AI 中文总结
该论文解决了Duval和Reiner关于单纯复形上拉普拉斯算子特征值的猜想，证明了纯复形的非零谱等于共轭顶点度分划当且仅当复形为移位复形，还得到了相关特征值的尖锐界及组合判据。
AI 中文摘要
设K是一个有限k维单纯复形，其中k≥1，记其(k-1)维上拉普拉斯算子为$\u200b\tilde{\u200b}\text{L}_{k-1}(K)\u200b$。我们解决了Duval和Reiner猜想的等号情形：对于纯复形，$\u200b\tilde{\u200b}\text{L}_{k-1}(K)\u200b$的非零谱等于$d^{\text{v}}(K)^{\text{T}}$（共轭顶点度分划）当且仅当该复形同构于一个移位复形。事实上，我们证明了更强的结论：仅二阶幂和相等就足以迫使复形具有移位性，且两者的差值等于面的失败初等移位数量的两倍。接下来，我们刻画Duval–Reiner界$\u200b\tilde{\u200b}\text{L}_{k-1}(K)\u200b$的第一特征值$\u03bb_1(\u200b\tilde{\u200b}\text{L}_{k-1}(K)\u200b)\u200b\ue000 d_1(K)+k$的等号情形，其中$d_1(K)$是脊的最大上度。我们引入“清洁脊”这一概念，它是支配顶点的高维类似物，并将其与关联的带符号面邻接矩阵的带符号Sachs展开相结合，得到一个必要且充分的组合判据。当带符号面邻接图连通且平衡（平衡等价于k维下的不可定向性）时，该判据简化为所有面共享一个公共脊的简单条件。最后，只要K至少有两个面，我们就证明了尖锐界$\u03bb_2(\u200b\tilde{\u200b}\text{L}_{k-1}(K)\u200b)\u200b\ue000 d_2(K)+k-1$，这为图的第二指标Brouwer–Haemers界提供了高维类似物；而该界向更高特征值指标的自然逐项扩展在m=3时就失效了，例如……
英文摘要
Let $K$ be a finite $k$-dimensional simplicial complex with $k\ge1$, and let $\Lup_{k-1}(K)$ be its $(k-1)$-dimensional up-Laplacian. We resolve the equality case of a conjecture of Duval and Reiner: for a pure complex, the nonzero spectrum of $\Lup_{k-1}(K)$ equals ${d^{\mathrm v}(K)}^{\T}$, the conjugate vertex-degree partition, if and only if the complex is isomorphic to a shifted complex. In fact, we prove more: equality of the second power sums alone already forces shiftedness, and the gap between them equals twice the number of failed elementary shifts of facets. We next characterize equality in the Duval--Reiner bound $λ_1(\Lup_{k-1}(K))\ge d_1(K)+k$, where $d_1(K)$ is the maximum upper degree of a ridge. We introduce the \emph{clean ridge}, a higher-dimensional analogue of a dominating vertex, and combine it with a signed Sachs expansion for the associated signed facet-adjacency matrix to obtain a necessary and sufficient combinatorial criterion. When the signed facet-adjacency graph is connected and balanced---with balance equivalent to disorientability in dimension $k$---the criterion reduces to the simple condition that all facets share a common ridge. Finally, whenever $K$ has at least two facets, we prove the sharp bound \[ λ_2\bigl(\Lup_{k-1}(K)\bigr)\ge d_2(K)+k-1. \] This gives a higher-dimensional analogue of the second-index Brouwer--Haemers bound for graphs. The natural termwise extension to higher eigenvalue indices fails already at $m=3$, as a s
