Let R be the orthogonal matrix corresponding to an operation in O(3). If
R is a proper rotation, then both vectors →V and axial vectors →A are transformed in the same way
→V→→V′=R→V
→A→→A′=R→A
if R is an improper rotations, then,
→V→→V′=R→V
→A→→A′=−R→A
And let V be the vector representation of O(3) and P be the axial-vector representation of O(3). Thus we can write,
→V→→V′=V(R)→V
→A→→A′=P(R)→A
My question is, considering V and P as representations of C4v group, how can I find their composition in terms of irreps ?
One can use the character tablegiven in this link http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=404&option=4