Understanding the polytabloid associated with a Young tabeau

Suppose I have the Young tableau $T$:

$[1][2][3][4]$

$[5][6]$

With corresponding tabloid $\{T\}$ (the equivalence class of Young tableaux).

We define the notion of the column stabilizer for a Young tableau (denoted $C(T)$) and we define the polytabloid associated to the Young tableau $T$ as follows:

I'm struggling to understand what the polytabloid is. Is it a vector? Or at least expressible or representable as a column vector?

This polytabloid lies in the free Abelian group generated by the tabloids. (Does your book have a notation for this group?) (In your example, with tableaux of shape $4\,2$ that would be a free Abelian group of rank $15$.) That's the additive group with one generator for each tabloid $\{T\}$. Then as the formula suggests, starting with the tableau $T$, one applies all possible permutations $\pi$ fixing the columns in $T$, resulting in tableaux $\pi\ast T$. Then take the associated tabloids $\{\pi\ast T\}$, weight them according to the sign of $\pi$ and add them all up in this free Abelian group. That sum is what you have pointed out in your diagram.
(I've used slightly different notation: $\{\pi\ast T\}$ rather than $\pi\ast\{T\}$.)