Three spin-one angular-momentum decomposition

Coupling three distinguishable spin-one systems gives

1 \otimes 1 \otimes 1 = 0 \oplus 3 (1) \oplus 2 (2) \oplus 3,

where the labels denote total spin and coefficients denote multiplicities. The dimensions are respectively

1

,

3 \cdot 3

,

2 \cdot 5

, and

7

, summing to

27

.

Table of contents
- Symmetric three-spin-one subspace Three spin-one angular-momentum decomposition

Symmetric three-spin-one subspace

The completely symmetric subspace of three spin-one systems has dimension

(3 3 + 3 - 1) = 10

and decomposes into one total-spin-

3

multiplet and one total-spin-

1

Sym^{3} (1) = 3 \oplus 1.

The highest-weight product state generates the seven-dimensional spin-

3

multiplet; its three-dimensional invariant complement is spin

1

.

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 33A / Solution

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