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 an integer ${\displaystyle n}$ 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

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

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:

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

The immanant shares several properties with determinant and permanent. In particular, the immanant is multilinear in the rows and columns of the matrix; and the immanant is invariant under simultaneous permutations of the rows or columns by the same element of the symmetric group.

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

The necessary and sufficient conditions for the immanant of a Gram matrix to be ${\displaystyle 0}$ are given by Gamas's Theorem.

• D. E. Littlewood; A.R. Richardson (1934). "Group characters and algebras". Philosophical Transactions of the Royal Society A. 233 (721–730): 99–124. Bibcode:1934RSPTA.233...99L. 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.