函子分离性取决于群阶在环中可逆性。

Some simple biset functors

Let $p$ be a prime number, let $H$ be a finite $p$-group, and let $\mathbb{F}$ be a field of characteristic 0, considered as a trivial $\mathbb{F} \mathrm{Out}(H)$-module. The main result of this paper gives the dimension of the evaluation $S_{H,\mathbb{F}}(G)$ of the simple biset functor $S_{H,\mathbb{F}}$ at an arbitrary finite group $G$. A closely related result is proved in the last section: for each prime number $p$, a Green biset functor $E_p$ is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation $E_p(G)$ is a free abelian group of rank equal …

范畴的双集函子

Biset functors for categories

本文将群上的双集函子推广到任意有限范畴，建立 Burnside 环、简单函子、幺半结构与纤维化等理论，并通过 Boolean 格联系对应函子。

Idempotents of double Burnside algebras, L-enriched bisets, and decomposition of p-biset functors

Let R be a (unital) commutative ring, and G be a finite group with order invertible in R. We introduce new idempotents in the double Burnside algebra RB(G,G), indexed by conjugacy classes of minimal sections of G, i.e. pairs (T,S) of subgroups of G, where S is a normal subgroup of T contained in the Frattini subgroup of T. These idempotents are orthogonal, and their sum is equal to the identity. It follows that for any biset functor F over R, the evaluation F (G) splits as a direct sum of specific R-modules indexed by minimal sections of G, up to conjugation. The restriction of these construct…

Shifted functors of linear representations are semisimple

We prove that, for any fields $k$ and $\mathbb{F}$ of characteristic $0$ and any finite group $T$, the category of modules over the shifted Green biset functor $(kR_{\mathbb{F}})_T$ is semisimple.