On the structure of quantum permutation groups

Abstract

The quantum permutation group of the set Xn = {1, . . . , n} corresponds to the Hopf algebra Aaut(Xn). This is an algebra constructed with generators and relations, known to be isomorphic to ℂ(Sn) for n ≤ 3, and to be infinite dimensional for n ≥ 4. In this paper we find an explicit representation of the algebra A aut(Xn), related to Clifford algebras. For n = 4 the representation is faithful in the discrete quantum group sense. © 2006 American Mathematical Society.