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.
be the corresponding irreducible representation-theoretic character of the symmetric group
is defined as the expression The determinant is a special case of the immanant, where
, 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.
matrices, there are three irreducible representations of
, as shown in the character table: As stated above,
produces the permanent and
produces the determinant, but
produces the operation that maps as follows: 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