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
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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 sumwith 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 (written as ) 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
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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, thenIn 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 (identity or the product of three 1-cycles), (transpositions or the products of one 2-cycle and one 1-cycle) and (3-cycles). 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 (which can be shown by computing its characters), the computation above gives the Schur polynomial of three variables corresponding to the partition .
References
- MacDonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 112. ISBN 9780198739128.
- Macdonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 63. ISBN 9780198739128.
- Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 349. ISBN 9780521789875.
- Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 352. ISBN 9780521789875.
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