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Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).

For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function of the order ideal and let $\widetilde{x}_I \in \mathbb{R}^{n+1}$ be the lift of $x_I$ where we make the last coordinate be $1$. Let $\mathcal{C}_P \subseteq \mathbb{R}^{n+1}$ be the convex cone generated by all the $\widetilde{x}_I$ for $I \in J(P)$, and consider the semigroup $S(\mathcal{C}_P) := \mathcal{C}_P \cap \mathbb{Z}^{n+1}$ and its semigroup algebra $k[S(\mathcal{C}_P)]$.

($\mathcal{C}_P$ is the cone of the affinization of the "order polytope" of $P$.)

Let us say that $I, I' \in J(P)$ are compatible if either $I\subseteq I'$ or $I'\subseteq I$.

Question: Has anyone considered the following analogy, suggesting $k[S(\mathcal{C}_P)]$ is similar to a cluster algebra? :

Clusters <-> Maximal collections of pairwise compatible elements of $J(P)$ (these are in bijection with linear extensions of $P$)

Cluster monomials <-> Monomials $\widetilde{x}_{I_1}^{i_1} \cdots \widetilde{x}_{I_m}^{i_m} \in k[S(\mathcal{C}_P)]$ with $I_1,\ldots,I_m$ pairwise compatible

Cluster mutation <-> ???

The point is that monomials of the form above give a basis for $k[S(\mathcal{C}_P)]$, just like for finite type cluster algebras the cluster monomials give a basis.

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  • $\begingroup$ Something that could possibly be relevant: Higashitani showed in "Two poset polytopes are mutation-equivalent" (arxiv.org/abs/2002.01364) that Stanley's transfer map between the order and chain polytopes of a poset can be realized by a series of "combinatorial mutations" of polytopes (whatever that means...) $\endgroup$
    – Sam Hopkins
    Commented Aug 1, 2022 at 14:53
  • $\begingroup$ Somewhat related previous question of mine: mathoverflow.net/questions/373256/… $\endgroup$
    – Sam Hopkins
    Commented Aug 1, 2022 at 17:14
  • $\begingroup$ Maybe it's better to work more simply with the cone generated by the $x_I$ in $\mathbb{R}^n$; then I think the relations are $x_I x_J = x_{I \cap J} x_{I \cup J}$. In this form these are maybe what are called "Hibi rings." I think this just amounts to throwing away the empty order ideal, so shouldn't make too much of a difference. $\endgroup$
    – Sam Hopkins
    Commented Aug 1, 2022 at 22:54
  • $\begingroup$ If I understand your notations, the algebra $k[S(\mathcal C_P)]$ is actually what is more commonly known as the Hibi ring/algebra and its generating relations are the ones you specify (generators of the Hibi ideal, a.k.a. the toric ideal of the order polytope of $P$). The ring generated by the (exponentials of the) $x_I$ can have more relations. $\endgroup$
    – Igor Makhlin
    Commented Aug 2, 2022 at 2:15
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    $\begingroup$ Your question reminds me of Lam-Pylyavsky's LP algebras. There's a special kind of LP algebras called graph LP algebras (arxiv.org/abs/1206.2612) that can be encoded by a directed graph, where clusters are nested collections of subsets of vertices (nested means pair-wise compatible or disjoint). I don't remember a lot about order polytopes now but maybe, if G is the comparability graph of P, then the graph LP algebra of G could say something about $k[S(\mathcal{C}_P)]$. $\endgroup$
    – Sylvester W. Zhang
    Commented Nov 19, 2022 at 6:24

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