V V TTejse TT fjsf

11 »3(e) »4(e) 11 •/is(/)»t(f) I is(e)ik »k »t(f)

The parenthesized expression is the product of the G-matrices e and f. Replacing this expression with a new single G-matrix of parameters along the conjoined edge ef proves that CV(Tn-1) * CV(T3) C CV(Tn). Now expanding the reparameterization given in Lemma 16.8 as a sum on the vertex u we obtain the other inclusion. □

Now we define a variety GV(4ri, 4r2, 4r3) which plays a large role when we extend invariants.

Definition 16.16 For l — 1, 2, 3 let (1j) be a string of indices of length ri. Let ¡M be an arbitrary G-matrix of size 4ri where the rows are indexed by {( 0 ), ( 0 ), ( J ), (1 )} and the columns are indexed by the 4ri indices (jj). Define the parameterization Q — ^ri r2 r3 (1M, 2M, 3M) by

if ) + J + ^(js) — 0 and Qj3 — 0 if J + + *(js) — 1. The projective variety that is the Zariski closure of the image of ^ri r2 r3 is denoted GV(4ri, 4r2, 4r3). The affine cone over this variety is CGV(4ri, 4r2, 4r3).

Remark 16.17 By the definition of GV(4ri, 4r2, 4r3), we have that any Q € GV(4ri, 4r2, 4r3) is a G-tensor. Furthermore GV(4,4,4) is equal to the variety defined by the SSM on the 3-leaf claw tree K1>3.

Besides the fact that GV(4ri, 4r2, 4r3) is equal to the SSM when r1 — r2 — r3 — 1, the importance of this variety for the strand symmetric model comes from the fact that GV(4ri, 4r2, 4r3) contains the SSM for any binary tree as illustrated by the following proposition.

Proposition 16.18 Let T be a binary tree and v an interior vertex. Suppose that removing v from T partitions the leaves of T into the three sets {1,..., r1}, {r1 + 1, ■ ■ ■ , r1 + r2}, and {r1 + r2 + 1..., r1 + r2 + r3}. Then the SSM on T is a subvariety of GV (4ri, 4r2, 4r3).

In the proposition, the indices in the Fourier coordinates for the SSM are grouped in the natural way according to the tripartition of the leaves.

Proof In the parametric representation qjij2...jn — TTejs(e)

(»v)€H e perform the sum associated to the vertex v first. This realizes the G-tensor Q

as the sum over the product, of entries of three G-tensors. □

Our goal for the remainder of this section is to prove a result analogous to Theorem 7 in [Allman and Rhodes, 2004a]. This theorem will provide a method for explicitly determining the ideal of invariants for G V(4ri, 4r2, 4r3) from the ideal of invariants for GV(4, 4, 4). Denote by GM(2l, 2m) the set of 2l x 2m G-matrices. A fundamental observation is that if r3' > r3 then

CGV(4ri, 4r2, 4r3) = CGV(4ri, 4r2, 4r3) * GM(4r3, 4r3).

Thus, we need to understand the * operation when V and W are "well-behaved" varieties.

Lemma 16.19 Let V C GM(2l, 4) be a variety with V * GM(4, 4) = V. Let I be the vanishing ideal of V. Let K be the ideal of 3 x 3 G-minors of the 2l x 2m G-matrix of indeterminates Q. Let Z be 2m x 4 G-matrix of indeterminates and

Then K + L is the vanishing ideal of W = V * GM(4,2m).

By a G-minor we mean a minor which involves only the non-zero entries in the G-matrix Q.

Proof A useful fact is that

Let J be the vanishing ideal of W. By the definition of W, all the polynomials in K must vanish on it. Moreover, if f (Q * A) is a polynomial in L, then it vanishes at all the points of the form P*B, for any P € V and B € GM(4, 2m). Indeed, as P * B * A € V and f € I we have f (P * B * A) = 0. As all the points of W are of this form, we obtain the inclusion K + L C J. Our goal is to show that J C K + L.

