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 Incoming links: Past exam of the groups and representations course of the University of Oxford / 2022 / 3 / e / Solution

Past exam of the groups and representations course of the University of Oxford / 2022 / 4 / b / Solution 2026-10-25

For the special unitary group

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}

is the exterior power

Λ^{k} V

, with highest weight

ω_{k}

(k N)

. Contraction with the invariant volume tensor identifies

V^{*} ≅ Λ^{N - 1} V

and

Λ^{2} V^{*} ≅ Λ^{N - 2} V

. This gives all the requested Dynkin labels, Young tableaux, dimensions and tensors:

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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