Extended \(G\)-vertex colored partition algebras as centralizer algebras of symmetric groups

Abstract

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].