Integer decomposition property with a partial order

Question

Let $\mathcal{P}$ be a convex lattice polytope in $\mathbb{R}^n$. We say that $\mathcal{P}$ has the integer decomposition property (or "is IDP") if for all $k\in \mathbb{N}$ and $\alpha \in k\mathcal{P}\cap\mathbb{Z}^n$, there are $\alpha_1,\ldots,\alpha_k \in \mathcal{P}\cap\mathbb{Z}^n$ such that $\alpha=\alpha_1+\cdots+\alpha_k$. IDP polytopes are a well-studied class with connections to commutative algebra, algebraic geometry, etc.

Here's a new/nonstandard notion. Let's say $\mathcal{P}$ is IDP$\leq$ if there exists a partial order $\leq$ on $\mathcal{P}\cap\mathbb{Z}^n$ such that for all $k\in \mathbb{N}$ and $\alpha \in k\mathcal{P}\cap\mathbb{Z}^n$, there is a unique multichain $\alpha_1 \leq \cdots \leq \alpha_k \in \mathcal{P}\cap\mathbb{Z}^n$ with $\alpha=\alpha_1+\cdots+\alpha_k$.

For example, the order polytope $\mathcal{O}(P)$ and the chain polytope $\mathcal{C}(P)$ of a poset $P$ are IDP$\leq$ where $\leq$ is the natural distributive lattice order (on order filters/antichains).

Question: Are there other interesting families of IDP$\leq$ polytopes?

Motivation: this property gives a canonical/algorithmic choice of decomposition for all latice points of dilates. Also then the zeta polynomial of $(\mathcal{P}\cap\mathbb{Z}^n,\leq)$ is the Ehrhart polynomial of $\mathcal{P}$.

Answer 1

A slightly more general combinatorial family that satisfies this is the family of $s$-lecture hall polytopes. These can be thought of as a weighted version of order polytopes. For a reference see theorem 1 in "Some Algebraic Properties of Lecture Hall Polytopes" by P. Brändén, L. Solus.

In general this sort of set up is precisely the combinatorial setup underlying standard monomial theory. If you look at the semigroup ring of the cone over $(\mathcal P,1)$ in $\mathbb R^{n+1}$, your condition is equivalent to asking that the set of standard monomials (those corresponding to $\mathbb z^{\alpha_1}\cdots \mathbb z^{\alpha_k}$ with $\alpha_1\le\cdots \le \alpha_k$) form a basis. This is why you will most likely see this sort of thing used in the context of Gelfand-Tseltin polytopes, polytopes from general root systems, or other situations with a representation theoretic/geometric flavor. One application I know where this gets used a lot is in construction toric degenerations of associated flag varieties, etc.

A closely related concept that leads to this situation is that of algebras with straightening laws. This is where one has access to a straightening rule that expresses the product of incomparable monomials as a linear combination of standard monomials. An author that has written a lot on this sort of thing is Takayuki Hibi. For example the paper "Chain polytopes and algebras with straightening laws" discusses straightening algeba structures on chain/order polytopes.

Answer 2

Have you checked the family of marked order polytopes? These include the classical Gelfand-Tsetlin polytopes, and I think I can construct such a partial order in case of GT-polytopes.

Let $T \in kP_\lambda$, where $T$ is a GT-pattern, which is in bijection with some SSYT in $SSYT(k \lambda,m)$. We want to find a partial order, such that $T=T_1 + \dotsb + T_k$ in a unique manner. Well, the $T_i$ are elements in $SSYT(\lambda,m)$, and $+$ is interpreted as concatenate-and-sort rows. So, if we take the partial order on $SSYT(\lambda,m)$ where $T \leq T'$ iff for each $i \leq j$, column $i$ from $T$ together with column $j$ from $T'$ form a valid SSYT when placed next to each other.

We can then find the unique decomposition $T=T_1 + \dotsb + T_k$ by letting $T_i$ be every $k$th column from $T$, starting with the $i$th one.

I think one can extend this construction to all marked order polytopes.

Answer 3

Just stumbled upon this question, pardon the necropost. There exist broad families of poset polytopes generalizing order and chain polytopes. These are known to have your IDP$\le$ property.

The first are the chain-order polytopes introduced by Fang and Fourier. Each such polytope is given by a partition of a poset into two subsets, the two extremal cases are the order and chain polytopes. For a concise definition of the specific non-marked(!) version I'm referring to see, for instance, Section 2 here (we use the unfortunate term "interpolating polytopes", please ignore it).

This family is further extended by the relative poset polytopes. Such a polytope is given by a poset $(P,<)$ and a second order $<'$ on $P$ weaker than $<$, see Definition 2.1.1 here. Curiously enough, we explicitly state that the IDP$\le$ property is satisfied (Corollary 2.1.7). In all of these polytopes lattice points are parametrized by order ideals so ordering them is not an issue.

Now, both chain-order polytopes and relative poset polytopes have marked versions where one distinguishes a subset $A$ of the poset and attaches a real number to every element of $A$. Such are the Gelfand-Tsetlin polytopes and FFLV polytopes, for instance. However, I'm not sure that the IDP$\le$ property is straightforwardly generalized to this setting. The problem is that as soon as one has non-vertex lattice points uniqueness tends to fail: if $x$, $x+v$ and $x+2v$ are lattice points in $\mathcal P$ and $x$ and $x+2v$ are comparable, then one has two distinct multichains for $2x+2v\in2\mathcal P$.

In particular, I'm afraid that this causes issues with both of the earlier answers, please correct me if I'm wrong. For Per's answer already for $\mathfrak{sl}_2$ and $\lambda=2\omega$ the point $\begin{smallmatrix}4 && 0\\ & 2 & \end{smallmatrix}$ in $2P_\lambda$ decomposes as both $\begin{smallmatrix}2 && 0\\ & 0 & \end{smallmatrix}+\begin{smallmatrix}2 && 0\\ & 2 & \end{smallmatrix}$ and $\begin{smallmatrix}2 && 0\\ & 1 & \end{smallmatrix}+\begin{smallmatrix}2 && 0\\ & 1 & \end{smallmatrix}$. As for the $s$-lecture hall polytopes in Gjergji's answer, no order is specified but it seems natural to assume that $0$ is the minimal element. This would lead to lack of uniqueness whenever at least one $s_i>1$.