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This problem concerns the topic representation theory of Lie Algebras.

The main purpose of the exercise is to study the form of the fundamental dominant weights of a Lie Algebra.

I would be very glad if someone could take a look at my approach! Thank you in advance.


Let V be a finite dimensional vector space over a field F.

Let L be a semisimple Lie Algebra equal to the special linear algebra, $\ L=sl(l+1,F)$ with dimension $\ l+1$.

Let the Cartan subalgebra H of L equal the intersection of the set of diagonal matrices and L, $\ H=d(l+1,F) \bigcap L.$

Let $\ \mu_1,...,\mu_{l+1} $ be the coordinate functions on H, relative to the standard basis of the general linear algebra $\ gl(l+1,F)$.

Then $\ \sum {\mu_i=0} $, and $\ \mu_1,...,\mu_{l} $ form a basis of the dual space H*, while the set of roots $\ \alpha_i=\mu_i-\mu_{i+1} (1\leq i\leq l)$ is a base $\ \Delta $ for the root system $\ \Phi $.

Verify that the Weyl Group W acts on H* by permuting the $\ \mu_i$; in particular, the reflection with respect to $\ \alpha_i $ interchanges $\ \mu_i,\mu_{i+1}$ and leaves the other $\ \mu_j $ fixed.

Then show that the fundamental dominant weights relative to $\ \Delta $ are given by $\ \lambda_k=\mu_1+...+\mu_k (1\leq k\leq l)$.


I have gathered some information about the given notions:

I know that the special linear algebra is the set of endomorphisms of V with trace zero. The standard basis of the special linear algebra contains all $\ e_{ij} (i \neq j), h_i = e_{ii} - e_{i+1,i+1}(1\leq i\leq l)$, for a total of $\ l+(l+1)^{2} -(l+1)$ matrices.

Since L is semisimple, the Cartan subalgebra equals the maximal toral subalgebra which is abelian.

The Weyl group is generated by the reflections $\ \sigma_{\alpha} ,\alpha \in \Phi $. The reflection $\ \sigma_{\alpha}$ sends $\ \alpha $ to $\ - \alpha $, so $\ \sigma_{\alpha}(\mu) = \mu- (2(\mu,\alpha)/(\alpha,\alpha)) \cdot \alpha $.

About the fundamental dominant weights:

If $\ \Delta $ = {$\ \alpha_1,...,\alpha_l $ }, then the vectors $\ (2 \alpha_i)$ \ $\ (\alpha_i , \alpha_i)$ again form a basis. Let $\ \lambda_1,...,\lambda_l $ be the dual basis: $\ (2 (\lambda_i, \alpha_j)$ \ $\ (\alpha_j , \alpha_j)) = \delta_{ij} $. Moreover, all $\ \langle \lambda_i, \alpha \rangle $ (with $\ \alpha \in \Delta ) $ are nonnegative integers and $\ \sigma_i \lambda_j = \lambda_j - \delta_{ij} \alpha_i $.


This is my effort in solving the first part of the exercise:

The problem is: "Verify that the Weyl Group W acts on H* by permuting the $\ \mu_i$; in particular, the reflection with respect to $\ \alpha_i $ interchanges $\ \mu_i,\mu_{i+1}$ and leaves the other $\ \mu_j $ fixed."

H* is spanned by $\ \mu_1,...,\mu_l $.

$\ \sigma_{\alpha_i} (\mu_j)= \mu_j - (2 (\mu_j , \alpha_i) / (\alpha_i , \alpha_i))\alpha_i $

$\ = \mu_j - (2 (\mu_j , \mu_i - \mu_{i+1}) / (\mu_i - \mu_{i+1} , \mu_i - \mu_{i+1}))(\mu_i - \mu_{i+1}) $


Until now I have not taken into consideration the premises given in the exercise about the Lie Algebra and the Cartan subalgebra. I wonder if this would affect the approach to the solution.


This is my effort in solving the second part of the exercise:

The problem is:

"Then show that the fundamental dominant weights relative to $\ \Delta $ are given by $\ \lambda_k=\mu_1+...+\mu_k (1\leq k\leq l)$." To find out of which form the fundamental dominant weights are, consider the weight space: $\ V_{\mu} = $ { $\ v\in V | h.v= \mu (h)v $ for all $\ h \in H $ }.

For the fixed basis { $\ e_1,...,e_{l+1} $ }, one should represent $\ v \in V_\mu $ as a combination of the elements of the basis.

Unfortunately, I don't know which operation to use for combining the elements because, as I have said above, I didn't know how to use the premises.

I think that this is how the proof is supposed to go on in general:

Applying the formula $\ h.v= \mu (h)v $ should yield $\ h.v=(\mu_1+...+\mu_p)(h)v, (1\leq p\leq l+1) $

The heightest weight should be of the form $\ \mu_1+...+\mu_q, (1\leq q\leq l+1) $, so that the fundamental dominant weights are of the form $\ \lambda_k =\mu_1+...+\mu_k, (1\leq k\leq l) $.

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So now your calculation of $\sigma_{\alpha_i}(\mu_j)$ has started out right. I have problems seeing what is holding you back. Just to make sure: Do you know that $(\vec{x},\vec{y})$ is the usual inner product? In a freshman course it was called the dot product. Originally it was defined differently, but at this point in a Lie-algebra book typically the weights and the roots have been mapped to a Euclidean space. –  Jyrki Lahtonen Jul 11 '11 at 5:32

1 Answer 1

Hints:

1) You are supposed to show that $\sigma_{\alpha_i}$ permutes the vectors $\mu_1,\mu_2,\ldots,\mu_{\ell+1}$ in a certain way, so why are you studying the effect ot $\sigma_{\alpha_i}$ on the roots?

2) Check the definition of a fundamental dominant weight. You wrote down the definition of a dominant weight.

3) Compute $\langle \lambda_i,\alpha_j\rangle$ for all $i,j$.

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Thank you very much for your hints. I tried to incorporate 1) and 2) into my section on the definitions and on my work on the first part of the exercise. Unfortunately I didn't know how to go on. On 3): < $\ \lambda_i , \alpha_j $ > = $\ \delta_{ij} $. Could you please help me from here on? –  nikki Jul 10 '11 at 13:30
    
Start with $(\mu_i,\mu_j)=\delta_{ij}$. So $\sigma_{\alpha_i}(\mu_j)=?$. Or to make it more visualizable: what would $\sigma_{\alpha}$ be in $\mathbf{R}^2$, if $\alpha=\vec{i}-\vec{j}$? –  Jyrki Lahtonen Jul 10 '11 at 15:14
    
In R² $\ \sigma $ sends $\ \alpha $ to $\ -\alpha $, so $\ \sigma_{\alpha} = j-i $. Hm, obviously I can't visualize what happens if $\ \sigma_{\alpha} $ is applied to $\ \mu_j $, apart from the abstract definitions... –  nikki Jul 10 '11 at 15:39
    
$\sigma_\alpha$ is a linear mapping from $\mathbf{R}^2$ to itself. Where does the reflection send: $\mu_1=\vec{i}$, $\mu_2=\vec{j}$, the point $(x,y)$? –  Jyrki Lahtonen Jul 10 '11 at 16:04
    
As you said: $$\sigma_{\alpha}(\mu) = \mu- 2\,\frac{(\mu,\alpha)}{(\alpha,\alpha)} \, \alpha. $$ Apply this for $\alpha=\vec{i}-\vec{j}$ and $\mu=\mu_1=\vec{i}$ as well as $\mu=\mu_2=\vec{j}$. –  Jyrki Lahtonen Jul 10 '11 at 21:21

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