Integer decomposition property with a partial order

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 $$\mathcal{P}$$.

• Forgive the terrible notation. Oct 4 '20 at 4:06

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.

• Another very nice answer! I guess there is not exactly a name for this situation though ("the standard monomials form a basis of the semigroup ring of the cone over the polytope" is not so snappy). I also wonder if anything interesting can be said about zeta polynomials of posets beyond distributive lattices from this Ehrhart perspective. Oct 4 '20 at 19:59

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.