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{{Short description|Mathematical function generalizing the determinant and permanent}}
{{distinguish|immanent|imminent (disambiguation){{!}}imminent|eminent (disambiguation){{!}}eminent}}
In mathematics, the '''immanant''' of a [[matrix (mathematics)|matrix]] was defined by [[Dudley E. Littlewood]] and [[Archibald Read Richardson]] as a generalisation of the concepts of [[determinant]] and [[Permanent (mathematics)|permanent]].

Let <math>\lambda=(\lambda_1,\lambda_2,\ldots)</math> be a [[Partition (number theory)|partition]] of an integer <math>n</math> and let <math>\chi_\lambda</math> be the corresponding irreducible [[Group representation|representation-theoretic]] [[Character (group theory)|character]] of the [[symmetric group]] <math>S_n</math>. The ''immanant'' of an <math>n\times n</math> [[Matrix (mathematics)|matrix]] <math>A=(a_{ij})</math> associated with the character <math>\chi_\lambda</math> is defined as the expression

:<math>\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)}.</math>

==Examples==

The determinant is a special case of the immanant, where <math>\chi_\lambda</math> is the [[alternating character]] <math>\sgn</math>, of ''S''<sub>''n''</sub>, defined by the [[parity of a permutation]].

The permanent is the case where <math>\chi_\lambda</math> is the [[trivial character]], which is identically equal to&nbsp;1.

For example, for <math>3 \times 3</math> matrices, there are three irreducible representations of <math>S_3</math>, as shown in the character table:
{| class="wikitable" style="text-align:right;"
|-
! <math>S_3</math>
! <math>e</math>
! <math>(1\ 2)</math>
! <math>(1\ 2\ 3)</math>
|-
| <math>\chi_1</math>
| 1
| 1
| 1
|-
| <math>\chi_2</math>
| 1
| −1
| 1
|-
| <math>\chi_3</math>
| 2
| 0
| −1
|}
As stated above, <math>\chi_1</math> produces the permanent and <math>\chi_2</math> produces the determinant, but <math>\chi_3</math> produces the operation that maps as follows:

:<math>\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \rightsquigarrow 2 a_{11} a_{22} a_{33} - a_{12} a_{23} a_{31} - a_{13} a_{21} a_{32}</math>

==Properties==
The immanant shares several properties with determinant and permanent. In particular, the immanant is [[Multilinear form|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 polynomial|Schur functions]] in the [[representation theory of the symmetric group]].

The necessary and sufficient conditions for the immanant of a [[Gram matrix]] to be <math>0</math> are given by [[Gamas's Theorem]].

==References==

* {{cite journal | author=D. E. Littlewood | authorlink=Dudley E. Littlewood |author2=A.R. Richardson |authorlink2=Archibald Read Richardson  | title=Group characters and algebras | journal=[[Philosophical Transactions of the Royal Society A]] | year=1934 | volume=233 | pages=99–124 | doi=10.1098/rsta.1934.0015 | issue=721–730 | bibcode=1934RSPTA.233...99L | doi-access= }}
* {{cite book | author=D. E. Littlewood | authorlink=Dudley E. Littlewood | title=The Theory of Group Characters and Matrix Representations of Groups | edition=2nd | year=1950 | publisher=Oxford Univ. Press (reprinted by AMS, 2006) | page=81 }}

[[Category:Linear algebra]]
[[Category:Matrix theory]]
[[Category:Permutations]]