Apologies for my English in advanced..
The following is a part from James' proof for the branching rule on the symmetric group:
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 $e_t$?