Hive Polytopes with Arbitrary Integral Right-Hand Sides Have Integral Points Exactly up to Rank 5: A Counterexample to a Conjecture of De Loera and McAllister

Knutson and Tao proved that a nonempty hive polytope has an integral vertex. De Loera and McAllister conjectured that this extends to arbitrary integral right-hand sides: if \(B\) and \(R\) are the matrices of the boundary equations and of the rhombus inequalities of hive patterns of rank \(r\), every nonempty polytope \(\{h\)\({}: Bh\)\({}=b,\ Rh\)\({}\le c\}\) with integral \(b\) and \(c\) should have an integral vertex, and the matrix \(\bigl(\begin{smallmatrix}B&0\\R&I\end{smallmatrix}\bigr)\) should have a unimodular triangulation (Oberwolfach Report 39/2004; Experiment. Math. 15 (2006), Conjecture 4.5). We show that for every \(r\ge6\) there is an integral \(c\) for which \(\{h\)\({}: Bh\)\({}=0,\ Rh\)\({}\le c\}\) is a single point with four coordinates equal to \(1/2\). Hence for \(r\ge6\) the conjecture fails in both printed forms: the cone generated by the columns of the matrix has no unimodular cover, and neither has their convex hull. For \(r=6\) the example is derived in a few lines from fifteen of the \(45\) rhombus inequalities. For \(r\le5\) every nonempty polytope of this form contains an integral point. This is contained in what the sources report as verified by computation; we prove it independently, by a reduction to finitely many cases and an exact computation. So the question whether such polytopes always contain an integral point, record OWR-741-001 of the UnsolvedMath collection, has the answer yes for \(r\le5\) and no for \(r\ge6\). The sources state that the conjecture was verified for \(r\le6\). In the indexing of their definitions the example has \(r=6\); a hive pattern printed there with the label \(r=5\) has five rows, which is rank \(4\) in that indexing. We cannot tell from the texts which indexing the verification used. The journal version of the paper of De Loera and McAllister was not accessible to us and was not read; we used the arXiv version. No counterexample was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.