Some remarks on algebras with straightening laws
The author considers the following problem. Given a finite poset (partially ordered set) \(H\), which ASL's (algebra with straightening law) exist on \(H\)? A moduli space for the ASL's on \(H\) is constructed as an algebraic scheme over an algebraically closed field \(k\) whose closed points correspond to ASL's on \(H\) over \(k\). If \(\text{rank} H = 2\) (the rank is the maximal number of elements in a chain in \(H)\), the moduli space is given quite explicitly with equations. Let \(k[H]\) be the discrete ASL on \(H\), i.e. the Stanley-Reisner ring where all ``right hand sides'' of the straightening relations are 0. Then it is shown that for any ASL on \(H\) the inequality \(\text{depth} k [H] \leq \text{depth} A \leq \text{depth} k[H] + 1\). \textit{Hibi} has conjectured: If there is an ASL on \(H\) which is an integral domain, then \(k[H]\) is Cohen-Macaulay. A counterexample is given, but the conjecture is proved in some special cases.
scientific article; zbMATH DE number 643794
scientific article