bootstraping a space from its tensor square

Consider the tensor square r?r of an irreducible group representation r with itself, and decompose it as irreducible representations. What can we said about the circumstance of finding the initial representation in the list? Or perhaps about finding its conjugate, as for instance in E6:


What groups have representations having this “bootstraping”? Can the irrep appear in both parts, symmetric and alternating, of the tensor square? Does it appear in an unique way, or can it be extracted from different combinations of the roots?

Similarly, consider the tensor square A?2 of an algebra. Are there situations where the new algebra does contain the initial one as a subalgebra in a non trivial way? This seems to generalise the question of generating an algebra from a finite number of elements and its n-times product, call it A×n, but perhaps it is not more general… still I wonder what can be said generically about such action. What I am expecting is that some ideal J can be chosen in
A?2 such that the quotient recovers the initial algebra. Or some similar mechanism, anyway.

The motivation of the post to be in BSM is, of course, my old observation that by choosing five quarks, out of all the set of three particle generations of the standard model, and pairing them we seem to be able to recover the full three generations, and I wondering if this phenomena could be tracked to some peculiar property in mathematical representation. Thinking it also in algebraic terms is interesting because the attempts to get generations out of the exceptional jordan algebra h3(O) or its twin h3(C?O) have some extra matter in the diagonal, an issue that also happens in the naive pairing.

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