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How old is the use of the name ’intertwiner’? I find it very obscure (and ugly) as a piece of terminology since it is simply a morphism of the representations. (I learnt representation theory from Ledermann and the term was nowhere to be seen in his courses! The courses were beautifully given, very categorical in a general sense.)
But if you talk in terms of matrices, it really looks like intertwiner…
Old books define intertwiner between representations $T$ and $T'$ as an invertible linear operator $U$ such that $T'(g) = U T(g) U^{-1}$ for all $g$. As one sees, $T(g)$ is sandwiched between $U$ and $U^{-1}$ (and this becomes a bit more involved at the level of matrix elements). Hence intertwiner.
I also agree strongly! I have always been confused about why people use such a weird word for such a simple thing. In my idiolect, being “sandwiched between” is not much like being “intertwined” – the latter word sounds to me like a braid or a tangle.
My reason for raising the point is that I feel that intertwiner when it occurs in the nLab should be used as an alternative to the morphism of reps or whatever not as the main term. At present we do not have an entry for ’intertwiner’ and perhaps we should with a bit of gentle explanation. (In fact the Wikipedia article is quite good in this.)
What I found almost insulting was that sometimes people giving seminars used the term as if it was the only one that could be imagined for the concept, whilst they did not say what the concept was and the term does not explain anything. As I said Ledermann as I remember, did not talk of intertwiners but did talk about homomorphisms of representations.
the latter word sounds to me like a braid or a tangle
Right, various notions of braiding in algebra, for example the notion of braided monoidal category involves similar sandwiches when passing from the tensor product to the tensor product in the opposite order. Representations were invented by people who wanted to consider groups realized via matrices, this is algebraic and coordinate thinking. Representation theorists usually start their courses by saying that groups are not abstract objects but appear concretely and the main way is via matrices. Such an introduction is a reason why I never went much into representation theory, despite being surrounded by good representation theorists all my life, but went to noncommutative algebraic geometry, while not being surrounded by such. If one does not like to talk representation and representation terminology, then one should say module, and homomorphism of modules. The same people who as a rule talk intertwiners of representations, very rarely say intertwiner of modules, but rather (homo)morphism of modules. Representations are in their business usually thought in terms of characters, matrix elements, matrix blocks and so on, while when saying module, they usually take the picture of a generalization of a vector space.
I think that in $n$Lab we should have both passages and entries with modern perspective from the point of view of invariant notions but also should have classical entries or passages which are friendly to traditional users, physicists and so on, and make link to traditional notation. If we sterilize the environment, the outcome may become somewhat sterile as well.
Yes, U is.
But that means that U is the intertwiner irrespective of what T and T’ are! Does an intertwiner have a domain and codomain? I am still confused. (Zoran, you were wise to avoid representation theorist for whom all groups are groups of matrices!) Can someone provide a stub on intertwiner which can be expanded, but would avoid some of the confusion. (I did a search on the web and the usage of the term seems varied and sometimes inconsistent.)
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