The Branching Rule, Part 2

We pick up our proof of the branching rule. We have a partition $\lambda$ with inner corners in rows $r_1,\dots,r_k$ . The partitions we get by removing each of the inner corner $r_i$ is $\lambda^i$ . If the tableau $t$ (or the tabloid $\{t\}$ has its $n$ in row $i$ , then $t^i$ (or $\{t^i\}$ ) is the result of removing that $n$ .

We’re looking for a chain of subspaces

$\displaystyle0=V^{(0)}\subseteq V^{(1)}\subseteq\dots\subseteq V^{(k)}=S^\lambda$

such that $V^{(i)}/V^{(i-1)}=S^{\lambda^i}$ as $S_{n-1}$ -modules. I say that we can define $V^{(i)}$ to be the subspace of $S^\lambda$ spanned by the standard polytabloids $e_t$ where the $n$ shows up in row $r_i$ or above in $t$ .

For each $i$ , define the map $\theta_i:M^\lambda\to M^{\lambda^i}$ by removing an $n$ in row $r_i$ . That is, if $\{t\}\in$ latex M^\lambda$ has its $n$ in row $r_i$ , set $\theta_i(\{t\})=\{t^i\}$ ; otherwise set $\theta_i(\{t\})=0$ . These are all homomorphisms of $S_{n-1}$ -modules, since the action of $S_{n-1}$ always leaves the $n$ in the same row, and so it commutes with removing an $n$ from row $r_i$ .

Similarly, I say that $\theta_i(e_t)=e_{t^i}$ if $n$ is in row $r_i$ of $t$ , and we get $\theta_i(e_t)=0$ if it’s in row $r_j$ with $j<i$ . Indeed if $n$ shows up above row $r_i$ , then since it’s the bottommost entry in its column that column can have no entries at all in row $r_i$ . Thus as we use $C_t$ to shuffle the columns, all of the tabloids that show up in $e_t=C_t^-\{t\}$ will be sent to zero by $\theta_i$ . Similar considerations show that if $n$ is in row $r_i$ , then of all the tabloids that show up in $e_t$ , only those leaving $n$ in that row are not sent to zero by $\theta_i$ . The permutations in $C_t$ leaving $n$ fixed are, of course, exactly those in $C_{t^i}^-$ , and our assertion holds.

Now, since each standard polytabloid $e_{t^i}\in S^{\lambda^i}$ comes from some polytabloid $e_t\in M^{\lambda}$ , we see they’re all in the image of $\theta_i$ . Further, these $e_t$ all have their $n$ s in row $r_i$ , so they’re all in $V^{(i)}$ . That is, $\theta_i(V^{(i)})=S^{\lambda^i}$ . On the other hand, if $t$ has its $n$ above row $r_i$ , then $\theta(e_t)=0$ , and so $V^{(i-1)}\subseteq\mathrm{Ker}(\theta_i)$ .

So now we’ve got a longer chain of subspaces:

$\displaystyle0=V^{(0)}\subseteq V^{(1)}\cap\mathrm{Ker}(\theta_1)\subseteq V^{(1)}\subseteq\dots\subseteq V^{(k)}\cap\mathrm{Ker}(\theta_k)\subseteq V^{(k)}=S^\lambda$

But we also know that

$\displaystyle\dim\left(\frac{V^{(i)}}{V^{(i)}\cap\mathrm{Ker}(\theta_i)}\right)=\dim(\theta_i(V^{(i)}))=\dim(S^{\lambda^i})=f^{\lambda^i}$

So the steps from $V^{(i)}\cap\mathrm{Ker}(\theta_i)$ to $V^{(i)}$ give us all the $f^{\lambda^i}$ as we add up dimensions. Comparing to the formula we’re categorifying, we see that this accounts for all of $f^\lambda=\dim(S^\lambda)$ . And so there are no dimensions left for the steps from $V^{(i-1)}$ to $V^{(i)}\cap\mathrm{Ker}(\theta_i)$ , and these containments must actually be equalities!

$\displaystyle V^{(i-1)}=V^{(i)}\cap\mathrm{Ker}(\theta_i)$

And thus

$\displaystyle \frac{V^{(i)}}{V^{(i-1)}}=\frac{V^{(i)}}{V^{(i)}\cap\mathrm{Ker}(\theta_i)}\cong S^{\lambda^i}$

as asserted. The branching rule then follows.

January 28, 2011 - Posted by John Armstrong | Algebra, Representation Theory, Representations of Symmetric Groups

5 Comments »

[…] 3″? Didn’t we just finish proving the branching rule? Well, yes, but there’s another part we haven’t mentioned yet. Not […]

Pingback by The Branching Rule, Part 3 « The Unapologetic Mathematician | January 31, 2011 | Reply
Hi, I really like your proof where everything is explained so well. But I dont get why the V^(i) as you defined them here are S^(n-1) modules. I tried showing it and the actual point missing is that the map from S_{n-1} x V^(i) gives results in V^(i). I think i.e. linear combinations of the polytabloids you used for V^(i) do still have n in one of the required rows after a permutation of S_{n-1} was applied. But how can this be true for an arbitrary permutation?

Comment by pani puri | December 4, 2013 | Reply
Sorry, it’s been a long time since I looked at this (almost three years now), so I might be a little rusty. I think the problem might be that I’m asserting that the quotient space $V^{(i)}/V^{(i-1)}$ is an $S_{n-1}$-module, not that $V^{(i)}$ itself is. Does that help?

Comment by John Armstrong | December 4, 2013 | Reply
Hi, thanks a lot for your answer! But I think there I have the same problem since now I need that n stays in the same row…

Comment by pani puri | December 9, 2013 | Reply
What I wrote was wrong. Now I need that all n stay in their line or are in a higher one by considering only polytabloids where the n in t is in one special line.

Comment by pani puri | December 10, 2013 | Reply

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