immanent

Let $S_n$ denote the symmetric group on $n$ elements. Let $\chi :S_n\to ℂ$ 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