Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 2 / 1 / a / i

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- Solution i

Solution

Realize the defining sl2 Lie algebra Lie algebra representation as

C x \oplus C y

, with

H = x \partial_{x} - y \partial_{y}

,

E = x \partial_{y}

,

F = y \partial_{x}

. A basis of the symmetric power is

e_{i} = x^{4 - i} y^{i}

for

0 \leq i \leq 4

, with

H e_{i} = (4 - 2 i) e_{i}

. Hence the exterior square basis

w_{ij} = e_{i} \land e_{j}

,

i < j

, has weight

8 - 2 (i + j)

.

Collecting equal weights gives the complete weight-space decomposition:

weight 6420 - 2 - 4 - 6 basis w_{01} w_{02} w_{03}, w_{12} w_{04}, w_{13} w_{14}, w_{23} w_{24} w_{34} multiplicity 1122211

Every one of the ten exterior basis vectors appears exactly once. Thus $U = ⨁_{m = 6, 4, 2, 0, - 2, - 4, - 6} U_{m}$ , with the displayed bases and dimensions.

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