THE h-VECTOR OF A LADDER DETERMINANTAL RING COGENERATED BY 2× 2 MINORS IS LOG-CONCAVE

The sequence h0, h1, . . . , hs is called the h-vector of the ring. Orginally the question was to decide whether a given sequence can arise as the h-vector of some ring. In this sense the validity of the conjecture would imply that log-concavity was a necessary condition on the h-vector. It is now known however [12, 3] that Stanley’s conjecture is not true in general. Several natural weakenings have been considered, but are still open. For example, Aldo Conca and Jürgen Herzog conjectured that the h-vector would be log-concave for the special case where R is a ladder determinantal ring. (Note that ladder determinantal rings are Cohen-Macaulay, as was shown in [8, Corollary 4.10], but not necessarily Gorenstein.) We will prove the conjecture of Conca and Herzog in the simplest case, i.e., where R is a ladder determinantal ring cogenerated by 2× 2 minors, see Corollary 4.6. In the case of ladder determinantal rings the h-vector has a nice combinatorial interpretation. This follows from work of Abhyankar and Kulkarni [1, 2, 10, 11], Bruns, Conca, Herzog, and Trung [4, 5, 6, 8]. In the following paragraphs, which are taken almost verbatim from [9], we will explain these matters.