Peng Zhou

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====== 2024-09-11 quiver hecke algebra ======
After so many years, let me read up on the 'convolution algebra' presentation for quiver hecke algebra, because after all, hecke algebra originates from convolution on flag variety, and quiver hecke algebra is a generalization. 

===== Quiver Hecke algebra =====
Following the Varagnolo-Vasserot paper https://arxiv.org/pdf/0901.3992

Let $\Gamma$ be a quiver without self-loop, $I$ is the vertex set and $E$ the arrow set,  and $a_{ij}$ be the adjacency matrix from node $i$ to $j$. Let $C$ be the symmetric Cartan matrix obtained by setting $c_{ii}=2$, $c_{ij} = -a_{ij} - a_{ji}$. 

Fix a dimension vector $\nu \in \N I$. We get a vector space $V_\nu$.  
Then for each increasing path $\vec \nu$ in $\N I$ to $\nu$ from $0$, we have a (multi)-flag interlaced in $V_\nu$. 
The flag variety is denoted as $F_{\vec nu}$. Abstractly, they are all isomorphic to product of flags $\prod_\nu F(V_{\nu_i})$. 

So far, we have ignored the arrows. Let $E_V = \prod_{e \in E} Hom(V_{s(e)}, V_{t(e)})$. For any $x \in E_V$, and any (homogenous?) flag $\phi \in F(V)$, we say $x$ preserves $\phi$, if for each step $V^l_\phi$ in the flag $V_\phi^*$, $x$ restricts to a sub quiver rep on $V_\phi^l$. 

If the flag $V_\phi$ increase dimension one at a time, we call it a (graded?) full flag variety of $V$; otherwise a graded partial flag variety. Fix a flag type $\vec \nu$, consider all such pairs $\wt F_{\vec \nu} = \{(\phi, x)\}$. It project to $F_{\vec \nu} = \{\phi\}$, and also to $E_V = \{x\}$. The associated graded $V_\phi^l / V_\phi^{l-1}$ defines a collection of line bundles over $F_{\vec \nu}$ and pullback to $\wt F$.