Immanant

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)}.}