On the structure of quantum permutation groups

Abstract

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