# immanent

Let ${S}_{n}$ denote the symmetric group^{} on $n$ elements.
Let $\chi :{S}_{n}\to \u2102$ be a complex character^{}.
For any $n\times n$ matrix $A={({a}_{ij})}_{i,j=1}^{n}$ define the *immanent* of $A$ as

$${\mathrm{Imm}}_{\chi}(A)=\sum _{\sigma \in {S}_{n}}\chi (\sigma )\prod _{j=1}^{n}{A}_{j\sigma (j)}.$$ |

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

Title | immanent |
---|---|

Canonical name | Immanent |

Date of creation | 2013-03-22 14:05:43 |

Last modified on | 2013-03-22 14:05:43 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 17 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 20C30 |

Related topic | permanent |

Related topic | character |