Let $G$ be a finite group that acts on a finite set $X$. Let $K$ be a field and consider the associated permutation representation $G\to GL_{|X|}(K)$. Does the dimension of the kernel of the corresponding algebra homorphism $KG\to Mat_{|X|}(K)$ depend on $K$? In other words is this number something that can be deduced soley from the action of $G$ on $X$?
I know that it does not depend on $K$ when you consider the natural action of $S_n$ on $\{1,\dots,n\}$. This is not too difficult to see as the image is the set of matrices whose rows and columns all sum to the same constant. This always has dimension $1+(n-1)^2$ and so the kernel must have dimension $n!-(n-1)^2-1$. However, I've no idea how to tackle this problem in general.