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Apologies for my English in advanced..

The following is a part from James' proof for the branching rule on the symmetric group: enter image description here

It can be found in "The Representation Theory of the Symmetric Groups" by James in page 34.
Another proof, based on the above, can be found here: https://unapologetic.wordpress.com/2011/01/28/the-branching-rule-part-2/

I can't seem to understand the definition for $\theta_i$ on a polytabloid $e_t$.

What if $n$ shows up in row $r_{i+1}$?

e.g. if we take $e_t$ to be the polytabloid generated by the standard tabloid $1 2 3//4$.
we get $e_t = 1 2 3//4 - 4 2 3//1$.

Acting on $e_t$ with $\theta_1$ force us to remove $4$ from the first row, leaving us with a structure that is not a Young tableau.

So I tried writing $2 3 4//1$ (the representative of row class in which $4 2 3//1$ is) instead.
We get $\theta_1(e_t) = \theta_1(1 2 3//4 - 2 3 4//1) = 0 - 2 3//1$ which is not equal to $e_{t^1}$ or $0$..

Can you please help me make the definition of $\theta_i$ on $\{t\}$ coincide with its definition on $e_t$?

Khal
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  • 9.4 does not mention the effect of $\theta_i$ on $e_t$ when $n$ belongs to the $r_{i+1}$th rot of $t$. So what is the issue? – David A. Craven Jul 27 '21 at 13:15
  • I gave an example of a valid situation like that, for which it is not clear to me how $\theta_i$ acts. If it is clear to you, it would be helpful if you could clarify the solution for this specific example. Of course, if you have a more general solution, it would also be appreciated. – Khal Jul 27 '21 at 13:33
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    I'm saying, what is wrong with $\theta_1(e_t)=-23//1$? $4$ lies in row 2. You are acting with $\theta_1$. 9.4 does not make any statement about this situation. – David A. Craven Jul 27 '21 at 13:52
  • That's a surprisingly good point. Later in the proof he really doesn't use this fact. So in order for the definition to be valid, you're implying I should take the representative 234//1 as I did in the latter part, instead of 423//1, which I took in the first part? – Khal Jul 27 '21 at 14:11
  • You shouldn't be taking any representative. Removing 4 from the tabloid 432//1 yields the tabloid 23//1, or 32//1, they are the same thing. As tabloids don't care about the order of the elements in the list, of course you may remove 4 from 432 to end up with 32, not _32. – David A. Craven Jul 27 '21 at 14:17
  • Oh, I wasn't aware of that.. It's just that in the second row of the proof, James specifically states that $r_1,...,r_k$ refer only to row that removing a node from leaves a valid diagram. – Khal Jul 27 '21 at 14:26
  • And of course, thanks for the replies! – Khal Jul 27 '21 at 14:27
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    He says removing a node from $[\mu]$, a Young diagram, not ${t}$, a tabloid. – David A. Craven Jul 27 '21 at 14:28

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