# What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations.

I have come across the terminology "a complete string of weights" in my lecture course, but it is not defined what he means by either "string of weights" or "complete" here, unfortunately.

It comes up in a discussion of the weight space of the Lie algebra $\mathfrak{su}(3)$.

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I know that the Lie Group $SU(3)$ has rank 2, so we get a 2-dimensional weight-lattice.

Moreover, $\mathfrak{su}(3)$ has two commuting $\mathfrak{su}(2)$ Lie subalgebras. The Lie algebra $\mathfrak{su}(2)$ has rank 1, so it would only have a 1-dimensional weight-lattice.

So I am guessing that it means there is an $\mathfrak{su}(2)$ sub-weight-lattice of the $\mathfrak{su}(3)$ weight-lattice which is a sub-set or equal to the $\mathfrak{su}(2)$ weight-lattice, or some such.

(Apologies for the possibly poor choice of terminology!)

Is this on the correct track?

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I have some notions about completeness, in terms of Metric Spaces, for example, a metric space (M,d) is Complete if all Cauchy sequences converge in (M,d).

Moreover, in terms of Linear Algebra a set of $n$ vectors $\{ e_i \}$ with $e_i \in V$ for some vector space $V \subset \mathbb{R}^n$ is complete if the $\{ e_i \}$ span V. (Though I could not find a wikipedia article mentioning the completeness of a basis, only this one mentioning __in__completeness)

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Is this the same meaning for "a string of weights".

I have tried searching, wikipedia for example, but get no results for "String of Weights"

I am a maths-physicists, so I have only first and second year pure maths. Notably, no Lebesgue Theory and Measure Spaces theory etc. (As well as lots of other stuff besides!).

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Thank you

• I think a bit more context would be helpful. Can you give the full sentence/paragraph where this appears. The term string (of weights) appears in the study of crystal basis, but it may not be what your lecturer means. May 13, 2014 at 16:14
• I've heard the term used in the following constants: the string of weights is attached to a particular $sl_{2}$ triple (corresponding to a root $\alpha$) inside the Lie algebra and a particular weight $\omega$ in the representation, and refers to the set of weights of the form $\omega + n \alpha$. May 14, 2014 at 13:07
• Seconding what Siddharth Venkatesh said. It is possible that complete here means both A) the string is unbroken (no gaps in the range of values of the integer multiplier $n$) as well as B) the end points of the string are mirror images of each other w.r.t. the reflection determined by $\alpha$. I could be wrong here, though. The context should clarify many a thing. May 16, 2014 at 6:08