On the structure of quantum permutation groups

The quantum permutation group of the set $X_n=\{ 1,\ldots ,n\}$ corresponds to the Hopf algebra $A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $\mathbb{C} (S_n)$ for $n\leq 3$, and to be infinite dimensional for $n\geq 4$. In this paper we find an explicit representation of the algebra $A_{aut}(X_n)$, related to Clifford algebras. For $n=4$ the representation is faithful in the discrete quantum group sense.