phyloToricQuads -- compute the quadratic invariants of a group-based phylogenetic model

Synopsis

Usage:

phyloToricQuads(T,M)

phyloToricQuads(n,E,M)
Inputs:
- T, an instance of the type LeafTree,
- n, an integer, the number of leaves
- E, a list, the internal edges of the tree, given by one part of the bipartition on leaves
- M, an instance of the type Model,
Optional inputs:
- QRing => ..., default value null, optional argument to specify Fourier coordinate ring
- Random => ..., default value false
Outputs:
- a list, a generating set of the quadratic invariants

Description

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

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

Ways to use phyloToricQuads :

phyloToricQuads(LeafTree,Model)
phyloToricQuads(ZZ,List,Model)

For the programmer

The object phyloToricQuads is a method function with options.

phyloToricQuads -- compute the quadratic invariants of a group-based phylogenetic model

Synopsis

Description

See also

Ways to use phyloToricQuads :

For the programmer