Representations of the quasi-partition algebras

The quasi-partition algebras were introduced by Daugherty and the first author as centralizers of the symmetric group. In this article, we give a more general definition of these algebras and give a construction of their simple modules. In addition, we introduce two new algebras, we give linear bases and show that for specializations of their parameters, these new algebras are isomorphic to centralizer algebras. We provide a generalized Bratteli diagram that illustrates how the representation theory of the three algebras discussed in this paper are related. Moreover, we give combinatorial formulas for the dimensions of the simple modules of these algebras.