Abstract
We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based models such as the Jukes–Cantor and Kimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model set-theoretically.
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Acknowledgments
The first author thanks Seth Sullivant for his great EIDMA/DIAMANT course on algebraic statistics in Eindhoven. It was Seth who pointed out that a result like the one in Sect. 3 could be used to treat various existing tree models in a unified manner. We also thank the anonymous referees for many valuable suggestions to improve the exposition.
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J. Draisma has been supported by DIAMANT, an NWO mathematics cluster and J. Kuttler by an NSERC Discovery Grant.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Draisma, J., Kuttler, J. On the ideals of equivariant tree models. Math. Ann. 344, 619–644 (2009). https://doi.org/10.1007/s00208-008-0320-6
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DOI: https://doi.org/10.1007/s00208-008-0320-6