On the Ideals of Equivariant Tree Models

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 Shimura 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. 1. Set-up and theorems Central objects in algebraic statistics are what we propose to call spaced trees; in Section 3 we explain how a spaced tree gives rise to a statistical model. Recall that a tree T is a connected, undirected graph without circuits. A vertex of T is called a leaf if it has valency 1, and an internal vertex otherwise. We write vertex(T ), leaf(T ), internal(T ) for the sets of vertices, leaves, and internal vertices of T , respectively. Stars are trees of diameter at most 2, and a center of a star is a vertex at distance 1 to all other vertices—so if the star has more than 2 vertices, then its center is unique. In all that follows, we work over a ground field K that is algebraically closed and of characteristic zero. Definition 1.1. A spaced tree T is given by the following data: First, a finite undirected tree, also denoted T ; second, for every p ∈ vertex(T ) a finite-dimensional vector space Vp; third, a non-degenerate symmetric bilinear form (. | .)p on each Vp; and fourth, for every p ∈ internal(T ) a distinguished basis Bp of Vp which is orthonormal with respect to (. | .)p. The space Vp at a leaf p may also be given a distinguished basis Bp, orthonormal with respect to (. | .)p, in which case p is called a based leaf. An internal vertex will also be called based. Any connected subgraph of T is regarded as a spaced tree with the data that it inherits from T . Note that there is some redundancy in this definition: given the distinguished basis Bp at an internal vertex p one could define (. | .)p by the requirement that Bp be orthonormal. We will leave out the subscript p from the bilinear form when it is obvious from the context. In many applications in algebraic statistics, symmetry is imposed on the algebraic model. This notion is captured well by the following notion of a G-spaced tree. Fix, for once and for all, a finite group G. Definition 1.2. A G-spaced tree (or G-tree, for short) is a spaced tree in which the space Vp at every vertex p is a G-module, on which (. | .)p is G-invariant, and in which Bp is G-stable whenever p is a based vertex. The second author is supported by DIAMANT, an NWO mathematics cluster.