immanentImmanent2003-12-05 17:05:382007-08-23 18:50:46Definitionpermanentdeterminantcharactertrace% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\imm}{Imm}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 \emph{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 $A$.