In Dynkin label coordinates, subtracting

α_{i}

subtracts row

i

of the stated Cartan matrix. Starting from

(1, 0, 0)

gives the weight chain

(1, 0, 0) - α_{1} (- 1, 1, 0) - α_{2} (0, - 1, 1) - α_{3} (0, 0, - 1) .

These are the four weights of the defining representation of

A_{3} ≅ sl_{4}

ii

Solution

The highest weight

(0, 1, 0)

gives the six-dimensional second exterior power of the defining representation. Adding each unordered pair of the four defining weights from part i gives

(0, 1, 0), (1, - 1, 1), (1, 0, - 1), (- 1, 0, 1), (- 1, 1, - 1), (0, - 1, 0) .

There are no multiplicities, in agreement with the assumption in the question.

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