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 and let be the corresponding irreducible representation-theoretic character of the symmetric group . The immanant of an matrix associated with the character is defined as the expression
The determinant is a special case of the immanant, where is the alternating character , 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 matrices, there are three irreducible representations of , as shown in the character table:
| 1 | 1 | 1 | |
| 1 | −1 | 1 | |
| 2 | 0 | −1 |
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 are given by Gamas's Theorem.
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This is a glossary of representation theory in mathematics.
Gamas's theorem is a result in multilinear algebra which states the necessary and sufficient conditions for a tensor symmetrized by an irreducible representation of the symmetric group to be zero. It was proven in 1988 by Carlos Gamas. Additional proofs have been given by Pate and Berget.
