Frobenius characteristic map

Frobenius characteristic map gives the expansion of the Schur function in the power–sum basis:

(1)

where are characters of the irreducible representation of the symmetric group indexed by the partition evaluated at elements of cycle type indexed by the partition and are cycle type permutation centralizer sizes. The inverse transform reads as

(2)

Characters can be calculated from this map if one knows how to calculate Schur functions.

can we derive frobenius characteristic map from schur-weyl duality? use latex formatting

Yes. In fact the Frobenius–characteristic map is nothing but the –character on the tensor space , viewed as an -module via Schur–Weyl duality. Here is the standard derivation.

  1. Schur–Weyl duality gives a bi-module decomposition where is the Specht (irreducible) –module of shape , and the corresponding irreducible polynomial –module.
  2. Let be any class function. We define its Frobenius image by taking the trace of the commuting –action on . Concretely, if , then
(3)

where if has cycle‐type then

  1. On the other hand, from the decomposition in step 1 one sees and more generally Hence and in particular . This is precisely the classical Frobenius characteristic map

In this way the Frobenius map is seen to “come for free” from Schur–Weyl duality by taking –characters on .

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