PlanetMath: immanent

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

immanent

(Definition)

Let denote the symmetric group on 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 as

$\displaystyle {\mathrm{Imm}}_{\chi} (A)=\sum_{\sigma\in {S_n}} \chi(\sigma) \prod_{j=1}^n A_{j \, \sigma( j)}.$

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 . 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 .

"immanent" is owned by Mathprof. [ full author list (3) | owner history (3) ]

(view preamble | get metadata)