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 Incoming links: Hook-content formula

Write

[r_{1}, r_{2}, \dots]

r_{i}

, and depict an empty box by

□

. Appending the box of the fundamental representation to another single box gives a row or a column:

□ \otimes □ = (□ □) \oplus □ □, [1] \otimes [1] = [2] \oplus [1, 1] .

4 \cdot 5/2 = 10

and

4 \cdot 3/2 = 6

, so

4 \otimes 4 = 10_{s} \oplus 6_{a} .

For the conjugate representation, the invariant volume tensor identifies

\overset{ˉ}{4} ≅ Λ^{3} 4

, represented by a column of three boxes. Adding one box gives

[1] \otimes [1, 1, 1] = [2, 1, 1] \oplus [1, 1, 1, 1], □ \otimes □ □ □ = □ □ □ □ \oplus □ □ □ □ .

A column of four boxes is the determinant representation, which is trivial on

S U (4)

(1, 0, 1)

and dimension fifteen. For example the hook-content formula gives

dim [2, 1, 1] = \frac{4 \cdot 5 \cdot 3 \cdot 2}{4 \cdot 1 \cdot 2 \cdot 1} = 15.

Equivalently, a mixed tensor

M^{i}_{j}

M \mapsto U M U^{†}

and splits into its trace and traceless matrix parts. Therefore

4 \otimes \overset{ˉ}{4} = 1 \oplus 15,

where

15

Use the standard chain ordering of the five simple roots of

S U (6)

r_{1} \geq \dots \geq r_{6} = 0

a_{i} = r_{i} - r_{i + 1}

. A full column of six boxes can be removed because it is the trivial determinant representation. The required data are

representation 6 \overset{ˉ}{6} 211535 (a_{1}, a_{2}, a_{3}, a_{4}, a_{5}) (1, 0, 0, 0, 0) (0, 0, 0, 0, 1) (2, 0, 0, 0, 0) (0, 1, 0, 0, 0) (1, 0, 0, 0, 1) row lengths [1] [1, 1, 1, 1, 1] [2] [1, 1] [2, 1, 1, 1, 1] tensor v^{i} w_{i} S^{ij} = S^{ji} A^{ij} = - A^{ji} M^{i}_{j}, M^{i}_{i} = 0 dim 66211535 .

In the same order, the Young tableaux are

□, □ □ □ □ □, □ □, □ □, □ □ □ □ □ □ .

Each upper index transforms by

U

, and each lower index by

U^{- 1}

. In particular

M \mapsto U M U^{- 1}

is the complex Adjoint representation. The column of five boxes can be represented either by

w_{i}

B^{i_{1} \dots i_{5}}

, using the invariant six-index alternating tensor:

w_{j} = \frac{1}{5 !} ϵ_{j i_{1} \dots i_{5}} B^{i_{1} \dots i_{5}} .

Thus it really is the conjugate representation, rather than a second independent six-dimensional choice. Similarly converting the lower index of

M^{i}_{j}

into five alternating upper indices gives the displayed hook Young diagram; the removed trace would be its full six-box column component.

The dimension counts are

dim Sym^{2} C^{6} = \frac{6 \cdot 7}{2} = 21, dim Λ^{2} C^{6} = \frac{6 \cdot 5}{2} = 15, dim End_{0} (C^{6}) = 6^{2} - 1 = 35.

The first two count independent entries of symmetric tensors and antisymmetric tensors; the last removes one trace from a mixed tensor. The hook-content formula independently gives these dimensions from the Young diagrams.