This post concerns the representation theory of the symmetric group over the complex numbers. Recall that the irreducible representations of the symmetric group are indexed by partitions of . Let be the irreducible representation indexed by . I want to say some words about the theorem that the decomposition of the induced module is given by the decomposition into eigenspaces under the action of the Jucys-Murphy element.
First, the relevant Jucys-Murphy element is
The way it acts on is not as an element of but by This is well-defined since commutes with .
What this action defines is a natural transformation from the functor to itself. The induction functor is (bi)-adjoint to the restriction functor and this natural transformation is even simpler to construct on the adjoint side. Recall that if and are adjoint functors, then there is an isomorphism
Here refers to the natural transformations from to itself, and the map in this isomorphism is given by pre- and post-composition by the unit and counit of the adjunction.
And the way that yields a natural transformation from to itself is very simple, it’s just by its usual action as an element of . If you transport this natural transformation to a natural transformation of the induction functor via the method I just mentioned, then you get the formula mentioned above.
Now given a pair of adjoint functors and , a natural transformation from to (and hence from to ) and a complex number , we can define a functor by
and similarly for (this requires some linearity assumptions, but they’re satisfied here. Also you could take generalised eigenspaces if you wanted to, but in our application there is no difference).
When you do this, the functors and are adjoint:
Proof: Both and are the -eigenspace of the action of on .
Now apply this to our situation. We also use the following standard fact about the action of the Jucys-Murphy element (as developed e.g. in the Vershik-Okounkov approach):
Consider the decomposition
Then the Jucys-Murphy element acts by the scalar on , where is the content of the box added to to get .
Now translating this statement via the above yoga onto the adjoint side, we get
In the decomposition
the Jucys-Murphy element acts by the scalar on , where is the content of the box added to to get .