What are the (homogeneous adjacency) spectra of the ultracube and the complete hypergraph

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite $k$-uniform hypergraph $H=(V,E)$, with $k\geq2$ and $V=[n]$, define its homogeneous adjacency eigenvalues as those $\lambda\in\mathbb C$ for which some $x\in\mathbb C^n\setminus\{0\}$ satisfies \[\sum_{e\in E:\,i\in e}\prod_{j\in e\setminus\{i\}}x_j=\lambda x_i^{k-1}\quad(i\in[n]).\] This is the adjacency tensor with entry $1/(k-1)!$ on ordered indices forming an edge, and zero otherwise. The characteristic polynomial is the homogeneous resultant of these $n$ equations after moving the right-hand sides to the left, normalized to be monic in $\lambda$. The spectrum consists of its complex roots, counted with their algebraic multiplicities. The complete $k$-uniform hypergraph $K_n^{(k)}$ has every $k$-subset of $[n]$ as an edge. Let $E_k$ be the hypergraph on $[k]$ consisting of one edge. The $d$-dimensional $k$-uniform ultracube $Q_k^d$ has vertex set $[k]^d$; its edges are the $k$ vertices obtained by varying one coordinate over $[k]$ while fixing all others. Thus $Q_k^d$ is the Cartesian power of $E_k$.

Statement recorded in the supplied source.

Determine the complete homogeneous adjacency spectra of $K_n^{(k)}$ for all integers $n\geq k\geq2$ and of $Q_k^d$ for all $k\geq2,d\geq1$, including algebraic multiplicities. In particular, describe the ultracube eigenvalues which do not arise merely by summing eigenvalues of its one-edge factors. [S2]

Short English statement

Determine all homogeneous adjacency eigenvalues, with multiplicity, of complete uniform hypergraphs and Cartesian powers of a single hyperedge.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100029 (AMR-030-0029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Joshua Cooper and Aaron Dutle, Spectra of Uniform Hypergraphs (2011), Definitions 2.3--2.6 and 3.1; sections 4.3--4.4 and 4.6; section 5, questions (1) and (3). https://arxiv.org/abs/1106.4856.