Lower bound cluster algebras: presentations, Cohen-Macaulayness, and normality

TL;DR

This paper provides explicit presentations for lower bound cluster algebras and proves they are Cohen-Macaulay and normal using combinatorial and algebraic techniques.

Contribution

It introduces explicit presentations for lower bound cluster algebras and establishes their Cohen-Macaulayness and normality through Grobner degenerations and combinatorial methods.

Findings

Lower bound cluster algebras have explicit presentations.

They degenerate to Stanley-Reisner schemes of vertex-decomposable complexes.

All lower bound algebras are normal.

Abstract

We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Grobner degenerates to the Stanley-Reisner scheme of a vertex-decomposable ball or sphere, and is thus Cohen-Macaulay. Finally, we use Stanley-Reisner combinatorics and a result of Knutson-Lam-Speyer to show that all lower bound algebras are normal.