The n-Category Café

Parts 2 & 3 are dead links.

Over on G+, people started talking about this picture of the creation operator acting on the category of Schur functors:

Owen Maresh asked “is it useful to think of that as a Pascal’s Triangle like-object?” Somewhat to my surprise, Layra Idarani said the answer is yes!

We can count the dimension of a symmetric group representation labelled by one of the Young diagrams here by counting the number of paths from the top of the picture to that diagram.

So, to get the dimension of each representation, you just add up the dimensions of the representations that have arrows pointing to it! (You start off with dimension 1 for the Young diagram with no boxes.)

In more detail:

There is a Pascal-ish behavior for the hook numbers associated to the Young diagrams. For a box $b$ in a diagram, we find the hook length of $b$, which is just the number of boxes to the right of $b$ in the same row, plus the number of boxes below $b$ in the same column, plus 1 for $b$ itself. Then for a diagram $D$ with $n$ boxes in it, we define $H(D)$ to be $n!$ divided by the product of all the hook length for all the boxes in the diagram.

Going back to the picture above, if you write out the hook numbers for each of the Young diagrams, you’ll find that $H(D)$ is just the sum of the $H(D_i)$ for $D_i$ contained in D and having one fewer box.

So for instance,

■■■
■■
■

has a hook number of $6!/(5*3*1*3*1*1) = 16$, and contains

■■■
■■

with hook number $5!/(4*3*1*2*1) = 5$,

■■■
■
■

with hook number $5!/(5*2*1*2*1) = 6$, and

■■
■■
■

with hook number $5!/(4*2*3*1*1) = 5$

This is just the decategorification of the usual branching rules for representations of the symmetric groups, where a diagram $D$ with n boxes corresponds to a vector space $V(D)$ with dimension $H(D)$ equipped with an action of $S_n$, and restricting to an action of $S_{n-1}$ gives a decomposition of $V(D)$ into the representations of $S_{n-1}$ whose diagrams are contained in D, each with multiplicity 1.

Of course, unlike with Pascal’s triangle, as you increase the number of boxes, diagrams start to have more and more sub-diagrams so the addition gets a little hairy.

I should add that the annihilation operator on the category of Schur functors, described on my talk, simply amounts to a functor sending each representation of $S_n$ (for any $n$) to its restriction to $S_{n-1}$. The creation operator, shown in the picture above, is the adjoint functor. So, it does ‘induction’ to send representations of $S_{n-1}$ to representations of $S_n$.

What’s the simplest category whose category of spans is (dagger compact) equivalent to FdHilb?