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Viewing Version
10
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'immanent'
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| Title of object: |
immanent |
| Canonical Name: |
Immanent |
| Type: |
Definition |
| Created on: |
2003-12-05 17:05:38 |
| Modified on: |
2005-05-22 21:16:49 |
| Classification: |
msc:20C30 |
| Keywords: |
permanent, determinant, character, trace |
Revision comment (for changes between this and next version):
| Changes for correction #6684 ('typo'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\DeclareMathOperator{\imm}{Imm} |
Content:
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
\[
\imm_{\chi} (A)=\sum_{\sigma\in {S_n}} \chi(\sigma) \prod_{j=1}^n A(j,\sigma j)
\qquad .\]
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 pof the permutation (which is the character of the permutation group associated to the one-dimensional representation), $\imm_{\chi} (A)$ is the determinanant of $A$. |
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