武汉大学刘觉平群论第八次作业.docVIP

Exercise 8 in Group Theory Exercises 8-1. Another convenient orthonormal basis for can be chosen to be Hence e1 spans W, while e2 and e3 span . With respect to this basis, 1. deduce the reducible matrices representation of permutation group S3 based on three basis e1, e2 and e3. 2. verify each matrix can be reduced to a direct sum of a matrix and a matrix, i.e., the three-dimensional representation of permutation group S3 can be decomposed into the direct sum of a one-dimensional identity representation (in the subspace spanned by e1) and a two-dimensional irreducible representation in the subspace spanned by e2 and e3. 1.Solution: the three dimensional representation of is the permutation of the basis: when (123) acts on e1, e2 and e3 so 2. under the three-dimensional representation of , Sospans an invariant one-dimensional vector space under the representation of , ∴e2 and e3 span the maximum orthogonal vector space of the space spanned by. Since the scalar product of vectors and the vector space spanned byare invariant, the vector space spanned by e2 and e3 are also invariant, so the three dimensional space can be decomposed into the direct summation of an irreducible invariant one-dimensional subspace and an irreducible invariant two-dimensional subspace. So, is the restriction of to the subspace spanned by, is the subspace spanned by e2 and e3. Exercises 8-2. According to the results of Exercises 8-1,fill in the following blanks. Table for the irreducible representation of the permutation group S3 Element Identity Sign of permutation Two-dimensional e 1 (123) 1 (132) 1 (12) 1 (13) 1 (23) 1 (Notation:Although you may find the answers in your text book or in Exercises 8-3,you’d better make sure that you know the steps to solve this problem.) Solution: suppose a is a reflection, then if a is a reflection and b is a rotation, they have the relationship: thus the one-dimensional representations of are: e 1 1 (123) 1 1 (132) 1 1 (