Equivariant Hilbert series for hierarchical Models

Abstract

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say, m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

Shift Invariant Algebras, Segre Products and Regular Languages

“…Some further instances in which the Segre product of two regular languages is regular have been established in [9]. In fact, this is always true as we show now.…”

“…Further work is needed. For example, information on equivariant Hilbert series of filtrations determined by hierarchical models as investigated in [9] would be of interest. Note that we did not analyze the structure of the used finite automata in this paper.…”

“…Our main motivation is to address this problem by introducing a new method. The starting point is a rationality result on equivariant Hilbert series of some hierarchical models in algebraic statistics in [9]. In this paper we generalize this approach considerably.…”

Nonlinear algebra and applications

“…Lastly, the need of algebraic statistics to advance statistical questions has led to invention and development of tools in algebra; Huh's work [123] on rational maximum likelihood estimates produced an interesting classification of varieties, and work of Hillar and Sullivant [117] on finitely generated models up to a symmetric group action spawned the non-Noetherian theory in commutative algebra; see, e.g., [73,141,164,176,177].…”

Nonlinear Algebra and Applications

“…Lastly, the need of algebraic statistics to advance statistical questions has lead to invention and development of tools in algebra; Huh's work [117] on rational maximum likelihood estimates produced an interesting classification of varieties, and work of Hillar and Sullivant [111] on finitely generated models up to a symmetric group action spawned the non-Notherian theory in commutative algebra [71,135,157,168,169].…”