 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2025/4/b/solution

Let

V = C^{6}

S U (6)

. Its weights are

w_{i} = e_{i} - \frac{1}{6} \sum_{j = 1}^{6} e_{j}

(w_{i})_{a} = δ_{ia} - δ_{i, a + 1}

. Thus the complete list is

w_{1} w_{2} w_{3} w_{4} w_{5} w_{6} a_{1} 1 - 1 0000 a_{2} 01 - 1 000 a_{3} 001 - 1 00 a_{4} 0001 - 1 0 a_{5} 00001 - 1 .

Successive lowering operators subtract the corresponding rows of the Cartan matrix in a. Each weight multiplicity is one.

\overline{6} ≅ V^{*}

is the dual representation and has the negatives of these weights. Its highest weight is

- w_{6} = (0, 0, 0, 0, 1)

Wt (\overline{6}) = ⎩ ⎨ ⎧ (- 1, 0, 0, 0, 0), (1, - 1, 0, 0, 0), (0, 1, - 1, 0, 0), (0, 0, 1, - 1, 0), (0, 0, 0, 1, - 1), (0, 0, 0, 0, 1) ⎭ ⎬ ⎫ .

Every entry has weight multiplicity one.

Λ^{2} V = 15

(2 6) = 15

w_{1} + w_{2} = (0, 1, 0, 0, 0)

e_{i} \land e_{j}

for

i < j

, carrying weight

w_{i} + w_{j}

. The fifteen sums, all distinct and all of weight multiplicity one, are

ij 121314151623242526343536454656 a_{1} 01111 - 1 - 1 - 1 - 1 000000 a_{2} 1 - 1 0000111 - 1 - 1 - 1 000 a_{3} 01 - 1 001 - 1 00011 - 1 - 1 0 a_{4} 001 - 1 001 - 1 01 - 1 001 - 1 a_{5} 0001 - 1 001 - 1 01 - 1 1 - 1 0 .

The exclusion of

i = j

comes from the exterior square, where

e_{i} \land e_{i} = 0

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