## Representations of quivers

Let $Q$ be a Dynkin diagram of type $A,D$ or $E$ with vertex set $I$. For $\gamma ={\gamma }_{1}{\alpha }_{1}^{\vee }+\cdots +{\gamma }_{n}{\alpha }_{n}^{\vee }$, with ${\gamma }_{1},\dots ,{\gamma }_{n}\in {ℤ}_{\ge 0}$, define

 ${Q}_{\gamma }=\left\{M\in Q\mathrm{-mod}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}\mathrm{dim}\left(M\right)=\gamma \right\}$     and     ${G}_{\gamma }=\prod _{i=1}^{n}{\mathrm{GL}}_{{\gamma }_{i}}\left(ℂ\right)$.

## Preprojective algebras

Let $Q$ be a Dynkin diagram of type $A,D$ or $E$ with vertex set $I$. Let $\stackrel{‾}{Q}$ be the quiver with an extra edge ${a}^{*}$, in the opposite direction, for each edge $a$ in $Q$. Let $ℂ\stackrel{‾}{Q}$ be the path algebra of $\stackrel{‾}{Q}$. For each vertex $i$ of $Q$ let

 ${r}_{i}=\sum _{a\in Q,s\left(a\right)=i}{a}^{*}a-\sum _{a\in Q,e\left(a\right)=i}{a}^{*}a,$ (pprels)
where $s\left(a\right)$ is the source of $a$ and $e\left(a\right)$ is the target of $a$. The preprojective algebra is
 $\Lambda =\frac{ℂ\stackrel{‾}{Q}}{⟨{r}_{i}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}i\in I⟩}$. (ppalg)

For $\gamma ={\gamma }_{1}{\alpha }_{1}^{\vee }+\cdots +{\gamma }_{n}{\alpha }_{n}^{\vee }$, with ${\gamma }_{1},\dots ,{\gamma }_{n}\in {ℤ}_{\ge 0}$, define

 ${\Lambda }_{\gamma }=\left\{M\in \Lambda \mathrm{-mod}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}\mathrm{dim}\left(M\right)=\gamma \right\}$     and     ${G}_{\gamma }=\prod _{i=1}^{n}{\mathrm{GL}}_{{\gamma }_{i}}\left(ℂ\right)$.
The preprojective cycles of type $\gamma$ are the irreducible components
 ${\Lambda }_{b}\in \mathrm{Irr}\left({\Lambda }_{\gamma }\right)$. (ppcyc)
The character of a $\Lambda$-module $M$ is the element of the shuffle algebra given by
 $\mathrm{ch}\left(M\right)=\sum _{{i}_{1},\dots {i}_{d}}\chi \left({ℱ\left(M\right)}_{{i}_{1},\dots ,{i}_{d}}\right){f}_{{i}_{1}}\cdots {f}_{{i}_{d}}$, (Λch)
where ${ℱ\left(M\right)}_{{i}_{1},\dots ,{i}_{d}}$ is the subset of the full flag variety of $M$ of composition series of $M$ of type $\left({i}_{1},\dots ,{i}_{d}\right)$ and $\chi \left({ℱ\left(M\right)}_{{i}_{1},\dots ,{i}_{d}}\right)$ is the Euler characteristic of ${ℱ\left(M\right)}_{{i}_{1},\dots ,{i}_{d}}$.

Let

 $\stackrel{\sim }{ℳ}=\underset{\gamma }{\oplus }{M\left({\Lambda }_{\gamma }\right)}^{{G}_{\gamma }}$,      where
$M\left({\Lambda }_{\gamma }\right)$ is the space of constructible functions on ${\Lambda }_{\gamma }$ and ${M\left({\Lambda }_{\gamma }\right)}^{{G}_{\gamma }}$ is the subspace of constructible functions constant on the orbits of ${G}_{\gamma }$. Let
 ${f}_{i}=\mathrm{ch}\left({Z}_{{\alpha }_{i}^{\vee }}\right)$,      where    ${Z}_{{\alpha }_{i}^{\vee }}={\Lambda }_{{\alpha }_{i}^{\vee }}\cong \mathrm{pt}$,
and let $ℳ$ be the subalgebra of $\stackrel{\sim }{ℳ}$ generated by the ${f}_{i}$. By [Lu2, Theorem 12.13] the map
 $\begin{array}{ccc}U{𝔫}^{-}& \to & ℳ\\ {f}_{i}& ↦& {f}_{i}\end{array}$      is an algebra isomorphism.

