Frobenius characteristic map


In mathematics, especially representation theory and combinatorics, a Frobenius characteristic map is an isometric isomorphism between the ring of characters of symmetric groups and the ring of symmetric functions. It builds a bridge between representation theory of the symmetric groups and algebraic combinatorics. This map makes it possible to study representation problems with help of symmetric functions and vice versa. This map is named after German mathematician Ferdinand Georg Frobenius.

Definition

The ring of characters

Source:
Let be the -module generated by all irreducible characters of over. In particular and therefore. The ring of characters is defined to be the direct sum with the following multiplication to make a graded commutative ring. Given and, the product is defined to bewith the understanding that is embedded into and denotes the induced character.

Frobenius characteristic map

For, the value of the Frobenius characteristic map at, which is also called the Frobenius image of, is defined to be the polynomial

Remarks

Here, is the integer partition determined by. For example, when and, corresponds to the partition. Conversely, a partition of determines a conjugacy class in. For example, given, is a conjugacy class. Hence by abuse of notation can be used to denote the value of on the conjugacy class determined by. Note this always makes sense because is a class function.
Let be a partition of, then is the product of power sum symmetric polynomials determined by of variables. For example, given, a partition of,
Finally, is defined to be, where is the cardinality of the conjugacy class. For example, when,. The second definition of can therefore be justified directly:

Properties

Inner product and isometry

Hall inner product

Source:
The inner product on the ring of symmetric functions is the Hall inner product. It is required that . Here, is a monomial symmetric function and is a product of completely homogeneous symmetric functions. To be precise, let be a partition of integer, then In particular, with respect to this inner product, form a orthogonal basis:, and the Schur polynomials form a orthonormal basis:, where is the Kronecker delta.

Inner product of characters

Let, their inner product is defined to be
If, then

Frobenius characteristic map as an isometry

One can prove that the Frobenius characteristic map is an isometry by explicit computation. To show this, it suffices to assume that :

Ring isomorphism

The map is an isomorphism between and the -ring. The fact that this map is a ring homomorphism can be shown by Frobenius reciprocity. For and,
Defining by, the Frobenius characteristic map can be written in a shorter form:
In particular, if is an irreducible representation, then is a Schur polynomial of variables. It follows that maps an orthonormal basis of to an orthonormal basis of. Therefore it is an isomorphism.

Example

Computing the Frobenius image

Let be the alternating representation of, which is defined by, where is the sign of the permutation. There are three conjugacy classes of, which can be represented by, and . These three conjugacy classes therefore correspond to three partitions of given by,,. The values of on these three classes are respectively. Therefore:Since is an irreducible representation, the computation above gives the Schur polynomial of three variables corresponding to the partition.