Immanants generalize the notion of determinant and permanent, and it is defined as $$Imm_{\lambda}(A)=\sum_{\sigma \in S_n}\chi_{\lambda}(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}$$ where $\chi_{\lambda}$ corresponds to the irreducible characters of a symmetric group and partition $\lambda$ of $n$.
- When $\lambda = (1,1,\ldots, 1)$, $\chi_{\lambda}=sgn(\sigma)$ is a sign function, and $Imm_{\lambda}(A)=det(A)$ is the determinant.
- When $\lambda = (2,1,\ldots, 1)$, $\chi_{\lambda}=sgn(\sigma)(|Fix(\sigma)|-1)$. ($Fix(\sigma)$ is the number of fixed points of $\sigma$).
- When $\lambda = (n-1,1)$, $\chi_{\lambda}=|Fix(\sigma)|-1$.
- When $\lambda = (n)$, $\chi_{\lambda}=1$, and $Imm_{\lambda}(A)=per(A)$ is the permanent.
Does anyone know of the closed form for the character functions $\chi_{\lambda}$ for other partitions $\lambda$ of $n$?