The quadratic invariants are also referred to as the edge invariants of the model.
Each Fourier coordinate corresponds to a consistent coloring of the edges of tree T. For any given internal edge e of T, the consistent colorings can be obtained by coloring two smaller graphs and gluing them along e. This corresponds to a fiber product on the corresponding toric varieties. The quadratic invariants naturally arise from this process by gluing a pair of colorings of one small graph to a pair of colorings of the other small graph in two different ways.
The optional argument QRing can be passed the ring of Fourier coordinates. Otherwise the function will create a new ring.
i1 : T = leafTree(4,{{0,1}})
o1 = {{0, 1, 2, 3}, {set {0, 1}, set {0}, set {1}, set {2}, set {3}}}
o1 : LeafTree
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i2 : S = qRing(T, CFNmodel) o2 = S o2 : PolynomialRing |
i3 : phyloToricQuads(T, CFNmodel, QRing=>S)
o3 = {- q q + q q , q q -
0,0,1,1 1,1,0,0 0,0,0,0 1,1,1,1 0,0,1,1 1,1,0,0
------------------------------------------------------------------------
q q , q q - q q , -
0,0,0,0 1,1,1,1 0,0,1,1 1,1,0,0 0,0,0,0 1,1,1,1
------------------------------------------------------------------------
q q + q q , - q q +
0,0,1,1 1,1,0,0 0,0,0,0 1,1,1,1 0,1,1,0 1,0,0,1
------------------------------------------------------------------------
q q , q q - q q , q q
0,1,0,1 1,0,1,0 0,1,1,0 1,0,0,1 0,1,0,1 1,0,1,0 0,1,1,0 1,0,0,1
------------------------------------------------------------------------
- q q , - q q + q q }
0,1,0,1 1,0,1,0 0,1,1,0 1,0,0,1 0,1,0,1 1,0,1,0
o3 : List
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The object phyloToricQuads is a method function with options.