Three identical LC circuits sit at the corners of an equilateral triangle and a…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Group theory and symmetry · Group theory and symmetry

Invariance, Ch Representations, Schur's lemma and characters

Problem

Three identical LC circuits sit at the corners of an equilateral triangle and a fourth sits at its centre, all coupled through their capacitors, with the whole arrangement carrying

C_{3v}

symmetry (classes

E

2C_3

3\sigma_v

, so

|G|=6

). Take as coordinates the four capacitor charges, which the symmetry operations permute. The representation of

C_{3v}

on the four charges has character

\chi = (4, 1, 2)

on these classes. The

C_{3v}

character table is

A_1 = (1,1,1)

A_2 = (1,1,-1)

E = (2,-1,0)

on classes

(E, 2C_3, 3\sigma_v)

. Someone decomposes

\chi

as follows: "The decomposition formula is

a_i = \frac{1}{|G|}\sum_{\text{classes}} \chi(g)\chi_i(g)^*

. So

a_{A_1} = \frac{1}{6}(4\cdot 1 + 1\cdot 1 + 2\cdot 1) = \frac{7}{6}

a_{A_2} = \frac{1}{6}(4\cdot 1 + 1\cdot 1 + 2\cdot(-1)) = \frac{1}{2}

a_E = \frac{1}{6}(4\cdot 2 + 1\cdot(-1) + 2\cdot 0) = \frac{7}{6}

\chi

contains fractional amounts of each irreducible representation." ((a)) Confirm the character by counting the charges each operation leaves in place, and find the error in this working. ((b)) Give the correct decomposition of

\chi

into irreducible representations of

C_{3v}

, and check that the dimensions add up to

4

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