S (U (3) \times U (2)) = {diag (U_{3}, U_{2}) : det U_{3} det U_{2} = 1} \subset S U (5)

is locally

S U (3) \times S U (2) \times U (1)

, with only a finite central quotient distinguishing the global groups. The hypercharge direction in the fundamental representation is

Y = diag (- \frac{1}{3}, - \frac{1}{3}, - \frac{1}{3}, \frac{1}{2}, \frac{1}{2}),

which is traceless and commutes with the

S U (3)

and

S U (2)

blocks.

The canonically normalized

S U (5)

generator must satisfy

Tr (T_{Y}^{2}) = 1/2

. Since

Tr (Y^{2}) = 5/6

T_{Y} = \frac{3}{5} Y .

g_{5} T_{Y} B_{μ}

, whereas the Standard Model convention contains

g_{Y} Y B_{μ}

. Hence

g_{Y} = 3/5 g_{5}

. The non-Abelian generators already have the canonical normalization, so gauge coupling unification in the SU(5) grand unified theory predicts

g_{s}^{2} = g_{w}^{2} = g_{5}^{2} = \frac{5}{3} g_{Y}^{2} .

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