The representation theory of the wreath product of a finite group with the monoid of all partial functions on a finite set as an EI-category algebra

Let $G$ be a finite group. We provide a description of the ordinary quiver of the complex monoid algebra of the wreath product $G\wr {\mathrm{P}\mathrm{T}}_n$, where ${\mathrm{P}\mathrm{T}}_n$ denotes the monoid of all partial functions on an $n$-element set. This description depends on the multiplicities of simple $G$-modules appearing in the decomposition of tensor products of simple $G$-modules. We also prove that the global dimension of this algebra is $n-1$. Both results are obtained by analyzing the associated Ehresmann EI-category related to the monoid. Finally, we describe the quiver of the algebra of the wreath product of $G$ with the submonoid of all order-preserving partial functions.