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