Cellular algebras have been introduced by J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] in order to axiomatize certain familiar properties from representation theory of symmetric groups and their Hecke algebras, e.g., the existence of ‘Specht modules’. They have found cellular structures also for several algebras ‘defined by diagrams’ like Brauer algebras \(D_n\) and Temperley-Lieb algebras \(T_n\).
The paper under review deals with larger algebras, the partition algebras \(P_n\), which also have bases consisting of diagrams and which contain the algebras \(D_n\) and \(T_n\) as subalgebras. The main result states that partition algebras are cellular. Moreover, the cellular structure of \(P_n\) restricts to that of \(D_n\) and of \(T_n\).
The proof follows a strategy developed by S. König and C. C. Xi [J. Lond. Math. Soc., II. Ser. 60, No. 3, 700-722 (1999)], which in fact yields a stronger assertion, relating the algebra \(P_n\) with the group algebras of several symmetric groups.
A stronger condition than ‘cellular’ is ‘quasi-hereditary’. In this context, the author gives a sufficient condition for \(P_n\) to be quasi-hereditary. This condition has been shown by S. König and the author [in Math. Ann. 315, No. 2, 281-293 (1999; see Zbl 0939.16003 above)] to be necessary as well.