# Origin of the term “weight” in representation theory

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.

• 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 '14 at 15:57
• 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 '14 at 16:24