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

Table of contents
- a ii
  - Solution a
- b ii
  - Solution b
- c ii
  - Solution c

a

Solution

For the Adjoint representation, the root-space decomposition gives the eight nonzero weights

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

each of multiplicity one, and weight zero of multiplicity two from the Cartan subalgebra. Thus its weight diagram is the root diagram with a double zero weight, and its total dimension is

8 + 2 = 10

.

Figure 1.
Adjoint sp4 weights: the eight roots and the zero weight of multiplicity two
. Details.

b

Solution

In the defining representation, the diagonal element of the Cartan subalgebra acts on the four standard basis vectors with eigenvalues

r, s, - r, - s

. Therefore

wt (V) = {L_{1}, L_{2}, - L_{1}, - L_{2}}, each with multiplicity one .

Their weight diagram is the four-point cross below.

Figure 1.
The four defining-representation weights of sp4
. Details.

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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