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) $.