A Remark on the Actions of <em>C</em>(<em>S_n⁺</em>)

Talking about permutations, one might think of the usual permutation groups, permuting the set of n points, and that is a correct way of thinking. But, one may see a point as a 1 × 1 matrix with entries in any number field, and by extending this way of viewing a point, one may think of n × n matrices instead of points, and think about permuting these matrices. These matrices might have commuting or noncommuting entries. In the case where the entries are noncommuting and are satisfying in the relations of the matrix quantum groups, developed by Woronowicz, one gets what has been referred to as the quantum permutation (symmetry) group Sn+, introduced and studied by Wang. But since we still do not have any knowledge about the elements of Sn+ , hence instead of that, studying its function space would be quite logical! Here in this paper, we will try to study the actions of C(Sn+) on some matrix spaces, especially those related to finite groups of Lie type. In accordance, we also study the possible invariant spaces of these actions.