The aim of this thesis is to study discrete structures associated with group-based models. Group-based models are statistical models on phylogenetic trees that can be parametrized by polynomial maps. The algebraic varieties given by these polynomial maps are toric. By the well-known correspondence between toric varieties and polyhedral fans, we can associate discrete structures, such as lattice polytopes and affine semigroups, with group-based models.
We follow three main lines in this dissertation. In Chapters 2 and 3, we study the Hilbert polynomials of group-based models. This is motivated by a result of Buczyńska and Wiśniewski stating that the Hilbert polynomial of the Jukes-Cantor binary model on a trivalent tree does not depend on the shape of the tree. In Chapter 2, we give a simple combinatorial proof to this statement, and in Chapter 3, we show that the analogous statement does not hold for the Kimura 3-parameter model.
In Chapters 4 and 5, we study the phylogenetic semigroups on graphs that generalize the Jukes-Cantor binary model on trees. In Chapter 4, we study the maximal degrees of the minimal generators of these semigroups. In Chapter 5, we investigate the minimal generators of the phylogenetic semigroups on graphs with a few holes, extending the work of Buczyńska.
Finally, in Chapter 6, we establish a connection between Berenstein-Zelevinsky triangles from representation theory and group-based models. This is motivated by the recent work of Sturmfels, Xu, and Manon related to conformal block algebras.
2. Combinatorial Proof of a Theorem by Buczyńska and Wiśniewski
3. Hilbert Polynomial of the Kimura 3-Parameter Model
4. Degrees of Minimal Generators of Phylogenetic Semigroups
5. Low Degree Minimal Generators of Phylogenetic Semigroups
6. Group-Based Models and Berenstein-Zelevinsky Triangles
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