Immanant

For an $n\times n$ matrix, let $S$ denote any permutation $e_1$, $e_2$, ..., $e_n$ of the set of numbers 1, 2, ..., $n$, and let $\chi^{(\lambda)}(S)$ be the character of the symmetric group corresponding to the partition $(\lambda)$. Then the immanant $\vert a_{mn}\vert^{(\lambda)}$ is defined as

$$\vert a_{mn}\vert^{(\lambda)}=\sum \chi^{(\lambda)}(S)P_S$$

where the summation is over the $n!$ permutations of the Symmetric Group and

$$P_s=a_{1e_1}a_{2e_2}\cdots a_{ne_n}.$$

See also Determinant, Permanent

References

Littlewood, D. E. and Richardson, A. R. ``Group Characters and Algebra.'' Philos. Trans. Roy. Soc. London A 233, 99-141, 1934.

Littlewood, D. E. and Richardson, A. R. ``Immanants of Some Special Matrices.'' Quart. J. Math. (Oxford) 5, 269-282, 1934.

Wybourne, B. G. ``Immanants of Matrices.'' §2.19 in Symmetry Principles and Atomic Spectroscopy. New York: Wiley, pp. 12-13, 1970.