The structure of the partition algebras (Q1921914)

Some topics concerning the structure of the partition algebras \(P_n(Q)\) (Martin, 1990, 1994) for a complex number \(Q\) and natural number \(n\) being a generalization both of the Brauer algebra \(D_n(Q)\) (1973) and of the Temperley-Lieb algebra \(T_n(Q)\) (1971) are considered. It is shown that the algebras \(P_n(Q)\) are quasi-hereditary (in the sense of Cline et al, 1988; Dlab, Ringel, 1989) for \(Q\neq 0\). On this basis, for every partition \(\lambda\) a Weyl module of \(P_n(Q)\) is introduced which is irreducible if \(Q\) is generic and which has a basis independent of \(Q\). The description of the structure of this Weyl module for every \(Q\) and, hence, of the structure of indecomposable projective modules is given. The work is treated as a concrete application of some category theory ideas (Green, 1980).