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

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- Solution c

Solution

The tensor-product weight diagram is obtained by adding every ordered pair of defining weights. Each

\pm 2 L_{i}

occurs once, while each of the four weights

\pm L_{1} \pm L_{2}

occurs twice, from the two orders of the summands. Zero occurs four times, from the ordered pairs

(L_{i}, - L_{i})

and

(- L_{i}, L_{i})

for

i = 1, 2

. Hence

weight \pm 2 L_{1}, \pm 2 L_{2} \pm L_{1} \pm L_{2} 0 multiplicity 124

The dimensions sum to

4 + 8 + 4 = 16

, as required for

V \otimes V

.

Figure 1.
Tensor-square sp4 weights, including the double diagonal weights and four-dimensional zero space
. Details.

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