 Solution

ID: past-exam-of-the-groups-and-representations-course-of-the-university-of-oxford/2022/4/b/solution

S U (N)

, denote a one-column Young tableau with

k

empty boxes by

C_{k}

. Explicitly the columns needed here are

C_{1} = a, C_{2} = a a, C_{3} = a a a, C_{4} = a a a a, C_{5} = a a a a a .

A column

C_{k}

Λ^{k} V

ω_{k}

(k N)

. Contraction with the invariant volume tensor identifies

V^{*} ≅ Λ^{N - 1} V

and

Λ^{2} V^{*} ≅ Λ^{N - 2} V

algebra A_{5} A_{5} A_{5} A_{5} A_{4} A_{4} A_{4} A_{4} R 6 \overline{6} 15 \overline{15} 5 \overline{5} 10 \overline{10} highest label (1, 0, 0, 0, 0) (0, 0, 0, 0, 1) (0, 1, 0, 0, 0) (0, 0, 0, 1, 0) (1, 0, 0, 0) (0, 0, 0, 1) (0, 1, 0, 0) (0, 0, 1, 0) tableau C_{1} C_{5} C_{2} C_{4} C_{1} C_{4} C_{2} C_{3} dimension and tensor 6 : v^{I} 6 : v_{I} 15 : T^{I J} = - T^{J I} 15 : T_{I J} = - T_{J I} 5 : v^{i} 5 : v_{i} 10 : T^{ij} = - T^{ji} 10 : T_{ij} = - T_{ji}

Here

1 \leq I, J \leq 6

and

1 \leq i, j \leq 5

. Lower indices denote the dual representation, equivalent to the conjugate representation for these compact groups. The SU(6)

15

in this question is an exterior square; the SU(5)

15

in solution is a symmetric square. The ambient group is essential to distinguish them.

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