For an irreducible representation $V(\lambda)$, write
$$
\lambda=\sum_i m_i\omega_i.
$$
Form
$$
T=\bigotimes_iV(\omega_i)^{\otimes m_i}.
$$
The tensor product of highest-weight vectors is killed by all positive-root spaces and has weight $\lambda$. It therefore generates a highest-weight constituent isomorphic to $V(\lambda)$. By <Complete reducibility of semisimple Lie algebra representations>, this constituent is a subrepresentation of $T$. This proves the <generation by fundamental representations> statement.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 4 / c / Solution

If $\lambda=\sum_im_i\omega_i$ is dominant integral, the tensor product $\bigotimes_iV(\omega_i)^{\otimes m_i}$ contains $V(\lambda)$ as the irreducible summand generated by the tensor product of highest-weight vectors.


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