Let $S_n$ denote the symmetric group on $n$ elements. Let $\chi:S_n\to\mathbb C$ be a complex character. For any $n\times n$ matrix $A=(a_{ij})_{i,j=1}^n$ define the immanent of $A$ as $$ \imm_{\chi} (A)=\sum_{\sigma\in {S_n}} \chi(\sigma) \prod_{j=1}^n A_{j \, \sigma( j)} $$

Special cases of immanents are determinants and permanents -- in the case where $\chi$ is the constant character ($\chi (x) = 1$ for all $x \in S_n$ ), $\imm_{\chi} (A)$ is the permanent of $A$ . In the case where $\chi$ is the sign of the permutation (which is the character of the permutation group associated to the (non-trivial) one-dimensional representation), $\imm_{\chi} (A)$ is the determinant of $A$ .