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?

permanentandimmanentcome from?Determinantwas introduced by Gauss when dealing with quadratic forms, in in his context it does determine something... $\endgroup$"immanant"is a participle of a made-up compound verb,"in+mano"(likeemanofromex+mano), with not clear meaning. Otherwise it is a mispelling ofimmanent(fromin+maneo, likepermanentfromper+maneo), maybe to rhyme withdeterminant. $\endgroup$