Differences

This shows you the differences between two versions of the page.

--- blog:2024-12-20 [2024/12/21 07:18] – created pzhou
+++ blog:2024-12-20 [2024/12/21 08:28] (current) – pzhou
@@ Line 9: / Line 9: @@
 We have a few observations:
   - $End(T_{(k_1, k_2)}) = NH_{k_1} \otimes NH_{k_2} =: NH_{k_1, k_2}. $. On the other side, we can consider $(BB \to BP_{k_1, k_2})_* \C$ whose endomorphism is also $NH_{k_1, k_2}$.
+  - This parabolic subgroup $P_{k_1, k_2}$ is the automorphism subgroup that preserves the partial flag $\C^{k_1} \In \C^k$.
+  - We have different kinds of flags there. They are different quiver representations of $ \bullet \to \bullet$. For example, consider the injection $j: \C^{k_1}\into \C^k$ as an object in the quiver rep. The moduli stack for this object is $Gr(k_1, k) / GL(k) = pt/P_{k_1, k_2}$, this is because $GL(k)$ acts on $Gr(k_1, k)$ transitively, with stabilizer of a point $P_{k_1, k_2}$.
+  - How to think about $T_{2,3} \to T_{1,4}$? We have flag $Fl_1:=( \C^3 \to \C^5)$ and $Fl_2 := (\C^4 \to \C^5)$. We do have map from $Fl_1 \to Fl_2$, just like $T_{2,3} \to T_{1,4}$ admits maps.
+Our goal is to build a category living over the Higgs side space. But I don't know what is the ambient space.
+One guess is that, the space is still and always is $BGL_k$, and the T-brane sheaf is still $BB \to BG$ pushing forward. But this is like after doing stop removal. How to remember?
+Another way of thinking is, we don't have to use $[pt/GL_k]$, but $[(pt \to pt)/GL_k]$, and then do sheaves on it. What does it mean? A sheaf of vector spaces on a space $X$, is a functor from the open category to $dgVect$. A sheaf on a quotient stack $[X/G]$, or the category of sheaves on $[X/G]$ is the limit (equalizer in the infinite version)
+$$ Sh(X) \to Sh(X \times G) \to Sh(X \times G \times G) \to \cdots $$
+But what is $Sh(pt \to pt)$?