AI 中文总结
研究单纯复形，证明其\((r - 1)\)维上拉普拉斯谱由\((r - 1)\)维面的共轭度序列优超并建立布劳威尔型不等式，给出格罗恩 - 梅里斯 - 白定理等高维类似物，还指出杜瓦尔 - 赖纳猜想在\(r \geq 2\)时不成立。
AI 中文摘要
设\(K\)为\(r\)维单纯复形。我们证明其\((r - 1)\)维上拉普拉斯谱由其\((r - 1)\)维面的共轭度序列优超：\[ {\mathbf{\lambda}}_{r - 1}(K) \preccurlyeq {\mathbf d}_{r - 1}^\top(K). \] 我们还建立了一个布劳威尔型不等式：对于每个整数\(\ell \geq 1\)，\[ \sum_{i = 1}^{\ell}\lambda_{r - 1,i}(K) \leq \frac{r + 1}{2}f_r(K) + \frac{f_{r - 2}(K)}{r} \binom{\ell + 1}{2}, \] 其中\(\lambda_{r - 1,i}(K)\)表示谱\({\mathbf{\lambda}}_{r - 1}(K)\)中第\(i\)大的特征值，\(f_t(K)\)表示\(K\)的\(t\)维面的数量。这些结果提供了格罗恩 - 梅里斯 - 白定理和布劳威尔 - 科塔里 - 图多塞定理的高维类似物，并在\(r = 1\)时恢复了相应的图结果。我们表明，关于顶点共轭度序列优超的杜瓦尔 - 赖纳猜想在每个维度\(r \geq 2\)时都不成立。更确切地说，对于每个\(n \geq r + 5\)，我们构造了一个\(n\)个顶点的纯\(r\)维复形，它在第五个部分和处违反了猜想的不等式。
英文摘要
Let $K$ be an $r$-dimensional simplicial complex. We prove that the spectrum of its $(r - 1)$-dimensional up-Laplacian is majorized by the conjugate degree sequence of its $(r - 1)$-dimensional faces: \[ {\mathbfλ}_{r-1}(K) \preccurlyeq {\mathbf d}_{r-1}^\top(K). \] We also establish a Brouwer-type inequality: for every integer $\ell \geq 1$, \[ \sum_{i = 1}^{\ell}λ_{r-1,i}(K) \leq \frac{r + 1}{2}f_r(K) + \frac{f_{r - 2}(K)}{r} \binom{\ell + 1}{2}, \] where $λ_{r-1,i}(K)$ denotes the $i$-th largest eigenvalue in the spectrum ${\mathbfλ}_{r-1}(K)$, and $f_t(K)$ denotes the number of $t$-dimensional faces of $K$. These results provide higher-dimensional analogs of the Grone-Merris-Bai theorem and the Brouwer-Kothari-Tudose theorem and recover the corresponding graph results when $r=1$. We show that the Duval-Reiner conjecture on the majorization by the conjugate degree sequence of vertices fails in every dimension $r \geq 2$. More precisely, for every $n \geq r + 5$, we construct a pure $r$-dimensional complex on $n$ vertices that violates the conjectured inequality at the fifth partial sum.
