Spectral Bounds and Shifted Complexes: Eigenvalues of the Up-Laplacian via Face Degrees

Abstract

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 $\lambda_1(\Lup_{k-1}(K))\ge d_1(K)+k$, 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 \[ \lambda_2\bigl(\Lup_{k-1}(K)\bigr)\ge d_2(K)+k-1. \] 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