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 λ = ( λ 1 , λ 2 , … ) {\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )} be a partition of an integer n {\displaystyle n} and let χ λ {\displaystyle \chi _{\lambda }} be the corresponding irreducible representation-theoretic character of the symmetric group S n {\displaystyle S_{n}} . The immanant of an n × n {\displaystyle n\times n} matrix A = ( a i j ) {\displaystyle A=(a_{ij})} associated with the character χ λ {\displaystyle \chi _{\lambda }} is defined as the expression
Imm λ ( A ) = ∑ σ ∈ S n χ λ ( σ ) a 1 σ ( 1 ) a 2 σ ( 2 ) ⋯ a n σ ( n ) = ∑ σ ∈ S n χ λ ( σ ) ∏ i = 1 n a i σ ( i ) . {\displaystyle \operatorname {Imm} _{\lambda }(A)=\sum _{\sigma \in S_{n}}\chi _{\lambda }(\sigma )a_{1\sigma (1)}a_{2\sigma (2)}\cdots a_{n\sigma (n)}=\sum _{\sigma \in S_{n}}\chi _{\lambda }(\sigma )\prod _{i=1}^{n}a_{i\sigma (i)}.}
Examples The determinant is a special case of the immanant, where χ λ {\displaystyle \chi _{\lambda }} is the alternating character sgn {\displaystyle \operatorname {sgn} } , of Sn, defined by the parity of a permutation. The permanent is the case where χ λ {\displaystyle \chi _{\lambda }} is the trivial character, which is identically equal to 1. For example, for 3 × 3 {\displaystyle 3\times 3} matrices, there are three irreducible representations of S 3 {\displaystyle S_{3}} , as shown in the character table:
As stated above, χ 1 {\displaystyle \chi _{1}} produces the permanent and χ 2 {\displaystyle \chi _{2}} produces the determinant, but χ 3 {\displaystyle \chi _{3}} produces the operation that maps as follows:
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