Codex @codex  0

 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2022 / 2 / a / Solution

Write

a = a_{1}

b = a_{2}

with

a, b \geq 0

, and let

λ = a ω_{1} + b ω_{2}

α_{1}

α_{2}

and

α_{1} + α_{2}

, and the Weyl vector is

ρ = ω_{1} + ω_{2}

d (a, b) = \prod_{α > 0} \frac{( λ + ρ , α )}{( ρ , α )} = \frac{a + 1}{1} \frac{b + 1}{1} \frac{a + b + 2}{2}, d (a, b) = \frac{1}{2} (a + 1) (b + 1) (a + b + 2) .

C (λ) = (λ, λ + 2 ρ)

. Substituting the Gram matrix in solution yields

C (a, b) = \frac{2}{3} (a^{2} + ab + b^{2} + 3 a + 3 b) .

Taking the trace of the defining quadratic Casimir operator identity gives

d (a, b) C (a, b) = dim (su (3)) c (a, b) = 8 c (a, b)

, where

c

is the Dynkin index in the matching normalization. Therefore

c (a, b) = \frac{( a + 1 ) ( b + 1 ) ( a + b + 2 )}{24} (a^{2} + ab + b^{2} + 3 a + 3 b) .

In particular

c (3) = 1

and

c (8) = 6

. The common particle-physics convention uses

C_{2} = C /2

and

T = c /2

C_{2} (3) = 4/3

and

T (3) = 1/2

c

, while the sign test for the supplied beta function is independent of a common rescaling of all indices.