====== 2023-01-27 End of AIM workshop ======
  * [AK]: Vector Bundle on $\P^2$
  * [EG]: Affine Springer Fiber
  * [WL]: Ruling and Stratification

===== Vector Bundles on $\P^2$ =====
It is always a good idea to share your thoughts, it might induce more sparks. 

We follow Knutson and Sharpe. 

Consider the moduli space of rank $n$ vector bundles on $\P^2$. It is given by a disjoint union of components, labelled by $(\lambda, \mu, \nu) \in (X^*(T)/W)^3$, dominant weights
$$ (L_\lambda \times L_\mu \times L_\nu) / GL(n) $$
where $L_\chi$ is the equivariant line bundle over the flag variety $GL_n/B$. 

First, we recall Klyacho's description of toric vector bundle on $\P^2$. Consider $v_1, v_2, v_3 \in N$ the three ray generators of the toric fan. Let $D_1, D_2, D_3$ be the corresponding divisor. The subtorus $T_i$ for $v_i$ fixes $D_i$. Consider the vector bundle $E$ on $\P^2$, we have $T_i$ acting on $E|_{D_i}$, with weights. They give me a collection of hyperplanes. 

Now, we can consider the restriction of $E$ over the torus fixed points. Here is {{ :blog:alexander_a_klyachko.pdf |Klyacho's}} 'filtration description' of a toric vector bundle. 


So, why we have a filtration? We can say, if we do restriction to fixed, points, we get a weight decomposition of the fiber over there. More generally, we get a multi-polytope, as shown in [[https://web.ma.utexas.edu/users/sampayne/pdf/Moduli-toric-vector-bundles.pdf | Sam Payne's paper]]

How does this compare with the configuration space of decorated flags?