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
,
…
)
be a partition of
n
and let
χ
λ
be the corresponding irreducible representation-theoretic character of the symmetric group
S
n
. The immanant of an
n
×
n
matrix
A
=
(
a
i
j
)
associated with the character
χ
λ
is defined as the expression
I
m
m
λ
(
A
)
=
∑
σ
∈
S
n
χ
λ
(
σ
)
a
1
σ
(
1
)
a
2
σ
(
2
)
⋯
a
n
σ
(
n
)
.
The determinant is a special case of the immanant, where
χ
λ
is the alternating character
sgn
, of Sn, defined by the parity of a permutation.
The permanent is the case where
χ
λ
is the trivial character, which is identically equal to 1.
For example, for
3
×
3
matrices, there are three irreducible representations of
S
3
, as shown in the character table:
As stated above,
χ
1
produces the permanent and
χ
2
produces the determinant, but
χ
3
produces the operation that maps as follows:
(
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
)
⇝
2
a
11
a
22
a
33
−
a
12
a
23
a
31
−
a
13
a
21
a
32
Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group.
