The monoidal structure of the category of partial representations of finite groups
Arthur R. Alves Neto, Eliezer Batista, Javier Méndez
TL;DR
The paper studies the category of partial representations of a finite group $G$ as a multifusion monoidal category, linking it to subgroup representations through the Christmas Tree Theorem and, for abelian $G$, to subgroups via the Matryoshka Theorem. It shows that the partial group algebra $\Bbbk_{par}G$ is isomorphic to the groupoid algebra $\Bbbk \Gamma(G)$, hence carrying a Hopf algebroid structure; consequently, ${}_{\Bbbk_{par}G}\mathcal{M}$ is semisimple and its simple objects are $M_{(X,\alpha)}=M_X\otimes \mathbb{V}^{(\alpha)}$ with $X\in \mathcal{P}_e(G)$ and $\mathbb{V}^{(\alpha)}$ an irreducible of the isotropy subgroup $G_X$. Tensor products are computed as balanced tensor products over the base algebra $A_{par}$, yielding a decomposition into matrix blocks $M_n(\Bbbk G_X)$ with $n=|X|/|G_X|$. The Christmas Tree and Matryoshka theorems provide strong monoidal functors embedding subgroup representations into the partial representation category of $G$, revealing how global subgroup representations are encoded within partial representations of the whole group. This perspective suggests connections to module categories over fusion categories and indicates directions for generalizing to broader classes of abelian groups and presentations.
Abstract
In this work, we analyze the structure of the category of partial representations of a finite group $G$ as a multifusion category, providing an alternative way to describe simple objects and their tensor products. We describe the interconnection between the category of partial representations of a finite group and the category of global representations of its subgroups (the Christmas Tree's Theorem). Also, for a finite abelian group $G$, we prove that the category of partial representations of any of its subgroups can be embedded into the category of partial representations of $G$ (the Matryoshka's Theorem).