Immanant of a matrix

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Immanant redirects here; it should not be confused with the philosophical immanent.

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 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:

Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.

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