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

We give an explicit presentation for each lower bound cluster algebra. Using this presentation, we show that each lower bound algebra Gröbner 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.