Norm in the fundamental representations of Lie algebras

2011-02-14T00:24:20Z

Let $\mathfrak{g}$ be a complex simple Lie algebra, $\omega_0$ a dominant weight, $\rho$ a fundamental representation with highest weight $\Lambda$.

Fix some weight $w$ in this representation. Let $\sigma$ enumerate ways in which weight space $w$ can be reached from the highest weight, i.e. each $\sigma$ corresponds to the sequence of the form $$E_{i_n}^-\dots E_{i_1}^-|\Lambda\rangle$$ where $\alpha_{i_n}+\dots+\alpha_{i_1}=\Lambda-w$. On the way from $\Lambda$ to $w$ we encounter weights $$(\Lambda=w_1, w_2, \dots, w_n, w_{n+1}=w)$$ Form the following vector in the weight space of $w$: $$|v(w)\rangle=\sum_\sigma \prod_{a=1}^n \frac{1}{\langle w-w_a,\omega_0\rangle} E_{i_n}^-\dots E_{i_1}^-|\Lambda\rangle$$ Question one is: does there exist any simple formula for the norm $\langle v(w)|v(w)\rangle$?

Question two is: prove the identity $$\sum_w (-1)^n \langle v(w)|v(w)\rangle \prod_{\beta>0} \langle \omega_0,\beta\rangle^{-2\langle w,\beta>/<\beta,\beta>}=0$$ Some experimentation shows that the identity is correct, and when the weight space is one-dimensional the formula is $$\langle v(w)|v(w)\rangle=\left(\frac{constant}{\prod_{\beta_a>0} \langle \beta_a,\omega_0\rangle^{n_a}}\right)^2$$ where $\beta_a$ are positive roots and for each of them $n_a$ is the length of the sequence $$w-n_a\beta_a,\dots,w-\beta_a,w$$ How can something like that be proved? There should be some nice formula for weight space with dimension >1 as well, but the brute force method didnt help... Any ideas would be appreciated. Do there exist any similar expressions in the literature? Thank you!

Norm in the fundamental representations of Lie algebras - MathOverflow

Norm in the fundamental representations of Lie algebras