AI 中文总结
该研究刻画了高维控制定理的等号情形，证明r维单纯复形中控制关系等号的充要条件，发现等号仅在序列耗尽非零项后出现，推论得等号等价于复形含唯一r-单形，证明基于局部下拉普拉斯分解与Ky Fan不等式等号。
AI 中文摘要
Grone-Merris-Bai定理指出，简单图的拉普拉斯谱被其共轭度序列所控制。最近，Zhang、Song和Fan将该结果推广到单纯复形，建立了(r-1)维上拉普拉斯谱与共轭(r-1)度序列之间的控制关系。本文刻画了该高维控制定理部分和不等式中的所有等号情形。对于每个r≥2的r维单纯复形X，证明了∑_{i=1}^{q}λ_{r-1,i}(X)=∑_{i=1}^{q}d_{r-1,i}^⊤(X)当且仅当q≥max{rankB_r(X),Δ_{r-1}(X)}。因此，与图的情形不同，等号仅在两个序列都耗尽所有非零项后才会出现。作为推论，第一部分和的等号以及整个序列之间的等号都等价于X包含唯一的r-单形。该证明基于局部下拉普拉斯分解和Ky Fan不等式的等号情形。
英文摘要
The Grone--Merris--Bai theorem states that the Laplacian spectrum of a simple graph is majorized by its conjugate degree sequence. Recently, Zhang, Song, and Fan extended this result to simplicial complexes by establishing a majorization relation between the spectrum of the $(r-1)$-dimensional up-Laplacian and the conjugate $(r-1)$-degree sequence. In this paper, we characterize all equality cases in the partial-sum inequalities of this higher-dimensional majorization theorem. For every $r$-dimensional simplicial complex $X$ with $r\ge2$, we prove that \[ \sum_{i=1}^{q}λ_{r-1,i}(X) = \sum_{i=1}^{q}d_{r-1,i}^{\top}(X) \] if and only if \[ q\ge \max\{\operatorname{rank}B_r(X),Δ_{r-1}(X)\}. \] Thus, unlike the graph case, equality can occur only after both sequences have exhausted all their nonzero terms. As consequences, equality in the first partial sum and equality between the entire sequences are both equivalent to $X$ containing a unique $r$-simplex. The proof is based on the local down-Laplacian decomposition and the equality case of the Ky Fan inequality.
Comments17pages
