A new ``compressed straight'' basis for the polynomial ring Z[t i, j ] is constructed. This basis is then employed to study the lattice of representations generated by the representations associated with row-convex shapes under the operations of intersection and linear span. Applications to ascertaining the Cohen Macaulay property for rings associated to elements of this lattice are also given. 2000 Academic Press 1. INTRODUCTION This paper is devoted to developing the algebra and combinatorics associated to tableaux, like 3 6 8 , 8 2 4 whose shapes are more general than the Ferrer's diagram of a (skew) partition. While the skew shapes have a long history, closely related to the semistandard Young tableaux (see, e.g., [Sa91]) these generalized shapes were introduced by Akin, Buchsbaum, and Weyman in [ABW82]. Recently, they have received considerable attention from the point of view of combinatorics and representation theory by Reiner and Shimozono [RS95, RS96, RS96c, S96] and from a more geometric viewpoint by Magyar and Lakshmibai [M94, LM96]. Elements of the class of row-convex shapes, which this paper focuses on, first appeared in the work of Akin and Buchsbaum [AB85] on building