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

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