Timeline for "Cluster algebra" structure for finite distributive lattices
Current License: CC BY-SA 4.0
13 events
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| Nov 19, 2022 at 6:24 | comment | added | Sylvester W. Zhang | 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 $k[S(\mathcal{C}_P)]$. | |
| Aug 2, 2022 at 3:11 | comment | added | Igor Makhlin | Yes, sorry, I understand the question, I was just trying to give a bit of context concerning Hibi ideals etc. I don't immediately recall such a connection with cluster theory being explored, no. | |
| Aug 2, 2022 at 2:53 | comment | added | Sam Hopkins | Sure, I agree this fact is folklore, and immediate to prove once you hear it - but again I'm more asking if there is a developed connection to cluster theory since superficially it seems so similar. E.g. does anyone view the different linear extensions (clusters) as giving different "charts" on some variety, that we can pass between via "mutations" of some kind? | |
| Aug 2, 2022 at 2:48 | comment | added | Igor Makhlin | Just as a side note, there's a very nice and well-known example. If you consider the set of all increasing columns of integers from $[1,n]$ and order them by $I<J$ iff the corresponding two-column Young tableau is semistandard, then you obtain a distributive lattice. The set of monomials you consider is the basis in the Plücker algebra given by all semistandard tableaux. | |
| Aug 2, 2022 at 2:42 | comment | added | Igor Makhlin | I'm not sure whom to credit with the fact that products over weakly increasing tuples form a basis, I'd view this as folklore of sorts. My favorite way of seeing this is that the set of all other monomials span a monomial ideal which is an initial ideal of the Hibi ideal (see Proposition 2.1 in arxiv.org/pdf/2003.02916.pdf but I'm definitely not going to take credit for this). | |
| Aug 2, 2022 at 2:34 | comment | added | Igor Makhlin | So the $\mathrm{Proj}$ of the ring given by the relations $X_IX_J-X_{I\cap J}X_{I\cup J}$ (the Hibi ring) is the (projective, complete, normal) toric variety of the order polytope, which is the convex hull of your $x_I$. Subsequently, this ring can be realized as the ring generated by the exponentials of your $\tilde x_I$ (the semigroup ring of the cone, yes). | |
| Aug 2, 2022 at 2:20 | comment | added | Sam Hopkins | @imakhlin I am not an expert in all this terminology, but yes I agree that the Hibi ring is at least very close to what I am talking about. I am wondering if the fact that it has a "cluster monomial" style basis has been observed/exploited anywhere, or if this analogy has been developed further. (Incidentally, I'm a bit confused about calling this a toric ideal: usually I would think the toric variety corresponding to a polytope is the complete, projective variety corresponding to the normal fan of the polytope, but the thing I'm talking about is definitely affine and not complete.) | |
| Aug 2, 2022 at 2:15 | comment | added | Igor Makhlin | 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. | |
| Aug 1, 2022 at 22:54 | comment | added | Sam Hopkins | 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. | |
| Aug 1, 2022 at 17:14 | comment | added | Sam Hopkins | Somewhat related previous question of mine: mathoverflow.net/questions/373256/… | |
| Aug 1, 2022 at 15:02 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Aug 1, 2022 at 14:53 | comment | added | Sam Hopkins | 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...) | |
| Aug 1, 2022 at 14:48 | history | asked | Sam Hopkins | CC BY-SA 4.0 |