  We present an abstract framework for the axiomatic study of diagram algebras.
Algebras that fit this framework possess analogues of both the Murphy and
seminormal bases of the Hecke algebras of the symmetric groups. We show that
the transition matrix between these bases is dominance unitriangular. We
construct analogues of the skew Specht modules in this setting. This allows us
to propose a natural tableaux theoretic framework in which to study the
infamous Kronecker problem.
