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
Indeed, what does it mean to say that 
? It means that 
preserves the rows of the tableau 
. And therefore it acts trivially on the tabloid 
. That is: 
. But of course we know that 
, and thus we rewrite 
, or equivalently 
. This means that 
, and thus 
, as asserted.
Similarly, we can show that 
. 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 
.
Finally, I say that the polytabloid 
is the same as the polytabloid 
. Indeed, we compute
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December 24, 2010 -
Posted by John Armstrong |
Algebra, Representation Theory, Representations of Symmetric Groups
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