The Unapologetic Mathematician

Permutations and Polytabloids

We’ve defined a bunch of objects related to polytabloids. Let’s see how they relate to permutations.

First of all, I say that

$\displaystyle R_{\pi t}=\pi R_t\pi^{-1}$

Indeed, what does it mean to say that $\sigma\in R_{\pi t}$? It means that $\sigma$ preserves the rows of the tableau $\pi t$. And therefore it acts trivially on the tabloid $\{\pi t\}$. That is: $\sigma\{\pi t\}=\{\pi t\}$. But of course we know that $\{\pi t\}=\pi\{t\}$, and thus we rewrite $\sigma\pi\{t\}=\pi\{t\}$, or equivalently $\pi^{-1}\sigma\pi\{t\}=\{t\}$. This means that $\pi^{-1}\sigma\pi\in R_t$, and thus $\sigma\in\pi R_t\pi^{-1}$, as asserted.

Similarly, we can show that $C_{\pi t}=\pi C_t\pi^{-1}$. This is slightly more complicated, since the action of the column-stabilizer on a Young tabloid isn’t as straightforward as the action of the row-stabilizer. But for the moment we can imagine a column-oriented analogue of Young tabloids that lets the same proof go through. From here it should be clear that $\kappa_{\pi t}=\pi\kappa_t\pi^{-1}$.

Finally, I say that the polytabloid $e_{\pi t}$ is the same as the polytabloid $\pi e_t$. Indeed, we compute

$\displaystyle e_{\pi t}=\kappa_{\pi t}\{\pi t\}=\pi\kappa_t\pi^{-1}\pi\{t\}=\pi\kappa_t\{t\}=\pi e_t$

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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