Gorenstein property for phylogenetic trivalent trees. (English) Zbl 1483.14088
The authors investigate phylogenetic tree models: rooted trees with matrices of transitions between finitely many states assigned to edges. They consider group-based models, i.e., ones with symmetries determined by a finite abelian group action. They establish several properties of such models by investigating geometric and combinatorial objects associated to a tree model: a toric variety and a lattice polytope.
The article under review starts with the proof that for a tripod tree and a group of symmetries of even cardinality the associated algebraic variety is not projectively normal. Then the authors investigate model for \(\mathbb{Z}_3\) and claw trees, and describe facets of associated polytopes. The next result concerns the Gorenstein property for polytopes associated to a trivalent tree and a small group of symmetries: for \(\mathbb{Z}_2\times\mathbb{Z}_2\) the polytope is Gorenstein of index 4 and for \(\mathbb{Z}_3\) it is Gorenstein of index 3. Finally, the authors prove the normality of associated polytopes for claw trees and \(\mathbb{Z}_3\).
The article under review starts with the proof that for a tripod tree and a group of symmetries of even cardinality the associated algebraic variety is not projectively normal. Then the authors investigate model for \(\mathbb{Z}_3\) and claw trees, and describe facets of associated polytopes. The next result concerns the Gorenstein property for polytopes associated to a trivalent tree and a small group of symmetries: for \(\mathbb{Z}_2\times\mathbb{Z}_2\) the polytope is Gorenstein of index 4 and for \(\mathbb{Z}_3\) it is Gorenstein of index 3. Finally, the authors prove the normality of associated polytopes for claw trees and \(\mathbb{Z}_3\).
Reviewer: Maria Donten-Bury (Warszawa)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 13P25 | Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.) |
| 92D15 | Problems related to evolution |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 92B10 | Taxonomy, cladistics, statistics in mathematical biology |
| 05C05 | Trees |
Keywords:
group-based model; phylogenetic trivalent tree; toric variety; Fano variety; Gorenstein polytopeCitations:
Zbl 1147.14027Software:
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