Extended $G$-vertex colored partition algebras as centralizer algebras of symmetric groups

The Partition algebras $P_k(x)$ have been defined in [M1] and [Jo]. We introduce a new class of algebras for every group $G$ called "Extended $G$-Vertex Colored Partition Algebras," denoted by $\widehat{P}_{k}(x,G)$, which contain partition algebras $P_k(x)$, as subalgebras. We generalized Jones result by showing that for a finite group $G$, the algebra $\widehat{P}_{k}(n,G)$ is the centralizer algebra of an action of the symmetric group $S_n$ on tensor space $W^{\otimes k}$, where $W=\mathbb{C}^{n|G|}$. Further we show that these algebras $\widehat{P}_{k}(x,G)$ contain as subalgebras the "$G$-Vertex Colored Partition Algebras ${P_{k}(x,G)}$," introduced in [PK1].