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#Ehrhart theory

Ehrhart theory studies the lattice points in dilates of rational and lattice polytopes. For a lattice polytope \(P,\) the basic counting function is \[E_P(n)=|nP\cap \setZ^d|.\] This is the Ehrhart polynomial of \(P.\) Its generating function is the Ehrhart series, usually written in the form \[\sum_{n\geq 0} E_P(n)t^n = \frac{h^*_P(t)}{(1-t)^{d+1}},\] where \(h^*_P(t)\) is the \(h^*\)-polynomial. See [Sta93, BS07] and the polytope page for the standard definitions and basic theorems.

Example

For the unit interval \(P=[0,1],\) \[E_P(n)=n+1,\qquad \sum_{n\geq 0}(n+1)t^n=\frac{1}{(1-t)^2}.\] Thus \(h^*_P(t)=1.\) For the unit square \([0,1]^2,\) \[E_P(n)=(n+1)^2,\qquad \sum_{n\geq 0}(n+1)^2t^n=\frac{1+t}{(1-t)^3},\] so \(h^*_P(t)=1+t.\)

Example

For the \(d\)-dimensional cube \([0,1]^d,\) the \(h^*\)-polynomial is the Eulerian polynomial: \[\begin{array}{c|l} d & h^*_{[0,1]^d}(t)\\ \hline 1 & 1\\ 2 & 1+t\\ 3 & 1+4t+t^2\\ 4 & 1+11t+11t^2+t^3. \end{array}\] The coefficients form the Eulerian number triangle A008292. This is one of the quickest ways Ehrhart theory meets permutation statistics.

#Reciprocity and duality

Ehrhart–Macdonald reciprocity relates lattice points in \(nP\) to lattice points in the interior of \(nP,\) by evaluating the Ehrhart polynomial at negative integers. For a \(d\)-dimensional lattice polytope, \[E_P(-n)=(-1)^d |nP^\circ\cap \setZ^d|.\] In terms of the Ehrhart series, palindromicity of the \(h^*\)-vector is tied to the Gorenstein condition. This is one reason Ehrhart theory often behaves like a discrete shadow of Poincaré duality.

#Inequalities for \(h^*\)-vectors

The \(h^*\)-vector \[h^*(P)=(h^*_0,h^*_1,\dotsc,h^*_d)\] of a \(d\)-dimensional lattice polytope is not an arbitrary nonnegative integer vector. The first restrictions are the basic ones: \[h^*_0=1,\qquad h^*_1=|P\cap \setZ^d|-d-1,\qquad \sum_{i=0}^{d} h_i^*=\operatorname{Vol}(P),\] where \(\operatorname{Vol}(P)\) is the normalized volume. If \(s\) is the degree of \(h^*_P(t),\) then \(h^*_s\) counts lattice points in a suitable interior dilation of \(P,\) and in particular \[h^*_d=|P^\circ\cap \setZ^d|.\] This gives the elementary inequality \[h^*_d\leq h^*_1.\]

There are also less immediate inequalities. T. Hibi proved [Hib94] that \[h^*_{d-1}+h^*_{d-2}+\dotsb+h^*_{d-i} \leq h^*_2+h^*_3+\dotsb+h^*_{i+1}, \qquad 1\leq i\leq d-1.\] If \(P\) has an interior lattice point, equivalently \(h^*_d\gt{}0,\) then Hibi also proved \[h^*_1\leq h^*_i,\qquad 1\leq i\leq d-1.\] Another family, due to R. P. Stanley [Sta91], says that if \(s=\deg h^*_P(t),\) then \[h^*_0+h^*_1+\dotsb+h^*_i \leq h^*_s+h^*_{s-1}+\dotsb+h^*_{s-i}, \qquad 0\leq i\leq s.\] These inequalities are useful quick checks when an experimental \(h^*\)-vector has been computed.

The two-dimensional case has a particularly concrete form. Scott’s theorem [Sco76] characterizes the possible \(h^*\)-polynomials \(1+h^*_1t+h^*_2t^2\) of lattice polygons by the following alternatives: \[h^*_2=0,\qquad h^*_2\leq h^*_1\leq 3h^*_2+3,\qquad \text{or}\qquad (h^*_1,h^*_2)=(7,1).\] J. Treutlein extended the necessary part of this statement to lattice polytopes of degree at most two [Tre10].

Finally, G. Balletti and A. Higashitani [BH18] proved a universal version of Scott’s inequality. If \(P\) is any lattice polytope, in any dimension and of any degree, and \[h^*_P(t)=1+h^*_1t+h^*_2t^2+\dotsb \qquad\text{satisfies}\qquad h^*_3=0,\] then one of the following holds: \[h^*_2=0,\qquad h^*_1\leq 3h^*_2+3,\qquad \text{or}\qquad (h^*_1,h^*_2)=(7,1).\] The hypothesis \(h^*_3=0\) is essential: their paper gives a five-dimensional example with \(h^*_P(t)=1+8t+t^2+8t^3,\) which violates Scott’s inequality once the \(h^*_3\)-term is present. Stapledon’s work [Sta09, Sta16] gives further families of inequalities, often sharper but less uniform in appearance.

