On a Conjecture of R. P. Stanley; Part I - Monomial Ideals
Joachim Apel
In 1982 Richard P. Stanley conjectured that a graded module M over a graded k-algebra R can be decomposed in a particular way in a direct sum of finitely many free modules over suitable subalgebras of R. Besides homogeneity conditions the most important restriction such a decomposition has to satisfy is that the subalgebras must have at least dimension depth(M). We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will show that any monomial ideal of depth at most 2, any monomial ideal in at most 3 variables, and any monomial ideal which is generic with respect to one variable possesses an involutive basis providing a Stanley decomposition of the ideal.