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.

Let A be a finite-dimensional Frobenius cellular algebra with cell datum (Λ,M,C,i). Take a nondegenerate bilinear form f on A. In this paper, we study the relationship among i, f and a certain Nakayama automorphism α. In particular, we prove that the matrix associated with α with respect to the cellular basis is uni-triangular under a certain condition.

Let R be an integral domain and A a cellular algebra over R with a cellular basis {CλS,T∣λ∈Λ and S,T∈M(λ)}. Suppose that A is equipped with a family of Jucys–Murphy elements which satisfy the separation condition in the sense of Mathas [‘Seminormal forms and Gram determinants for cellular algebras’, J. reine angew. Math.619 (2008), 141–173, with an appendix by M. Soriano]. Let K be the field of fractions of R and AK=A⨂ RK. We give a necessary and sufficient condition under which the centre of AK consists of the symmetric polynomials in Jucys–Murphy elements. We also give an application of our result to Ariki–Koike algebras.

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.