Group Theory 5.1 One-Dimensional Representations

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

A_n and its coset consist the factor group S_n/A_n, which is isomorphic to C_2, thus symmetry group has two one-dimensional representations. These one-dimensional representations can also been derived with idempotents.

 

Gist

Theorem: The symmetrizer s=\sum_pp and the anti-symmetrizer a=\sum_p(-1)^pp, where the summation runs over all of the p\in S_n, are primitive idempotents.

Thus, the primitive idempotents can generate irreducible representation space, which is the left ideals generated by s and a. Since qs=s and qa=(-1)^qa for all q\in S_n, the left ideals are both one dimensional and the matrix elements are 1 and (-1)^q 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? e_{\mu}e_{\mu}=e_{\mu} is intrinsically satisfied.

How to prove sa=0?

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