I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). While we have basic theoretical understanding of representation theory, we are using the tools in this section in a primarily practical way, without a deep understanding of their validity.
We know that the Jucys-Murphy elements act on the algebra generated by the standard tableaux, but the action is not the natural one. My understanding is that the action is relatively well-known: it is the one that allows all of the standard tableaux to be eigenvectors with eigenvectors easily determined by the axial distance. To describe this action, we were given something like the following construction. My main question is: Does this construction produce the correct action?
Use the last-letter ordering on the Young tableaux to uniquely order them $t_i$. Given a partition $\lambda$, define a matrix $M=\rho^\lambda((m-1, m))$ for all transpositions $((m-1,m)$ as having zeros in all entries except:
- $M_{ii}=\pm 1$ if $m-1$ and $m$ are in the same row ($+$) or column ($-$).
- If $i<k$ and swapping $m-1$ and $m$ in $t_i$ produces $t_k$, then $M_{ii}=-M_{kk}=-\delta$ and $M_{ik}=M_{ki}=\sqrt{1-\delta^2}$, where $\delta$ is the axial distance from $m-1$ to $m$.
What's bugging me about this construction is purely practical: it seems like it's not the most useful thing, considering that the Jucys-Murphy elements are composed of entirely different transpositions. We're not supposed to need a computer algebra system, but this was such an enormous part of the material we covered, it's probably going to come up. But I would hope for a less computationally intensive method than multiplying six or more $4\times 4$ or $5\times 5$ matrices