Since V * gM(4,4) = V, we must also have W * GM(2m, 2m) = W. This implies that there is an action of Gl(C,m) x Gl(C,m) on W and hence, any graded piece of J, the vanishing ideal of W, is a representation of Gl(C, m) x Gl(C, m). Let Jd be the dth graded piece of J. Since Gl(C, m) x Gl(C, m) is reductive, we just need to show each irreducible subspace M of Jd belongs to K + L. By construction, K + L is also invariant under the action of Gl(C, m) x Gl(C, m) and, hence, it suffices to show that there exists a polynomial f € M such that f € K + L.

Let f € M be an arbitrary polynomial in the irreducible representation M. Let P be a 2l x 4 G-matrix of indeterminates. Suppose that for all B € gM(4,2m), f (P * B) = 0. This implies that f vanishes when evaluated at any G-matrix Q which has rank 2 in both components. Hence, f € K. If f € K there exists a B € GM(4, 2m) such that fB(P) := f (P * B) # 0.

Renaming the P indeterminates we can take D to be a matrix in G(2m, 4) formed by 1s and 0s such that fB(Q * D) ^ 0. Since f € J, we must have /b(P) € I. Therefore /b(Q * D) € L. Let B' = D * B € GM(2m, 2m). Although B' € G1(C, m) x G1(C, m), the representation M must be closed and hence /(Q * B') = fs(Q * D) £ M which completes the proof. □

Proposition 16.20 Generators for the vanishing ideal of GV(4ri, 4r2, 4r3) are explicitly determined by generators for the vanishing ideal of GV(4,4,4).

Proof Starting with GV(4,4,4), apply the preceding lemma three times. Now we will explain how to compute these polynomials explicitly. For 1 = 1,2,3 let Z be a 4r x 4 G-matrix of indeterminates. This G-matrix Z acts on the 4ri x 4r2 x 4r3 tensor Q by G-tensor multiplication in the 1th coordinate. For each f € gens(I), where I is the vanishing ideal of GV(4,4,4), we construct the polynomials coeffZf (Q * Z, * Z2 * Z3). That is, we construct the 4 x 4 x 4 G-tensor Q * Z, * Z2 * Z3, plug this into f and expand, and extract, for each Z monomial, the coefficient, which is a polynomial in the entries of Q. Letting f range over all the generators of I determines an ideal L.

We can also flatten the 3-way G-tensor Q to a G-matrix in three different ways. For instance, we can flatten it to a 4ri x 4r2+r3 G-matrix grouping the last two coordinates together. Taking the ideal generated by the 3 x 3 G-minors in these three flattenings yields an ideal K. The ideal K + L generates the vanishing ideal of GF(4ri, 4r2,4r3). □

16.4 Extending invariants

In this section we will show how to derive invariants for arbitrary trees from the invariants introduced in Section 16.2. We also introduce the degree 3 determinantal flattening invariants which arise from flattening the n-way G-tensor associated to a tree T under the SSM along an edge of the tree. The idea behind all of our results is to use the embedding of the SSM into the variety G V (4ri, 4r2, 4r3).

Let T be a tree with n-taxa on the SSM and let v be any interior vertex. Removing v creates a tripartition of the leaves into three sets of cardinalities ri, r2 and r3, which we may suppose, without loss of generality, are the sets {1,..., ri}, {ri + 1,..., ri + r2}, and {r, + r2 + 1,..., r, + r2 + r3}.

Proposition 16.21 Let f (m., n., o., i., j.be one of the degree 3 invariants for the 3-taxa tree Ki,3 introduced in Proposition 16.10. For each l = 1, 2, 3 we choose sets of indices mi, ii € {0,1}ri, ni, ji € {0,1}r2, and oi, ki € {0,1}r3 such that a(mi) = mi a(ni) = n and a(oi) = oj. Then f (m., n., o., i., j., k.)

mi ni oi qiljlki min202

qm2nioi qi2jiki m2n202

qiij2k2 qi2J2k2 flmin203 flm2n203 qiij2k3 qi2J2k3

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