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

Abstract

The Partition algebras Pk(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 Pbk(x,G), which contain partition algebras Pk(x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra Pbk(n,G) is the centralizer algebra of an action of the symmetric group Sn on tensor space W⊗k , where W = C n|G| . Further we show that these algebras Pbk(x,G) contain as subalgebras the “G-Vertex Colored Partition Algebras Pk(x,G)," introduced in [PK1].