TITLE: What are the applications of immanants?
QUESTION [36 upvotes]: Definitions of determinant:
$\det(A) = \sum_{\sigma \in S_n}\operatorname{sgn} \sigma\prod_{i}a_{i, \sigma(i)}$
and permanent:
$\mathrm{per}(A) = \sum_{\sigma \in S_n}\prod_{i}a_{i, \sigma(i)}$
admit a generalization in the form of immanant:
$\mathrm{Imm}_{\lambda}(A) = \sum_{\sigma \in S_n}\chi_{\lambda}(\sigma)\prod_{i}a_{i, \sigma(i)}$
where $\lambda$ labels irreducible representations of $S_n$ and $\chi_{\lambda}$ is the character. Determinant and permanent are easily seen to be special cases of $\mathrm{Imm}_{\lambda}$.
While determinants are ubiquitous in mathematics and permanents also have many application, esp. in combinatorial problems, other kinds of immanants seem to be rarely used. Are there any problems where use of $\mathrm{Imm}_{\lambda}$ other than $\det$ and $\operatorname{per}$ is natural?

REPLY [12 votes]: A problem with applying immanants is that after expressing something in terms of immanants, you might not be closer to calculating them and you can't easily manipulate expressions of immanants.  For example, suppose you want to count Hamiltonian cycles in a graph (positivity solves the Hamiltonian cycle problem). You notice that this is expressible as a linear combination of immanants of the adjacency matrix:
$$\sum_{\sigma \in S_n} f(\sigma)\prod_{i}a_{i, \sigma(i)} $$
where $f(\sigma) = 1$ when $\sigma$ is an $n$-cycle and $0$ otherwise. Since $f$ is constant on conjugacy classes, we can decompose it as a linear combination of characters. However, the Hamiltonian cycle problem is NP-complete, so it would be surprising if there were a quick way to compute this linear combination of immanants. Either there are many terms to compute, or these terms are hard to compute, or both, even if you are just trying to determine whether the result is positive.