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 formula_1 be a partition of formula_2 and let formula_3 be the corresponding irreducible representation-theoretic character of the symmetric group formula_4. The "immanant" of an formula_5 matrix formula_6 associated with the character formula_3 is defined as the expression
Examples.
The determinant is a special case of the immanant, where formula_3 is the alternating character formula_10, of "S""n", defined by the parity of a permutation.
The permanent is the case where formula_3 is the trivial character, which is identically equal to 1.
For example, for formula_12 matrices, there are three irreducible representations of formula_13, as shown in the character table:
As stated above, formula_14 produces the permanent and formula_15 produces the determinant, but formula_16 produces the operation that maps as follows:
Properties.
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 permutations of the rows or columns. 
Littlewood and Richardson studied the relation of the immanant to Schur functions in the representation theory of the symmetric group.