Block Stanley Decompositions II. Greedy Algorithms, Applications, and Open Problems

Abstract

Stanley decompositions are used in applied mathematics (dynamical systems) and \(\mathfrak {sl}_2\) invariant theory as finite descriptions of the set of standard monomials of a monomial ideal. The block notation for Stanley decompositions has proved itself in this context as a shorter notation and one that is useful in formulating algorithms such as the “box product”. Since the box product appears only in dynamical systems literature, we sketch its purpose and the role of block notation in this application. Then we present a greedy algorithm that produces incompressible block decompositions (called “organized”) from the monomial ideal; these are desirable for their likely brevity. Several open problems are proposed. We also continue to simplify the statement of the Soleyman–Jahan condition for a Stanley decomposition to be prime (come from a prime filtration) and for a block decomposition to be subprime, and present a greedy algorithm to produce “stacked decompositions”, which are subprime.