# immanent

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

 $\operatorname{Imm}_{\chi}(A)=\sum_{\sigma\in{S_{n}}}\chi(\sigma)\prod_{j=1}^{n% }A_{j\,\sigma(j)}.$

Special cases of immanents are determinants and permanents — in the case where $\chi$ is the constant character ($\chi(x)=1$ for all $x\in S_{n}$), $\operatorname{Imm}_{\chi}(A)$ 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), $\operatorname{Imm}_{\chi}(A)$ is the determinant of $A$.

Title immanent Immanent 2013-03-22 14:05:43 2013-03-22 14:05:43 Mathprof (13753) Mathprof (13753) 17 Mathprof (13753) Definition msc 20C30 permanent character