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Victor
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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 $$\sum_w (-1)^n \langle v(w)|v(w)\rangle \prod_{\beta>0} \langle w,\beta\rangle^{-2\langle w,\beta>/<\beta,\beta>}=0$$$$\sum_w (-1)^n \langle v(w)|v(w)\rangle \prod_{\beta>0} \langle \omega_0,\beta\rangle^{-2\langle w,\beta>/<\beta,\beta>}=0$$ 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!

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 $$\sum_w (-1)^n \langle v(w)|v(w)\rangle \prod_{\beta>0} \langle w,\beta\rangle^{-2\langle w,\beta>/<\beta,\beta>}=0$$ 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!

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!

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Victor
Victor
  • 41
  • 3

Norm in the fundamental representations of Lie algebras

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 $$\sum_w (-1)^n \langle v(w)|v(w)\rangle \prod_{\beta>0} \langle w,\beta\rangle^{-2\langle w,\beta>/<\beta,\beta>}=0$$ 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!