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

Abstract

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 vertexdecomposable 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.A more interesting and morally correct statement is that each of these spaces possesses a stratification such that each stratum naturally has a cluster algebra in its ring of functions.2 More specifically, we consider lower bound algebras defined by a quiver in the body of the paper, and consider the more general context of geometric type in Appendix B.

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

“…We would like to remark that this paper is not the first to make a connection between cluster algebras and Gröbner theory. In [41] Muller, Rajchgot and Zykoski obtained presentations for lower bounds of cluster algebras using Gröbner theory. Further, it is worth noticing that the theory of universal coefficients for cluster algebras is particularly well-developed for finite and surface type cluster algebras, see [44].…”

Geometric vertex decomposition and liaison

“…, 𝑦 𝑛 ] and ideal 𝐾 𝑄 ⊆ 𝑅 𝑄 such that the lower bound algebra L 𝑄 associated to Q can be expressed as L 𝑄 = 𝑅 𝑄 /𝐾 𝑄 . Fix the lexicographical monomial order with [32,Theorem 1.7] and the proof of [32,Theorem 3.3], in < 𝐾 𝑄 is the Stanley-Reisner ideal of a simplicial complex Δ on the vertex set {𝑦 1 , . .…”

“…Remark 5.4. It follows from [32,Theorem 1.7] that 𝐾 𝑄 is homogeneous if and only if Q has no frozen vertices and Q has exactly two arrows entering each vertex and two arrows exiting each vertex.…”

Classification of singularities of cluster algebras of finite type: the case of trivial coefficients

“…In [BMRS15], Benito, Muller, Rajchgot, and Smith proved that locally acyclic cluster algebras are strongly F-regular (when defined over a field of prime characteristic) and that they have at worst canonical singularities (over a field of characteristic 0). Further, Muller, Rajchgot, and Zykoski [MRZ18] showed that the lower bound cluster algebra (which is an approximation of a given cluster algebra obtained by a suitable truncation of the construction process) is Cohen-Macaulay and normal.…”