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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-3/1/b/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 1 / b / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

Take the simple roots

L_{1} - L_{2}

and

L_{2} - L_{3}

. A weight represented by

(k_{1}, k_{2}, k_{3})

is in the Closed dominant Weyl chamber exactly when

k_{1} \geq k_{2} \geq k_{3}

. Among nonnegative triples of total four, the possibilities are

(4, 0, 0), (3, 1, 0), (2, 2, 0), (2, 1, 1)

. Using

L_{1} + L_{2} + L_{3} = 0

, their explicit weight space bases are

weight 4 L_{1} 2 L_{1} - L_{3} - 2 L_{3} L_{1} basis s_{11} \otimes s_{11} s_{11} \otimes s_{12}, s_{12} \otimes s_{11} s_{11} \otimes s_{22}, s_{22} \otimes s_{11}, s_{12} \otimes s_{12} s_{11} \otimes s_{23}, s_{23} \otimes s_{11}, s_{12} \otimes s_{13}, s_{13} \otimes s_{12}

Each listed tensor has the required summed weight, and the monomial basis from part (i) shows linear independence and exhausts all possible tensors of that weight. Thus these are all the requested bases.

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