# Group Theory 5.1 One-Dimensional Representations

Symmetry group is just the permutation group, which has a non-trivial invariant subgroup consisting of all even permutations. (An even permutation is one which is equivalent to an even number of simple transpositions.) has a coset consisting of all odd permutations.

and its coset consist the factor group , which is isomorphic to , thus symmetry group has two one-dimensional representations. These one-dimensional representations can also been derived with idempotents.

#### Gist

Theorem: The symmetrizer and the anti-symmetrizer , where the summation runs over all of the , are primitive idempotents.

Thus, the primitive idempotents can generate irreducible representation space, which is the left ideals generated by and . Since and for all , the left ideals are both one dimensional and the matrix elements are and respectively.

#### Analysis

In fact, Group Theory in Physics has introduced a definition of essentially idempotent, which is just idempotents with an additional normalization constant. But what dose the normalization means? is intrinsically satisfied.

How to prove ?