Presentations of Semigroup Algebras of Weighted Trees

TL;DR

This paper provides explicit presentations for certain subalgebras related to phylogenetic tree models, connecting algebraic invariants with toric degenerations of geometric objects like Grassmannians.

Contribution

It introduces new presentations for subalgebras of invariants in phylogenetic models, linking them to toric degenerations of classical algebraic varieties.

Findings

Presentations for subalgebras of invariants of binary symmetric models.

Connections between these algebras and toric degenerations of Grassmannian-related rings.

Insights into the algebraic structure of phylogenetic tree models.

Abstract

We find presentations for subalgebras of invariants of the coordinate algebras of binary symmetric models of phylogenetic trees studied by Buczynska and Wisniewski in \cite{BW}. These algebras arise as toric degenerations of rings of global sections of weight varieties of the Grassmanian of two planes associated to the Pl\"ucker embedding, and as toric degenerations of rings of invariants of Cox-Nagata rings.