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

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The ring of characters

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Source:[1]

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 be with the understanding that   is embedded into   and   denotes the induced character.

Frobenius characteristic map

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

 

Remarks

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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

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Inner product and isometry

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Hall inner product

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Source:[2]

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

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Let  , their inner product is defined to be[3]

 If  , then

 

Frobenius characteristic map as an isometry

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

Ring isomorphism

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The map   is an isomorphism between   and the  -ring  . The fact that this map is a ring homomorphism can be shown by Frobenius reciprocity.[4] 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

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Computing the Frobenius image

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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

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  1. ^ MacDonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 112. ISBN 9780198739128.
  2. ^ Macdonald, Ian Grant (2015). Symmetric functions and Hall polynomials. Oxford University Press; 2nd edition. p. 63. ISBN 9780198739128.
  3. ^ Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 349. ISBN 9780521789875.
  4. ^ Stanley, Richard (1999). Enumerative Combinatorics: Volume 2 (Cambridge Studies in Advanced Mathematics Book 62). Cambridge University Press. p. 352. ISBN 9780521789875.