Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 3 / ii

Table of contents
- Solution ii

Solution

For

v \in V_{λ}

and

w \in W_{μ}

, the tensor product of Lie algebra representations acts by

H (v \otimes w) = H v \otimes w + v \otimes H w = (λ (H) + μ (H)) v \otimes w .

Thus its weight space of weight

ν

is

(V \otimes W)_{ν} = ⨁_{λ + μ = ν} V_{λ} \otimes W_{μ}, dim (V \otimes W)_{ν} = \sum_{λ + μ = ν} (dim V_{λ}) (dim W_{μ}) .

Substituting this into the formal character and using

e (λ + μ) = e (λ) e (μ)

proves

char (V \otimes W) = (char V) (char W) .

Every sum is finite for the Lie algebra representations under consideration.

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