(第十一次群论作业).doc

The 11th Assignment Assignment about rotation group and decomposition of the self-direct product of groups Problem 1 The multiplication table of the group is given as follows. E A B C F K M N E A B C E A B C A E M K B N E M C K N E F K M N N C B F K F C A M A F B F K M N F M K N K C F A M F A B N B C F E B A C B E N M C N K E A M E K Write the character table for the group . If we know that the representation matrices of the generators and in a two-dimensional irreducible representation are and two sets of basis functions and respectively transform according to this two-dimensional representation of combine the product function such that the basis function belongs to the irreducible representation of . 1. 特征表 1 1 1 1 1 1 1 1 1 -1 1 -1 1 1 2 -2 0 0 0 2. Problem 2 Define a matrix by Where and , , are standard Pauli’s matrices. For an arbitrary unitary matrix as follows , (1) we can define a transformation , (2) for vector , obeying following rule (3) Then 1, write the expression of the transformation matrix , corresponding to , in terms of and . 2, prove that is a real orthogonal matrix and . 3, search for three pairs of values of and which can satisfies following mappings 4, calculate matrix, and prove is just the rotation of a rigid body with Euler’s angles, i.e., . 1. 2 故是实正交变换 故命题得证 3 4. = = 4. 故与欧拉相同 Problem 3 Prove that there exists a homomorphism mapping from to , search for the kernel of this homomorphism. What is the factor group of this homomorphism mapping ? 对于任意 1 right left