A new \compressed straight" basis for the polynomial ring Z[ti;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.

Let K be a field and X an m×n matrix of indeterminates over K. Let K[X] denote the polynomial ring generated by all the indeterminates Xij . For a given positive integer r ≤ min{m, n}, we consider the determinantal ideal Ir+1 = Ir+1(X) generated by all r + 1 minors of X if r < min{m, n} and Ir+1 = (0) otherwise. Let Rr+1 = Rr+1(X) be the determinantal ring K[X]/Ir+1. Determinantal ideals and ri...