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# 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

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

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 $A$.

Keywords:

permanent, determinant, character, trace

Related:

permanent, character

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

20C30*no label found*

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