On the ideals of equivariant tree models
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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.
KeywordsBilinear Form Toric Variety Root Distribution Equivariant Model Internal Vertex
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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- 3.Casanellas, M., Fernández-Sánchez, J.: The geometry of the Kimura 3-parameter model. Adv. Appl. Math. (2007, to appear). Preprint available from http://arxiv.org/abs/math/0702834
- 4.Casanellas M., Sullivant S.: The strand symmetric model. In: Algebraic Statistics for Computational Biology. Cambridge University Press, Cambridge (2005)Google Scholar
- 5.Derksen H., Kemper G.: Computational Invariant Theory, vol. 130 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2002)Google Scholar
- 7.Eriksson, N., Ranestad, K., Sturmfels, B., Sullivant, S.: Phylogenetic algebraic geometry. In: Projective varieties with unexpected properties, pp. 237–255. Walter de Gruyter GmbH & Co. KG (2005)Google Scholar
- 12.Kraft, H., Procesi, C.: A Primer in Invariant Theory (unpublished). Text available from http://www.math.unibas.ch/~kraft/Papers/KP-Primer.pdf
- 17.Weyl H.: The Classical Groups, their Invariants and Representations. Princeton University Press, Princeton, NJ (1939)Google Scholar