I am having trouble with this problem and the referred exercise 13.12.
Here is the statement of the exercise along with the screenshot of the table below: For $L$ of type $G_2$, use Table 2 of (22.4) to determine all weights and their multiplicities for $V (λ), λ = λ_1 + 2λ_2$. Compute dim $ V (λ) = 286$. [Cf. Exercise 13.12.]
My initial understanding was to look at row (12), but the sum of dimensions is not 286. How one goes about this computation? Does one need to compute the orbit of (12) under the Weyl group? If yes, how does one do it and use it to compute the dimension?
Can one use the Weyl Dimension formula where $\lambda=m_1λ_1 + m_2λ_2$? $$\dim V(\lambda) = 1/120 (m_1+1)(m_2+1)(m_1+m_2+2)(m_1+2m_2+3)(m_1+3m_2+4)(2m_1+3m_2+5)$$
