# Immanant of a matrix

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