Equality Cases for the Face-Degree Majorization Theorem on Simplicial Complexes

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.