Immanant

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 ${\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )}$ be a partition of ${\displaystyle n}$ and let ${\displaystyle \chi _{\lambda }}$ be the corresponding irreducible representation-theoretic character of the symmetric group ${\displaystyle S_{n}}$. The immanant of an ${\displaystyle n\times n}$ matrix ${\displaystyle A=(a_{ij})}$ associated with the character ${\displaystyle \chi _{\lambda }}$ is defined as the expression

${\displaystyle \operatorname {Imm} _{\lambda }(A)=\sum _{\sigma \in S_{n}}\chi _{\lambda }(\sigma )a_{1\sigma (1)}a_{2\sigma (2)}\cdots a_{n\sigma (n)}.}$

The determinant is a special case of the immanant, where ${\displaystyle \chi _{\lambda }}$ is the alternating character ${\displaystyle \operatorname {sgn} }$, of S_n, defined by the parity of a permutation.

The permanent is the case where ${\displaystyle \chi _{\lambda }}$ is the trivial character, which is identically equal to 1.

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

${\displaystyle S_{3}}$	${\displaystyle e}$	${\displaystyle (1\ 2)}$	${\displaystyle (1\ 2\ 3)}$
${\displaystyle \chi _{1}}$	1	1	1
${\displaystyle \chi _{2}}$	1	−1	1
${\displaystyle \chi _{3}}$	2	0	−1

As stated above, ${\displaystyle \chi _{1}}$ produces the permanent and ${\displaystyle \chi _{2}}$ produces the determinant, but ${\displaystyle \chi _{3}}$ produces the operation that maps as follows:

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

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