Bases of quasi-hereditary covers of diagram algebras. (Q2842336)

This paper concerns quasi-hereditary covers of finite-dimensional algebras, in particular of those defined by diagram calculus, such as the Brauer algebra and the partition algebra. The classic example of such a quasi-hereditary cover is the Schur algebra as a cover of the group algebra of the symmetric group, and the idea here is to modify the tableau combinatorics used in that case to construct bases for permutation modules and a cellular basis for the Schur algebra; the author carries this out for the Brauer algebra, the walled Brauer algebra and the partition algebra, developing the appropriate notion of a row-standard tableau in each case and analysing homomorphisms between permutation modules.NEWLINENEWLINE This paper is very well written; in particular, the introduction is very readable and gives a clear overview of the setting. The background information is selected carefully, and the theory for the Schur algebra is recalled in clear detail as a model for the other algebras. There are a few mistakes (``principle'' instead of ``principal'', some confusion between left and right actions, and (as is obligatory in this area) confusion between ``tableau'' and ``tableaux''), but these do not detract from an excellent paper.