#### Vol. 12, No. 1, 2021

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Equivariant Hilbert series for hierarchical Models

### Aida Maraj and Uwe Nagel

Vol. 12 (2021), No. 1, 21–42
##### 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.

##### Keywords
hierarchical model, invariant filtration, equivariant Hilbert series, finite automaton, regular language
##### Mathematical Subject Classification 2010
Primary: 05A15, 13P25, 68W30
##### Milestones
Received: 4 November 2019
Revised: 31 July 2020
Accepted: 31 August 2020
Published: 9 April 2021
##### Authors
 Aida Maraj Max Planck Institute for Mathematics in the Sciences Leipzig Germany Uwe Nagel Department of Mathematics University of Kentucky Lexington, KY United States