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 Incoming links: Low-dimensional SU3 tensor representations

Use upper indices for the defining representation of SU(3) and lower indices for its dual representation. A Young diagram with row lengths

(ℓ_{1}, ℓ_{2})

(p, q) = (ℓ_{1} - ℓ_{2}, ℓ_{2})

; a full column of height three is a determinant representation and can be removed for SU(3). The Weyl dimension formula reduces to

dim (p, q) = \frac{( p + 1 ) ( q + 1 ) ( p + q + 2 )}{2} .

1, 3, \overline{3}, 6, \overline{6}, 8

. The following box arrays draw their Young diagrams; filling boxes by tensor-slot labels gives representative Young tableaux:

representation 13 \overline{3} 6 \overline{6} 8 diagram \emptyset i i i i i i i i i i i i (ℓ_{1}, ℓ_{2}) (0, 0) (1, 0) (1, 1) (2, 0) (2, 2) (2, 1) (p, q) (0, 0) (1, 0) (0, 1) (2, 0) (0, 2) (1, 1) dim 133668

Their associated tensors, in the same order, can be taken as follows:

$1$ : a scalar $s$ , unchanged by SU(3).
$3$ : a vector $v^{i}$ , with $v^{i} \mapsto U^{i}_{j} v^{j}$ .
$\overline{3}$ : an antisymmetric tensor $A^{[ij]}$ , equivalently a covector $w_{k} = \frac{1}{2} ϵ_{kij} A^{ij}$ , with $w_{i} \mapsto w_{j} (U^{- 1})^{j}_{i}$ .
$6$ : a symmetric tensor $S^{(ij)}$ , with $S^{ij} \mapsto U^{i}_{k} U^{j}_{ℓ} S^{k ℓ}$ .
$\overline{6}$ : a symmetric covariant tensor $S_{(ij)}$ , with $S_{ij} \mapsto S_{k ℓ} (U^{- 1})^{k}_{i} (U^{- 1})^{ℓ}_{j}$ . In the four-upper-index Young tableau description, use two antisymmetric index pairs with symmetry under exchange of pairs, and convert each pair to a lower index with the invariant $ϵ$ tensor.
$8$ : a mixed tensor $T^{i}_{j}$ with $T^{i}_{i} = 0$ , transforming as $T \mapsto U T U^{- 1}$ . This is the Adjoint representation. Its nine components obey one independent trace constraint.

1, 3, 3, 6, 6, 8

by direct component counting. To check that none were missed, if

p + q \geq 3

then

(p + 1) (q + 1) \geq p + q + 1

, so the displayed dimension is at least

(p + q + 1) (p + q + 2) /2 \geq 10

. The pairs with

p + q \leq 2

are precisely the six listed pairs.