Immanant of a matrix

Immanant redirects here; it should not be confused with the philosophical immanent.

In mathematics, the immanant of a matrix was defined by Dudley E. Littlewood and Archibald Read Richardson as a generalisation of the concepts of determinant and permanent.

Let $\lambda=(\lambda_1,\lambda_2,\ldots)$ be a partition of $n$ and let $\chi_\lambda$ be the corresponding irreducible representation-theoretic character of the symmetric group $S_n$ . The immanant of an $n\times n$ matrix $A=(a_{ij})$ associated with the character $\chi_\lambda$ is defined as the expression

{\rm Imm}_\lambda(A)=\sum_{\sigma\in S_n}\chi_\lambda(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}\cdots a_{n\sigma(n)}.

The determinant is a special case of the immanant, where $\chi_\lambda$ is the alternating character $\sgn$ , of S_n, defined by the parity of a permutation.

The permanent is the case where $\chi_\lambda$ is the trivial character, which is identically equal to 1.

For example, for $3 \times 3$ matrices, there are three irreducible representations of $S_3$ , as shown in the character table:

$S_3$	$e$	$(1\ 2)$	$(1\ 2\ 3)$
$\chi_1$	1	1	1
$\chi_2$	1	−1	1
$\chi_3$	2	0	−1

As stated above, $\chi_1$ produces the permanent and $\chi_2$ produces the determinant, but $\chi_3$ produces the operation that maps as follows:

\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \rightsquigarrow 2 a_{11} a_{22} a_{33} - a_{12} a_{23} a_{31} - a_{13} a_{21} a_{32}

Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.

References

D. E. Littlewood; A.R. Richardson (1934). "Group characters and algebras". Philosophical Transactions of the Royal Society A 233 (721–730): 99–124. doi:10.1098/rsta.1934.0015.

D. E. Littlewood (1950). The Theory of Group Characters and Matrix Representations of Groups (2nd ed.). Oxford Univ. Press (reprinted by AMS, 2006). p. 81.

External links

Immanent at PlanetMath

This article is issued from Wikipedia - version of the Friday, October 23, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.