Time complexity of the immanant
The state of affairs as of 2013 is described in Mertens and Moore, The Complexity of the Ferminonants and Immanants of Constant Width. Let $\lambda$ be the partition corresponding to $\chi$.

Immanants are easy if the leftmost column of $\lambda$ contains $n - O(1)$ boxes (Barvinok; Bürgisser).
Immanants are hard if $\lambda_i - \lambda_{i+1} = \Omega(n^\delta)$ (Brylinsky and Brylinsky, improving on results of Hartmann and Bürgisser which applied only to hooks and rectangles).
The problem of computing the $\lambda$-immanant given $\lambda$ is hard even if $\lambda$ is restricted to have width 2, and promised to have at least $n^\delta$ boxes in the second column (Mertens and Moore; de Rugy-Altherre).

The paper of de Rugy-Altherre is subsequent to Mertens–Moore.