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=(λ1,λ2,\ldots)
n
\chiλ
Sn
n x n
A=(aij)
\chiλ
\operatorname{Imm}λ(A)=\sum
\sigma\inSn |
\chiλ(\sigma)a1\sigma(1)a2\sigma(2) … an\sigma(n)=
\sum | |
\sigma\inSn |
\chiλ(\sigma)
n | |
\prod | |
i=1 |
ai\sigma(i).
The determinant is a special case of the immanant, where
\chiλ
sgn
The permanent is the case where
\chiλ
For example, for
3 x 3
S3
S3 | e | (1 2) | (1 2 3) | |
---|---|---|---|---|
\chi1 | 1 | 1 | 1 | |
\chi2 | 1 | −1 | 1 | |
\chi3 | 2 | 0 | −1 |
\chi1
\chi2
\chi3
\begin{pmatrix}a11&a12&a13\ a21&a22&a23\ a31&a32&a33\end{pmatrix}\rightsquigarrow2a11a22a33-a12a23a31-a13a21a32
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
0
. D. E. Littlewood . Dudley E. Littlewood . The Theory of Group Characters and Matrix Representations of Groups . 2nd . 1950 . Oxford Univ. Press (reprinted by AMS, 2006) . 81 .