Order-preserving maps from a poset to a chain, the order polytope, and the Todd class of the associated toric variety.
The paper is concerned with three objects: a finite poset \(Q\), its order polytope \(P\), and the toric variety \(X\) associated to the outer normal fan of \(P\). It is known that the number \(| [n+1]^Q| \) of order-preserving maps from \(Q\) to the chain of \(n+1\) elements is equal to the number of lattice points in the \(n\)-fold dilation of \(P\). The main result of the paper is a new formula for \(| [n+1]^Q| \), obtained from this equality by applying techniques for counting lattice points. This formula could also be obtained if one knew that a certain cycle represents the Todd class of \(X\). The author conjectures that this is always the case for toric varieties arising from posets, and he proves the conjecture under the assumption that \(Q\) is a chain.
scientific article; zbMATH DE number 2003947