The Unapologetic Mathematician

Mathematics for the interested outsider

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

December 24, 2010 Posted by | Algebra, Representation Theory, Representations of Symmetric Groups | 2 Comments

   

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