The ${a}^{*}$-forgetting morphism is

 ${\pi }_{\gamma }:{\Lambda }_{\gamma }\to {Q}_{\gamma }$
According to [GLS, §10.3], it was proved by Lusztig [Lu1] that there are bijections
 $\begin{array}{ccccc}\left\{{\Lambda }_{b}\right\}& ↔& \left\{{G}_{\gamma }\phantom{\rule{.1em}{0ex}}\text{-orbits on}\phantom{\rule{.2em}{0ex}}{Q}_{\gamma }\right\}& ↔& \left\{\text{multisegments}\phantom{\rule{.2em}{0ex}}b\phantom{\rule{.2em}{0ex}}of weight\phantom{\rule{.2em}{0ex}}\gamma \right\}\\ \stackrel{‾}{{\pi }^{-1}\left({Q}_{b}\right)}& ←& {Q}_{b}& ←& b\end{array}$.

## The type ${A}_{n}$ quiver

Let ${Q}_{n}$ be

 $•\stackrel{\phantom{-}{a}_{1\phantom{-}}}{←}•\stackrel{\phantom{-}{a}_{2\phantom{-}}}{←}•\stackrel{\phantom{-}{a}_{3\phantom{-}}}{←}\phantom{\rule{.5em}{0ex}}\cdots \phantom{\rule{.5em}{0ex}}\stackrel{{a}_{n-2}}{←}•\stackrel{{a}_{n-1}}{←}•$
Then ${\stackrel{‾}{Q}}_{n}$ is
 $•\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}•\underset{\phantom{-}{a}_{2\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{2\phantom{-}}}{⇆}}•\underset{\phantom{-}{a}_{3\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{3\phantom{-}}}{⇆}}\phantom{\rule{.5em}{0ex}}\cdots \phantom{\rule{.5em}{0ex}}\underset{{a}_{n-2}^{*}}{\overset{{a}_{n-2}}{⇆}}•\underset{{a}_{n-1}^{*}}{\overset{{a}_{n-1}}{⇆}}•$
and the preprojective algebra $\Lambda$ is generated by ${a}_{1},\dots ,{a}_{n},{a}_{1}^{*},\dots ,{a}_{n}^{*},$ with relations
 ${a}_{1}{a}_{1}^{*}=0$, ${a}_{n-1}^{*}{a}_{n-1}=0$, and ${a}_{i}^{*}{a}_{i}={a}_{i+1}{a}_{i+1}^{*}$. $1$ $2$ $0$ $n-1$ $n$ $0$ $i-1$ $i$ $i+1$

A typical $\Lambda$-module can be viewed as a linear combination of the basis elements with "red" maps corresponding to the ${a}_{i}$ and "blue" maps corresponding to the ${a}_{i}^{*}$. If we view the basis elements as boxes, then the ${a}^{*}$-forgetting morphism takes

to

Example: Type ${A}_{2}$. In this case

 ${Q}_{2}\phantom{\rule{1em}{0ex}}\text{is}\phantom{\rule{1em}{0ex}}•\stackrel{\phantom{-}{a}_{1\phantom{-}}}{←}•\phantom{\rule{4em}{0ex}}\text{and}\phantom{\rule{4em}{0ex}}{\stackrel{‾}{Q}}_{2}\phantom{\rule{1em}{0ex}}\text{is}\phantom{\rule{1em}{0ex}}•\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}•$.
Then
 ${\Lambda }_{{\alpha }_{1}}=\left\{\stackrel{ℂ}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}\stackrel{0}{•}\right\}\simeq \mathrm{pt}$ and ${\Lambda }_{{\alpha }_{2}}=\left\{\stackrel{0}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}\stackrel{ℂ}{•}\right\}\simeq \mathrm{pt}$
and
 ${\Lambda }_{{\alpha }_{1}+{\alpha }_{2}}=\left\{\stackrel{ℂ}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}\stackrel{ℂ}{•}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{1},{a}_{1}^{*}\in ℂ\phantom{\rule{.5em}{0ex}}\text{with}\phantom{\rule{.5em}{0ex}}{a}_{1}{a}_{1}^{*}=0\right\}$
has two irreducible components
 ${\Lambda }_{\left(1,0,1\right)}=\left\{\left(a,0\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}a\in ℂ\right\}$ and ${\Lambda }_{\left(0,1,0\right)}=\left\{\left(0,{a}^{*}\right)\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}^{*}\in ℂ\right\}$
and ${G}_{{\alpha }_{1}+{\alpha }_{2}}={ℂ}^{×}×{ℂ}^{×}$ acts on ${\Lambda }_{{\alpha }_{1}+{\alpha }_{2}}$ with three orbits
 ${\Lambda }_{\left(1,0,1\right)}-\left\{\left(0,0\right)\right\},\phantom{\rule{3em}{0ex}}{\Lambda }_{\left(0,1,0\right)}-\left\{\left(0,0\right)\right\},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left\{\left(0,0\right)\right\}$.

