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--- blog:2023-07-24 [2023/07/25 06:12] – pzhou
+++ blog:2023-07-24 [2023/07/25 06:32] (current) – pzhou
@@ Line 3: / Line 3: @@
   * watched a youtube video [[https://www.youtube.com/watch?v=NOl0v54DaXo | 8 traits of successful people]]
   * try to fix the annoying keyboard on a macbook pro, which turns out to be [[https://support.apple.com/keyboard-service-program-for-mac-notebooks| not my (or my wife's) fault.]]
+  * found an interesting lecture note of Teleman on rep theory. Never really understood what is character formula.
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 You have a coordinate ring, of the torus, or of the plane. And you have the $W$ action. You pass to the $W$ invariant subring. Then, the original ring turns out to be a **free module** of finite rank over the W-invariant ring.
-You have a bunch of fundamental weight, they are element in the weight lattice, that are dual to the coroots. (ok, what are roots and coroots? Just do $gl_n$. you pick a cartan subalgebra, and let the cartan acts on Lie algebra. I guess I am just not used to having a family of commuting operator acting on something. (how about module over a commutative ring?) ok fine. ok, as h-mod, $\frak g$ lives over a bunch of points on $\frak h^*$. So far, these are canonical, we don't have Killing form. Let's assume 'semi-simple', which says, the root vectors span $\frak h^*$. Take a half-plane, and take some primitive roots, call them simple root.
+You have a bunch of fundamental weight, they are element in the weight lattice, that are dual to the coroots. (ok, what are roots and coroots? Just do $gl_n$. you pick a cartan subalgebra, and let the cartan acts on Lie algebra. I guess I am just not used to having a family of commuting operator acting on something. (how about module over a commutative ring?) ok fine. ok, as h-mod, $\frak g$ lives over a bunch of points on $\frak h^*$. So far, these are canonical, we don't have Killing form. Let's assume 'semi-simple', which says, the root vectors span $\frak h^*$. Take a half-plane, and take some primitive roots, call them simple root.  read this note https://math.mit.edu/~dav/roots.pdf for what is coroot)
-cut the crap. read this note https://math.mit.edu/~dav/roots.pdf
+So, we have $R(G) = R(T)^W$ and $\C[G]^G = \C[T]^W$. The first one is the 'integral form'. What does a 'coordinate ring over $\Z$' even mean? And, how much information did I lose?