Perturbation method for determining the group of invariance of hierarchical models

Tomonari Sei, Satoshi Aoki, Akimichi Takemura

Research output: Contribution to journalArticlepeer-review


We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342-358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.

Original languageEnglish
Pages (from-to)375-389
Number of pages15
JournalAdvances in Applied Mathematics
Issue number4
Publication statusPublished - 2009 Oct
Externally publishedYes


  • Computational algebraic statistics
  • Group action
  • Stabilizer
  • Sudoku
  • Wreath product

ASJC Scopus subject areas

  • Applied Mathematics


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