h-vectors of Gorenstein* simplicial posets

As is well known, h-vectors of simple (or simplicial) convex polytopes are characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations and some other inequalities. Simple convex polytopes determine Gorenstein* simplicial posets and h-vectors are defined for simplicial posets. It is known that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville equations and that every component in the h-vectors must be non-negative. In this paper we will show that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley and complete characterization of those h-vectors. Our proof is purely algebraic but the idea of the proof stems from topology.Comment: 12 page

Normalized Weyl-type ⋆\star-product on K\"ahler manifolds

We define a normalized Weyl-type $\star$-product on general K\"{a}hler manifolds. Expanding this product perturbatively we show that the cumbersome term, which appears in a Berezin-type product, does not appear at least in the first order of $\hbar$. This means a normalization factor, which is introduced by Reshetikhin and Takhtajan for a Berezin-type product, is unnecessary for our Weyl-type product at that order.Comment: to be published in Mod. Phys. Lett.