The immanant of a matrix
Encyclopedia
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
be a partition of 
and let 
be the corresponding irreducible representation-theoretic
character of the symmetric group
. The immanant of an 
matrix 
associated with the character 
is defined as the expression
The determinant is a special case of the immanant, where
is the alternating character 
, of Sn, defined by the parity of a permutation.
The permanent is the case where
is the trivial character, which is identically equal to 1.
Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group
.
Dudley E. Littlewood
Dudley Ernest Littlewood was a British mathematician known for his work in group representation theory....
and Archibald Read Richardson
Archibald Read Richardson
Archibald Read Richardson was a British mathematician known for his work in algebra.He collaborated with Dudley E. Littlewood on invariants and group representation theory...
as a generalisation of the concepts of determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
and permanent
Permanent
The permanent of a square matrix in linear algebra, is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix...
.
Let
be a partition of and let be the corresponding irreducible representation-theoreticGroup representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
character of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
. The immanant of an matrix associated with the character is defined as the expressionThe determinant is a special case of the immanant, where
is the alternating character , of Sn, defined by the parity of a permutation.The permanent is the case where
is the trivial character, which is identically equal to 1.Littlewood and Richardson also studied its relation to Schur functions in the representation theory of the symmetric group
Representation theory of the symmetric group
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...
.
External links
- Immanent at PlanetMathPlanetMathPlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...