Crystals from KLR and preprojective algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 11 October 2012

Dynkin diagrams

$G$ a complex reductive algebraic group corresponds to

$$\begin{array}{ccc}({𝔥}_{\mathbb{Z}},W_0)& \text{with}& {𝔥}_z\text{a free}\mathbb{Z}\text{-module}\\ & & W_0\text{a finite subgroup of}GL\left({𝔥}_{\mathbb{Z}}\right)\text{generated by reflections}\end{array}$$

Let $C_0$ be a fundamental region for the action of $W_0$ on ${𝔥}_{ℝ}=ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}\text{.}$ Let

$${𝔥}^{{\alpha }_1},\dots ,{𝔥}^{{\alpha }_n}\text{be the walls of}{C}_0$$

Make a graph with vertices $1,\dots ,n$ and

$$\begin{array}{ccc} i j i j i j i j & \text{if}{𝔥}^{{\alpha }_i}\measuredangle {𝔥}^{{\alpha }_j}\text{is}& \begin{array}{c}\frac{\pi }{2}\\ \frac{\pi }{3}\\ \frac{\pi }{4}\\ \frac{\pi }{6}\end{array}\end{array}$$

Example Type $S{L}_3$

$$C0 𝔥α1∨ 𝔥α2∨ π3 1 2$$

The quiver Hecke algebra:

Choose an orientation.

$$\begin{array}{ccc}Q=& & I=\left\{\text{colours}\right\}=\left\{\text{vertices of}Q\right\}\text{.}\end{array}$$
$${𝔥}_{\mathbb{Z}}^{\ge 0}={\mathbb{Z}}_{\ge 0}\text{-span}\{{\alpha }_i\mid i\in I\}\text{.}$$

Let $\gamma \in {𝔥}_{\mathbb{Z}}^{\ge 0}\text{.}$ The quiver Hecke algebra $R_{\gamma }$ has generators $y_1,\dots ,y_d,e_u$ for $u\in I^{\gamma },$ ${\psi }_1,\dots ,{\psi }_{d-1}$ and relations $y_i{y}_j=y_j{y}_i,$ $e_u{e}_v={\delta }_{uv}{e}_u,$ ${\displaystyle 1=\sum _{u\in I^{\gamma }}{e}_u}$ and more... where

$$I^{\gamma }=\{u=(u_1,\dots ,u_d)\text{sequences of colors with}{\alpha }_{u_1}+\dots +{\alpha }_{u_d}=\gamma \}$$

with $\mathbb{Z}$-grading

$$\text{deg}\left(e_u\right),\text{deg}\left(y_i\right)=2,\text{deg}\left({\psi }_r{e}_u\right)=\{\begin{array}{cc}-2,& \text{if}{u}_r=u_{r+1}\\ -1,& \text{if} ur ur+1 \\ 0,& \text{if} ur ur+1 \end{array}$$

Let ${\displaystyle R=\underset{\gamma \in {\mathbb{Z}}_{\ge 0}}{⨁}{R}_{\gamma }}$

$$M\text{a}\mathbb{Z}\text{graded}R\text{-module so that}M=\underset{j\in \mathbb{Z}}{⨁}M\left[j\right]$$

Define

$$\text{char}\left(M\right)=\sum _{j\in \mathbb{Z}}\sum _{u\in I^{\gamma }}\text{dim}\left(e_{u}M\left[J\right]\right)q^j{f}_{u_1}\dots f_{u_d}\text{.}$$

(generating function in noncommutative $f_i,i\in I\text{).}$

(Khoranov-Lavda/Rouquier)

$$\begin{array}{cccccc}\text{char:}& \begin{array}{c}\text{Grothendieck}\\ \text{group}\end{array}& \left\{\begin{array}{c}\text{fin. dim'l}\mathbb{Z}\text{-graded}\\ R\text{modules}\end{array}\right\}& ⟶& U_q{n}^{-}& \text{(quantum group)}\\ & \begin{array}{c}\text{simple}\\ ℝ\text{-modules}\end{array}& L_b& ⟼& \text{char}\left(L_b\right)& \text{(canonical basis)}\end{array}$$

Define ${\stackrel{\sim }{f}}_{i}b$ by

$$L_{{\stackrel{\sim }{f}}_{i}b}=\text{head}\left({\text{Ind}}_{R_{{\alpha }_i}\otimes R_{\gamma }}^{R_{{\alpha }_i+\gamma }}\left(L_b\right)\right)$$

As directed graphs with labels $\stackrel{{\stackrel{\sim }{f}}_i}{⟶}$

$$\begin{array}{ccc}\left\{\begin{array}{c}\text{simple}\mathbb{Z}\text{-graded}\\ R\text{-modules}{L}_b\end{array}\right\}& \stackrel{\sim }{⟶}& \left\{\begin{array}{c}\text{MV polytopes}\\ b\end{array}\right\}\text{.}\end{array}$$

Preprojective algebras

$$\begin{array}{cc}\begin{array}{cc}Q=& \end{array}& \begin{array}{cc}\bar{Q}=& \end{array}\end{array}$$

Idea: Replace beads by vector spaces.

$$\begin{array}{cc} ℂn1 ℂn2 ℂn3 & \begin{array}{c}{ℂ}^{n}j\text{corresponds to}\\ n_j\text{beads on runner}j\text{.}\end{array}\end{array}$$

The data of

1. a vector space for each vertex
2. a linear transformation for each edge

is a representation of $Q$ (or $\bar{Q}\text{).}$

In the case of $\bar{Q}$ require

$$\sum _{ i a ∈Q }{a}^{*}a=\sum _{ i a ∈Q }a{a}^{*},\text{for each}i\in I\text{.}$$

Example: Type ${GL}_n$

$$\begin{array}{cc}\begin{array}{cc}Q=& a1 a2 a3 an-2 \end{array}& \begin{array}{cc}\bar{Q}=& a1 a2 an-2 a1* a2* an-2* \end{array}\end{array}$$

and we require

$$a_1{a}_1^{*}=0,a_{i-1}^{*}{a}_{i-1}=a_i{a}_i^{*}\text{for}i=2,\dots ,n-2,a_{n-2}^{*}{a}_{n-2}=0$$

Let

$$\Lambda =\left\{\text{isomorphism classes of representations of}\bar{Q}\text{satisfying}\left(PP\right)\right\}$$

As discrete $d$ graphs with labels $\stackrel{\sim }{f_i},$

$$\begin{array}{ccc}\left\{\begin{array}{c}\text{irreducible components}\\ {\Lambda }_b\text{of}\Lambda \end{array}\right\}& \stackrel{\sim }{⟶}& \left\{\begin{array}{c}\text{MV polytopes}\\ b\end{array}\right\}\end{array}$$

Notes and References

These are a typed version of lecture notes for the fifth in a series of lectures given at the Brazil Algebra Conference, Salvador, 19 July 2012.

page history