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 be a partition of an integer n and let 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 is defined as the expression

Examples

The determinant is a special case of the immanant, where is the alternating character \sgn, of Sn, defined by the parity of a permutation. The permanent is the case where 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: As stated above, \chi_1 produces the permanent and \chi_2 produces the determinant, but \chi_3 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 simultaneous permutations of the rows or columns by the same element of the symmetric group. Littlewood and Richardson studied the relation of the immanant to Schur functions in the representation theory of the symmetric group. The necessary and sufficient conditions for the immanant of a Gram matrix to be 0 are given by Gamas's Theorem.