Crystals from KLR and preprojective algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 11 October 2012

Dynkin diagrams

G a complex reductive algebraic group corresponds to

(𝔥,W0) with 𝔥za free-module W0a finite subgroup of GL(𝔥) generated by reflections

Let C0 be a fundamental region for the action of W0 on 𝔥=𝔥 . Let

𝔥α1,, 𝔥αn be the walls ofC0

Make a graph with vertices 1,,n and

i j i j i j i j if𝔥αi 𝔥αjis π2 π3 π4 π6

Example Type SL3

C0 𝔥α1 𝔥α2 π3 1 2

The quiver Hecke algebra:

Choose an orientation.

Q= I={colours}= {vertices ofQ}.
𝔥0=0 -span{αiiI}.

Let γ𝔥0. The quiver Hecke algebra Rγ has generators y1,,yd,eu for uIγ, ψ1,,ψd-1 and relations yiyj=yjyi, euev=δuv eu, 1=uIγ eu and more... where

Iγ= { u=(u1,,ud) sequences of colors withαu1++ αud=γ }

with -grading

deg(eu), deg(yi)=2, deg(ψreu)= { -2, ifur= ur+1 -1, if ur ur+1 0, if ur ur+1

Let R=γ0Rγ

MagradedR -module so thatM=j M[j]


char(M)= j uIγ dim(euM[J]) qjfu1 fud.

(generating function in noncommutative fi,iI).


char: Grothendieck group { fin. dim'l-graded Rmodules } Uqn- (quantum group) simple -modules Lb char(Lb) (canonical basis)

Define fib by

Lfib= head ( Ind RαiRγ Rαi+γ (Lb) )

As directed graphs with labels fi

{ simple-graded R-modulesLb } { MV polytopes b } .

Preprojective algebras

Q= Q=

Idea: Replace beads by vector spaces.

n1 n2 n3 njcorresponds to njbeads on runnerj.

The data of

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

is a representation of Q (or Q).

In the case of Q require

i a Q a*a= i a Q aa*,for eachiI.

Example: Type GLn

Q= a1 a2 a3 an-2 Q= a1 a2 an-2 a1* a2* an-2*

and we require

a1a1*=0, ai-1*ai-1= aiai*for i=2,,n-2, an-2*an-2=0


Λ= {isomorphism classes of representations ofQsatisfying(PP)}

As discrete d graphs with labels fi,

{ irreducible components ΛbofΛ } { MV polytopes b }

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.

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