I Invariant symbol implies existence of singlet representation

AndrewGRQTF

I don't understand what the last paragraph of the attached page means. Why does the Kronecker delta being an invariant symbol mean that the product of a representation R and its complex conjugate representation has the singlet representation with all matrices being zero?

Doesn't the number zero always form a trivial one-dimensional representation of any group, because when plugged into the equation $[T ^a , T ^b] = i f^{\text{abc}} T^c$ it trivially satisfies it?

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Orodruin

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It is not a matter of the possible existence of a one-dimensional representation. It is about the product representation of R and its complex conjugate containing a singlet representation when decomposed into irreducible representations. The singlet representation is the trivial representation.

AndrewGRQTF

It is not a matter of the possible existence of a one-dimensional representation. It is about the product representation of R and its complex conjugate containing a singlet representation when decomposed into irreducible representations. The singlet representation is the trivial representation.
Why does the Kronecker being invariant mean that the $R \otimes \overline{R}$ can be decomposed into a direct sum of the singlet representation and other irreps?

Orodruin

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Because being invariant is the fundamental property of being in the trivial representation.

"Invariant symbol implies existence of singlet representation"

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