2026/07/22 by Jing Huang
#math.CO
Let \(F⊆\binomVq\) be a \(q\)-uniform family on a finite vertex set \(V\). Write \(sr(F)\) for the sum of the \(r\) largest eigenvalues of its simplicial up-Laplacian and \(dF(v)\) for the degree of \(v∈ V\). Then Dr(F)=∑v∈ Vmin\dF(v),r\ is the \(r\)-th partial sum of the conjugate degree sequence of \(F\). The majorization assertion in the Duval--Reiner conjecture [Trans. Amer. Math. Soc., 2002] states that \(sr(F)≤ Dr(F)\) for every \(q\)-uniform family \(F\) and every \(r≥1\). We disprove this assertion in two complementary senses: for every \(r≥5\), there is a strict counterexample at index \(r\) in some uniformity, while every uniformity \(q≥3\) admits a strict counterexample at some index \(r≥5\). In contrast, we prove the universal inequality \(s2(F)≤ D2(F)\) and classify all equality cases. The counterexamples are obtained from two \(3\)-uniform seeds with explicitly computed characteristic polynomials through defect-preserving ridge-whiskering and set-complement duality. For the second partial sum, core completion reduces the problem to the boundary matrix of a complete simplex, where Ky Fan variational and compression arguments yield both the inequality and its equality classification.