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.
| Original language | English |
|---|---|
| Pages (from-to) | 21-42 |
| Number of pages | 22 |
| Journal | Algebraic Statistics |
| Volume | 12 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2021 |
Bibliographical note
Publisher Copyright:© 2021, Mathematical Sciences Publishers. All rights reserved.
Funding
We would like to thank the anonymous referee for helpful comments. Both authors were partially supported by Simons Foundation grants #317096 and #636513. MSC2010: 05A15, 13P25, 68W30. Keywords: hierarchical model, invariant filtration, equivariant Hilbert series, finite automaton, regular language.
| Funders | Funder number |
|---|---|
| Simons Foundation | 68W30, MSC2010, 636513, 317096, 13P25 |
Keywords
- equivariant Hilbert series
- finite automaton
- hierarchical model
- invariant filtration
- regular language
ASJC Scopus subject areas
- Algebra and Number Theory
- Applied Mathematics
- Statistics and Probability