Homework 2 Questions with Solutions - Symmetry Problems in Physics | PHYS 711, Assignments of Physics

Download Homework 2 Questions with Solutions - Symmetry Problems in Physics | PHYS 711 and more Assignments Physics in PDF only on Docsity! Physics 711, Symmetry Problems in Physics Fall 2005 Homework Solutions for assignment 2 Georgi 1.D. From D2(g) = SD1(g)S −1 and D2(g)A = AD1(g), both ∀g, D2(g)A = AS −1D2(g)S, and D2(g)AS−1 = AS−1D2(g), so by Schur’s lemma AS−1 = λ1, or A = λS, where S is a nonsingular n× n matrix and n = dimD1,2. Group of integers 1, 2, ..., n− 1, n prime, mod n is a group of order n− 1. The product is (k2, k1) → k2k1, mod n. The powers of any element g in a finite group must recur. Let gs = g be the first recurrance for any element g. Then gs−1 = e. Taking cosets with respect to the subgroup generated by these powers, s− 1 must divide n− 1; i.e. (n− 1)/(s− 1) = t, where t is an integer. Then g(s−1)t = gn = e. In our case, e = 1, g = k, and kn−1 = 1, mod n. Character table of S3 Rep|Class 111 3 12 (3) 1 1 1 (111) 1 1 −1 (21) 2 −1 0 Normalized character table of S3 Rep|Class 111 3 12 (3) 1/ √ 6 1/ √ 3 −1/√2 (111) 1/ √ 6 1/ √ 3 1/ √ 2 (21) √ 2/3 −1/√3 0 χa,α = √ kα/N , N = 6, kα = 1, 111; 2, 3; 3, 12.