Immanent

The immanent is a quantity of a matrix defined by Dudley Littlewood and Archibald Richardson . It is a generalization of the determinant as well as the permanent .

Let be a partition of and the corresponding irreducible representational character of the symmetric group . The immanent with the character of a - matrix is defined as ${\displaystyle \lambda =\left(\lambda _{1},\lambda _{2},\dots \right)}$ ${\displaystyle n}$ ${\displaystyle \chi _{\lambda }}$ ${\displaystyle {\mathfrak {S}}_{n}}$ ${\displaystyle \chi _{\lambda }}$ ${\displaystyle n\times n}$ ${\displaystyle A=\left(a_{ij}\right)_{ij}}$

${\displaystyle \operatorname {Imm} _{\chi }(A):=\sum _{\sigma \in {\mathfrak {S}}_{n}}\chi (\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}}$

The permanent is the special case with the trivial character.

The determinant is the special case of the immanent with , the alternating character . ${\displaystyle \chi _{\lambda }=\operatorname {sgn} }$

For example, there are three irreducible representations of matrices , as the following table shows. ${\displaystyle 3\times 3}$ ${\displaystyle {\mathfrak {S}}_{3}}$

${\displaystyle {\mathfrak {S}}_{3}}$	${\displaystyle e}$	${\displaystyle (1\,2)}$	${\displaystyle (1\,2\,3)}$
${\displaystyle \chi _{1}}$	1	1	1
${\displaystyle \chi _{2}}$	1	−1	1
${\displaystyle \chi _{3}}$	2	0	−1

As mentioned above result and the permanent or the determinant; on the other hand, one gets the figure ${\displaystyle \chi _{1}}$ ${\displaystyle \chi _{2}}$ ${\displaystyle \chi _{3}}$

${\displaystyle \operatorname {Imm} _{\chi _{3}}{\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}\quad =\quad 2a_{11}a_{22}a_{33}-a_{12}a_{23}a_{31}-a_{13}a_{21}a_{32}}$

Littlewood and Richardson studied the relationship with Schur polynomials.

supporting documents

• Dudley Littlewood , Archibald Richardson : Group characters and algebras . In: Philosophical Transactions of the Royal Society A . 233, No. 721-730, 1934, pp. 99-124. doi : 10.1098 / rsta.1934.0015 .

• Dudley Littlewood : The Theory of Group Characters and Matrix Representations of Groups . 2nd Edition. Oxford Univ. Press (reprinted by AMS, 2006), 1950, p. 81 (English).