The Equivariant Hilbert Series of Hierarchical Models

A hierarchical model is realizable by a simplicial complex that describes the dependency relationships among random variables and the number of states of each random variable. Diaconis and Sturmfels have constructed toric ideals that provide useful information about the model. This talk concerns quantitative properties for families of ideals arising from hierarchical models with the same dependency relations and varying number of states. We introduce and study invariant filtrations of such ideals, and their equivariant Hilbert series. A condition that guarantees this multivariate series is a rational function will be presented. The key is to construct finite automata that recognize languages corresponding to invariant filtrations. Lastly, we show that one can similarly prove the rationality of an equivariant Hilbert series for some filtrations of algebras. This is joint work with Uwe Nagel.