Immanant Explained

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)

be a partition of an integer

n

and let

\chiλ

be the corresponding irreducible representation-theoretic character of the symmetric group

Sn

. The immanant of an

n x n

matrix

A=(aij)

associated with the character

\chiλ

is defined as the expression

\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).

Examples

The determinant is a special case of the immanant, where

\chiλ

is the alternating character

sgn

, of Sn, defined by the parity of a permutation.

The permanent is the case where

\chiλ

is the trivial character, which is identically equal to 1.

For example, for

3 x 3

matrices, there are three irreducible representations of

S3

, as shown in the character table:

S3

e

(1 2)

(1 2 3)

\chi1

111

\chi2

1−11

\chi3

20−1
As stated above,

\chi1

produces the permanent and

\chi2

produces the determinant, but

\chi3

produces the operation that maps as follows:

\begin{pmatrix}a11&a12&a13\a21&a22&a23\a31&a32&a33\end{pmatrix}\rightsquigarrow2a11a22a33-a12a23a31-a13a21a32

Properties

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

are given by Gamas's Theorem.

References

. 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 .