I would add the following, which is implicit in the presentation: this phenomenon of real representations is not specific to finite groups. Real irreducible representations of a group are always neatly divided into three types: real, complex or quaternionic. This is [Schur\’s lemma](https\://ncatlab\.org/nlab/show/Schur\%27s\+lemma\#statement) together with the fact that the real division algebras are exactly R, C and the quaternions H.
(Should ML interpretability people care about infinite groups to begin with—unlike mathematicians, who love them all? For once, models as well as datasets can exhibit (exact or approximate) continuous symmetries, and these symmetries be understood mathematically as actions of matrix Lie groups such as the group GL_n of all invertible matrices or the group O_n of n-dimensional rotations. Sometimes these actions are linear, so are themselves representations, and sometimes they can be studied by linearizing them. Using representation theory to study more general geometric group actions is one of those great tricks of mathematics which reduce complicated problems to linear algebra.)
Nice!
I would add the following, which is implicit in the presentation: this phenomenon of real representations is not specific to finite groups. Real irreducible representations of a group are always neatly divided into three types: real, complex or quaternionic. This is [Schur\’s lemma](https\://ncatlab\.org/nlab/show/Schur\%27s\+lemma\#statement) together with the fact that the real division algebras are exactly R, C and the quaternions H.
(Should ML interpretability people care about infinite groups to begin with—unlike mathematicians, who love them all? For once, models as well as datasets can exhibit (exact or approximate) continuous symmetries, and these symmetries be understood mathematically as actions of matrix Lie groups such as the group GL_n of all invertible matrices or the group O_n of n-dimensional rotations. Sometimes these actions are linear, so are themselves representations, and sometimes they can be studied by linearizing them. Using representation theory to study more general geometric group actions is one of those great tricks of mathematics which reduce complicated problems to linear algebra.)