by Exercise 4.4 (a), where is the conjugate partition of .
Note that (here). Note also that .
Try first a simple one like and generalize the relationship between and .
I'm stuck at the following exercise of the Fulton-Harris "Representation theory" (p.47 ex. 4.4 (c)):
Show that the representation of a partition of the symmetric group is the tensor product of the representation of the conjugate partition and the alternating representation.
Can anybody help me please?