## Polytabloids

Given any collection of permutations, we define two group algebra elements.

Notice that doesn’t have to be a subgroup, though it often will be. One particular case that we’ll be interested in is

where is the column-stabilizer of a Young tableau . If has columns , then . Letting run over is the same as letting run over for each from to . That is,

so we have a nice factorization of this element.

Now if is a tableau, we define the associated “polytabloid”

Now, as written this doesn’t really make sense. But it does if we move from just considering Young tabloids to considering the vector space of formal linear combinations of Young tabloids. This means we use Young tabloids like basis vectors and just “add” and “scalar multiply” them as if those operations made sense.

As an example, consider the tableau

Our factorization lets us write

And so we calculate

Now, the nice thing about is that if we hit it with any permutation , we get .