# Monthly Archives: June 2021

## Jucys-Murphy elements and induction

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 .

Filed under maths

## Oaxaca Photos (December 2019)

I picked up my old phone and decided to try turning it on again. And after charging it, I was surprised, it turned on for the first time in 18 months. I guess the remedy for fixing a water damaged phone is to just wait a long long time.

This means I got access to some photos that I thought were previously lost, and I’ll present some of them here today.

Our trip begins in Oaxaca City, where I was visiting Banff in Mexico. First up, we have a visit to Monte Albán, an archaeological site on top of a hill right next to the city itself.

Next we see a scene in the city. A wedding party is marching down the street.

Now there is a picture of myself, to convince you that I actually was there.

That picture and the rest of the pictures below were all taken at Hierve El Agua.

Part 2 of Mexican photos coming soon.