## Abstract

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize.

Original language | English |
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Pages (from-to) | 236-266 |

Number of pages | 31 |

Journal | Journal of Algebra |

Volume | 631 |

DOIs | |

State | Published - Oct 1 2023 |

### Bibliographical note

Funding Information:The second author was partially supported by Simons Foundation grant #636513.

Publisher Copyright:

© 2023 Elsevier Inc.

## Keywords

- Equivariant Hilbert series
- Finite automata
- Polynomial rings in infinitely many variables
- Regular languages
- Representation stability
- Segre languages
- Segre products
- Shift invariant algebras

## ASJC Scopus subject areas

- Algebra and Number Theory