#P-hardness proofs of matrix immanants evaluated on restricted matrices
Istvan Miklos, Cordian Riener
TL;DR
This work investigates the #P-hardness of computing matrix immanants on restricted inputs, showing that broad families of immanants remain intractable even for structured matrices. The authors leverage the Murnaghan–Nakayama rule and border-strip tableaux to understand irreducible characters, and they construct intricate graph gadgets to reduce perfect-matchings counts in 3-regular bipartite (and planar) graphs to immanant evaluations. They prove #P-hardness for 0-1 adjacency matrices when the shape $\lambda$ contains a large domino-tileable region, and for edge-weighted planar directed graphs when $\lambda=(\mathbf{1}+\lambda_d)$ with $|\lambda_d|=n^{\varepsilon}$ and domino tiling properties; the reductions rely on careful cycle-structure control and interpolation over auxiliary parameters. The results enrich the landscape of restricted-input hardness for immanants and raise open questions about related objects such as fermionants, TL-immanants, and partition-algebra recombinants.
Abstract
We establish the $\#P$-hardness of computing a broad class of immanants, even when restricted to specific categories of matrices. Concretely, we prove that computing $λ$-immanants of $0$-$1$ matrices is $\#P$-hard whenever the partition~$λ$ contains a sufficiently large domino-tileable region, subject to certain technical conditions. We also give hardness proofs for some $λ$-immanants of weighted adjacency matrices of planar directed graphs, such that the shape $λ= (\mathbf{1} + λ_d)$ has size $n$ such that $|λ_d| = n^\varepsilon$ for some $0 < \varepsilon < \frac{1}{2}$, and such that for some $w$, the shape $λ_d/(w)$ is tileable with $1 \times 2$ dominos.