and, when $\gamma =2{\alpha }_{1}+2{\alpha }_{2}$,

 ${\Lambda }_{2{\alpha }_{1}+2{\alpha }_{2}}=\left\{\stackrel{{ℂ}^{2}}{•}\underset{\phantom{-}{a}_{1\phantom{-}}^{*}}{\overset{\phantom{-}{a}_{1\phantom{-}}}{⇆}}\stackrel{{ℂ}^{2}}{•}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{1},{a}_{1}^{*}\in ℂ\phantom{\rule{.5em}{0ex}}\text{with}\phantom{\rule{.5em}{0ex}}{a}_{1}{a}_{1}^{*}=0\right\}$
has three irreducible components
 ${\Lambda }_{\left(1,1,1\right)}=\left\{\left(a,{a}^{*}\right)\in {\Lambda }_{\gamma }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}\mathrm{rk}\left(a\right)\le 1\phantom{\rule{.5em}{0ex}}\text{and}\phantom{\rule{.5em}{0ex}}\mathrm{rk}\left({a}^{*}\right)\le 1\right\},$
 $=$
 ${\Lambda }_{\left(2,0,2\right)}=\left\{\left(a,{a}^{*}\right)\in {\Lambda }_{\gamma }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}^{*}=0\right\}$ and ${\Lambda }_{\left(0,2,0\right)}=\left\{\left(a,{a}^{*}\right)\in {\Lambda }_{\gamma }\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}a=0\right\}$

As computed in [GLS, §5.6],

 $\mathrm{ch}\left({\Lambda }_{\left(1,0,1\right)}\right)={f}_{1}\circ {f}_{2}\phantom{\rule{3em}{0ex}}\text{and}\phantom{\rule{3em}{0ex}}\mathrm{ch}\left({\Lambda }_{\left(0,1,0\right)}\right)={f}_{2}\circ {f}_{1},$
and
 $\mathrm{ch}\left({\Lambda }_{\left(2,0,2\right)}\right)=\frac{1}{4}{f}_{1}\circ {f}_{1}\circ {f}_{2}\circ {f}_{2},\phantom{\rule{3em}{0ex}}\mathrm{ch}\left({\Lambda }_{\left(0,2,0\right)}\right)=\frac{1}{4}{f}_{2}\circ {f}_{2}\circ {f}_{1}\circ {f}_{1},$
and
 $\mathrm{ch}\left({\Lambda }_{\left(1,1,1\right)}\right)=\frac{1}{2}{f}_{2}\circ {f}_{1}\circ {f}_{1}\circ {f}_{2},=\frac{1}{2}{f}_{1}\circ {f}_{2}\circ {f}_{2}\circ {f}_{1},$

## Notes and References

This summary of the theory of quiver representations and preprojective algebras is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem. The theory began from the ideas of [Lu1-2] and has been developed at length in [GLS] and later papers of Geiss, Leclerc and Schroer.

## References

[GLS] C. Geiss, B. Leclerc and J. Schröer, Semicanonical bases and preprojective algebras, Ann. Sc. École Norm. Sup. 38 (2005), 193-253. (2003), 567-588, arXiv:math/0402448, MR2144987.

[Lu1] G. Lusztig, Quivers, perverse sheaves and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421.

[Lu2] G. Lusztig, Semicanonical bases arising from enveloping algebras, Adv. Math. 151 (2000), 129-139.