# 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 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.

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.