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In representation theory, there are the related concepts of weights and roots. Since both are kinds of generalised eigenvalues, and eigenvalues are roots of e.g. the characteristic polynomial, the word "root" makes sense to me (at least, the question is reduced to why zeros of polynomials / equations are called "roots".) But I wondered:

Who used the term weight (or poids, or Gewicht, or ...) for the first time? And for what (if any) specific reason?

This site does not know the word "weight" in this meaning. (But see the entry "radix" about roots (of equations).) The only thing I could find on the internet is this (unanswered) stackexchange question.

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It's likely that Elie Cartan originated the use of poids in representation theory of Lie groups. But it's harder to sort out the underlying rationale for the "weight" concept here. In the case of "root" there is a sense historically of having a sort of formal characteristic polynomial attached to the adjoint representation. –  Jim Humphreys Jan 18 at 15:57
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I think it's likely that this use of the term 'weight' in representation theory comes from the use of the term 'weighted homogeneous' for polynomials that are only homogeneous when the variables are assigned the appropriate 'weights'. I'm pretty sure that this use of 'weights' was common in the 19th century and possibly even before that. When you consider that the maximal torus will be acting on the weight vectors in a manner that is entirely analogous to weights of variables in a weighted homogeneous situations, this terminology seems quite natural. –  Robert Bryant Jan 18 at 16:24
    
Thanks again, both of you are right. –  Torsten Schoeneberg Jan 20 at 11:44

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Robert Bryant's comment motivates me to mention the "weighty" historical monograph Emergence of the Theory of Lie Groups (Springer, 2000) written by Thomas Hawkins. As usual with terminology such as "weight", the history reaches back into nineteenth century's invariant theory (Cayley, G. Kowalewski) but becomes most relevant to modern Lie theory in the work of Elie Cartan about a century ago. The early part of Chapter 8 in Hawkins' book is most pertinent but not easy reading.

Though Cartan's use of the term poids (weight, Gewicht) was not the earliest mathematical occurrence, it does seem to have been the first use in connection with what we now call weights of representations. There is also a long history involving the term "root" (and its offshoot "secondary root"), going back to antiquity, but here the work of Killing anticipates Cartan's more definitive treatment of semisimple Lie groups and what we now call their Lie algebras. The history is not at all easy to untangle, but I think Hawkins was thorough in his study of the development of ideas along with terminology.

Terminology in this particular subject should not be taken too seriously, I think, and sometimes the names given to things are either misleading or inappropriate (including concepts named after people). Still, we are stuck with the language, which is almost impossible to change.

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Thanks, I will have a look at this book. –  Torsten Schoeneberg Jan 18 at 18:30
    
@Torsten: The history of Lie theory goes back a long way, and Tom Hawkins has devoted much of his professional career to understanding it in detail. Of course, a modern researcher need not master so much of the history, but it's an interesting example of the evolution of ideas. –  Jim Humphreys Jan 18 at 18:37
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Before someone comments my editing the post with "poids" instead of "poid" : this is one of those singular french words taking an 's' (pun intended). –  Julien Puydt Jan 19 at 19:23
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@Julien: Thanks for the proofreading help. I'm not sure how I wrote "poid", but I should note that "nineteenth century invariant theory" is proper English usage though your version is not. (And for me "relevant" is about as good as "pertinent", though the former word is overused.) –  Jim Humphreys Jan 19 at 20:00
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Pp. 272 and 288 of Hawkins's book answer the question, consistent with Robert Bryant's comment. Cartan introduced "poids" in papers of 1909 / 1913. According to Hawkins, this was prompted by "Gewichte" in a 1902 paper by G. Kowalewski (an expository version of which is digizeitschriften.de/dms/img/?PPN=GDZPPN002118912). Kowalewski's usage, in turn, comes from 19th century invariant theory. -- Amusing sidenotes: 1) In his French paper in Crelle 1854, Cayley translates weight as "pesanteur". 2) On p. 288, Hawkins writes "poid" without "s", like, well, some of us have done here ... –  Torsten Schoeneberg Jan 19 at 22:04

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