杜瓦尔 - 赖纳猜想：反例与第二部分和不等式
The Duval--Reiner Conjecture: Counterexamples and the Second Partial-Sum Inequality
AI总结：
研究杜瓦尔 - 赖纳猜想中\(s_r(F)\leq D_r(F)\)的断言，通过特定方法得到反例反驳该断言，同时证明\(s_2(F)\leq D_2(F)\)并分类等式情形，核心方法包括利用特定种子、核心完备化及相关矩阵论证。
AI中文摘要：
设\(F\subseteq\binom{V}{q}\)是有限顶点集\(V\)上的\(q\) - 均匀族。用\(s_r(F)\)表示其单纯上拉普拉斯矩阵的\(r\)个最大特征值之和，\(d_F(v)\)表示\(v\in V\)的度。杜瓦尔 - 赖纳猜想中的优超断言称对每个\(q\) - 均匀族\(F\)和\(r\geq1\)，有\(s_r(F)\leq D_r(F)\)。本文从两方面反驳此断言：对\(r\geq5\)，在某些均匀性下存在\(r\)处的严格反例，且每个\(q\geq3\)在某些\(r\geq5\)处有严格反例。同时证明了\(s_2(F)\leq D_2(F)\)并对等式情形分类。反例通过特定方法得到，对于第二部分和，通过核心完备化将问题归结为完全单纯形的边界矩阵，利用相关论证得到不等式及其等式分类。
英文摘要：
Let \(F\subseteq\binom{V}{q}\) be a \(q\)-uniform family on a finite vertex set \(V\). Write \(s_r(F)\) for the sum of the \(r\) largest eigenvalues of its simplicial up-Laplacian and \(d_F(v)\) for the degree of \(v\in V\). Then $D_r(F)=\sum_{v\in V}\min\{d_F(v),r\}$ 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 \(s_r(F)\le D_r(F)\) for every \(q\)-uniform family \(F\) and every \(r\ge1\). We disprove this assertion in two complementary senses: for every \(r\ge5\), there is a strict counterexample at index \(r\) in some uniformity, while every uniformity \(q\ge3\) admits a strict counterexample at some index \(r\ge5\). In contrast, we prove the universal inequality \(s_2(F)\le D_2(F)\) 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.
