#P-hardness of computing matrix immanants are proved for each member of a broad class of shapes and restricted sets of matrices. The class is characterized in the following way... If a shape of size $n$ in it is in form $(w,\mathbf{1}+\lambda)$ or its conjugate is in that form, where $\mathbf{1}$ is the all-$1$ vector, then $|\lambda|$ is $n^{\varepsilon}$ for some $0<\varepsilon$, $\lambda$ can be tiled with $1\times 2$ dominos and $(3w+3h(\lambda)+1)|\lambda| \le n$, where $h(\lambda)$ is the height of $\lambda$. The problem remains \#P-hard if the immanants are evaluated on $0$-$1$ matrices. We also give hardness proofs of some immanants whose shape $\lambda = (\mathbf{1}+\lambda_d)$ has size $n$ such that $|\lambda_d| = n^{\varepsilon}$ for some $0<\varepsilon<\frac{1}{2}$, and for some $w$, the shape $\lambda_d/(w)$ is tilable with $1\times 2$ dominos. The \#P-hardness result holds when these immanants are evaluated on adjacency matrices of planar, directed graphs, however, in these cases the edges have small positive integer weights. read more

PDF Abstract
Computational Complexity
Representation Theory

Add Datasets
introduced or used in this paper