In <Dynkin label> coordinates, subtracting $\alpha_i$ subtracts row $i$ of the stated Cartan matrix. Starting from $(1,0,0)$ gives the weight chain
$$
(1,0,0)
\xrightarrow{-\alpha_1}(-1,1,0)
\xrightarrow{-\alpha_2}(0,-1,1)
\xrightarrow{-\alpha_3}(0,0,-1).
$$
These are the four weights of the defining representation of $A_3\cong\mathfrak{sl}_4$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 302 / 3 / d / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-302/3/d/i/solution