#Order polytopes and posets

For a finite poset \(P,\) the order polytope \(\mathcal O(P)\) packages order-preserving maps \(P\to[0,1].\) Its \(h^*\)-polynomial is the \(P\)-Eulerian polynomial; equivalently, it records descents of linear extensions. The chain polytope \(\mathcal C(P)\) has the same Ehrhart polynomial as \(\mathcal O(P),\) but its inequalities are controlled by chains rather than order relations.

Example

If \(P\) is an antichain on \(d\) elements, then \(\mathcal O(P)=[0,1]^d.\) Hence \[E_{\mathcal O(P)}(n)=(n+1)^d.\] Its \(h^*\)-polynomial is the Eulerian polynomial, whose coefficients are the Eulerian numbers A008292. Thus even the most basic order polytopes already produce familiar sequence data.

If \(P\) is a chain on \(d\) elements, then \(\mathcal O(P)\) is a simplex, and \(E_{\mathcal O(P)}(n)\) counts weakly increasing sequences \[0\leq a_1\leq a_2\leq \dotsb \leq a_d\leq n.\]

Example

Let \(P_\lambda\) be the Ferrers poset of a partition \(\lambda.\) Linear extensions of \(P_\lambda\) are standard Young tableaux of shape \(\lambda,\) and the \(h^*\)-polynomial of \(\mathcal O(P_\lambda)\) is the descent generating polynomial over those tableaux. For \(\lambda=(n,n),\) this gives the Narayana polynomial; the Narayana triangle is A001263. This is the Ehrhart-theoretic form of the example on the real-rooted tableaux page.

#Where it appears

Ehrhart theory appears in several parts of algebraic combinatorics:

#Conjectural Ehrhart positivity

A recurring phenomenon is that representation-theoretic GT-polytopes seem to have more positive Ehrhart polynomials than arbitrary polytopes. This should be treated as a conjectural pattern, since Ehrhart positivity is false for general lattice polytopes and even for many natural-looking families.

The general mechanism behind the results below is the ideal-chain decomposition for marked order polytopes. The skew-GT proof uses that same decomposition, but is not a direct application of its ideal/filter-closure criterion: the relevant ideals and filters of the GT poset are shifted skew shapes, whose Ehrhart positivity is not known in general [Sec. 1, JM26].

Theorem (Jochemko–Menon [Thm. 3.5, JM26]).

The Ehrhart polynomial of the skew Gelfand–Tsetlin polytope \(\gtp_{\lambda/\mu}\) has nonnegative coefficients. Equivalently, \[k\mapsto \schurS_{k\lambda/k\mu}(1^m)\] has nonnegative coefficients as a polynomial in \(k.\) K. Jochemko and K. Menon thereby prove the Alexandersson–Alhajjar conjecture stated on the Gelfand–Tsetlin page [AA19].

Their method also proves Ehrhart positivity for the unsliced row-interval flagged faces of skew GT polytopes [Sec. 3.2, JM26]. It does not imply positivity after fixing the tableau content: that operation extracts one stretched monomial coefficient and intersects the flagged face with the rational affine weight slice.

The King–Tollu–Toumazet conjecture predicts that every coefficient of the stretched Littlewood–Richardson polynomial \[t\mapsto c^{t\nu}_{t\lambda,t\mu}\] is nonnegative. This is known when the indexing partitions have at most four parts.

Theorem (Ferudun [Thm. 2, Fer26]).

If \(\lambda,\mu,\nu\) have length at most four, then every coefficient of \(t\mapsto c^{t\nu}_{t\lambda,t\mu}\) is nonnegative. Equivalently, the Ehrhart polynomial of the associated hive polytope is coefficientwise nonnegative.

Conjecture

The flagged skew Schur stretching polynomial \[k \mapsto K_{k\lambda/k\mu,k\nu}(\avec,\bvec)\] has nonnegative coefficients. This is the flagged generalization of the King–Tollu–Toumazet positivity conjecture [AO23].

Conjecture

For fixed \(\lambda\) and \(\sigma\in\symS_n,\) the polynomial \[k\mapsto \key_{k\lambda,\sigma}(1^n)\] is Ehrhart-positive; more precisely, the polynomial \(P_\sigma(\lambda_1,\dotsc,\lambda_n;k)\) from the GT-face model for key polynomials has nonnegative coefficients, and remains nonnegative after changing variables to partial sums \(a_j=\lambda_1+\dotsb+\lambda_j\) [AA19].

These conjectures are parallel: ordinary skew Schur functions, flagged skew Schur functions, and key polynomials all have GT-type models, and stretching the highest weights turns the relevant specialization into an Ehrhart polynomial or a closely related lattice-point count. The evidence suggests that the representation-theoretic inequalities impose positivity that is not visible for arbitrary polytopes.

#Typical questions

Common questions include:

  • whether the coefficients of \(E_P(n)\) are nonnegative;

  • whether \(h^*_P(t)\) is unimodal, real-rooted, or gamma-positive;

  • whether a rational polytope has period collapse;

  • whether a family admits a combinatorial interpretation for its \(h^*\)-coefficients.

For experimental families, the first rows of \(h^*\)-vectors are often among the best invariants to compare with OEIS. When an \(h^*\)-triangle is recognizable, recording the OEIS entry usually helps locate older enumerative interpretations, recurrences, and transforms.

These questions are often subtle: the \(h^*\)-vector of a lattice polytope is always nonnegative, but Ehrhart polynomial coefficients can be negative.

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