Timeline for Getting the Weyl dimension formula geometrically
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 14, 2010 at 15:10 | history | edited | David E Speyer | CC BY-SA 2.5 |
added 27 characters in body
|
| Jul 14, 2010 at 15:05 | comment | added | David E Speyer | Any other cool examples of algebraic varieties whose real tangent bundles are stably trivial? The only one that comes to my mind is abelian varieties. | |
| Jul 14, 2010 at 15:04 | comment | added | David E Speyer | Let $\alpha_1$, $\alpha_2$, ..., $\alpha_N$ be the Chern roots of $T_{\mathbb{C}} X$. Set $\rho=(1/2) \sum \alpha_i$. So the Todd class is $e^{\rho} \prod T(\alpha_i)$. Now, $\prod T(\alpha_i)$ is a symmetric function of the $\alpha$'s, and is invariant under negating any subset of the $\alpha$'s. So it must be a power series in the elementary symmetric functions of $\alpha_i^2$. However, those are the chern classes of $T_{\mathbb{C}} \oplus T_{\mathbb{C}}^*$, which we just argued were trivial. So, in this setting, the Todd genus is simply $e^{\rho}$. | |
| Jul 14, 2010 at 15:00 | comment | added | David E Speyer | Here is an attempt to explain why this happens. Let $X$ be an algebraic variety whose real tangent bundle is stably trivial. $K/T$ has this property, although I'm not sure what the quickest proof is. Writing $T_\mathbb{R}$ and $T_\mathbb{C}$ for the real and holomorphic tangent bundles, we have $T_\mathbb{R} \otimes \mathbb{C} \cong T_{\mathbb{C}} \oplus T_{\mathbb{C}}^*$ (topologically), so the Chern classes of $T_{\mathbb{C}} \oplus T_{\mathbb{C}}^*$ are trivial. (Continued...) | |
| Jul 14, 2010 at 14:45 | comment | added | David E Speyer | Here is one of the odd things to note here. The product $\prod_{\alpha \in \Phi^{+}} T(\alpha)$ is $W$-symmetric. By the standard presentation of the cohomology ring of $X$, this means that every high degree term is cohomologically trivial. So we are really just trying to compute $\int_X e^{\mu}$. This partially responds to Bugs' objection that the formula for the Todd genus is messy. (continued...) | |
| Jul 14, 2010 at 14:36 | history | answered | David E Speyer | CC BY-SA 2.5 |