Crystals from paths and MV polytopes

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

Last update: 8 October 2012

Crystals: The Path Model

Initial data: (𝔥,W0) where

  1. 𝔥 has -basis
  2. W0 a finite subgroup of GL(𝔥) generated by reflections

Let 𝔥=-span of the basis and

Buniv= { p:[0,1]𝔥 pis continuous, piecewise linear, p(1)𝔥, p(0)=0 }

A crystal is a subset B or Buniv closed under the root operators f1,,fn, e1,,en.

C0 is a fundamental chamber for W0 acting on 𝔥

𝔥α1,, 𝔥αn are the walls ofC0 𝔥αi p fip pos.side (towardsC0) 𝔥αi p f˜ip=0


eiq= { p, ifq=fip, 0, otherwise,

Example Type SL3

C0 𝔥α1 𝔥α2

A highest weight path is pBuniv with

p([0,1])= impC0-p,


𝔥+=𝔥 C0, 𝔥++=𝔥 C0and 𝔥+ 𝔥++ λρ+λ as-modules.


pis a highest weight patheip=0 fori=1,2,,n.

Let λ𝔥+ and pλ a highest wt. path with pλ(1)=λ.

B(λ)= { fi1 finpλ λ0 ,1i1,,id n }

The Weyl character

sλ=pB(λ) Xϕ(1). eip= { q, iffiq=p, 0, otherwise.

A crystal B is a subset of Buniv closed under the action of e1,,en and f1,,fn.

f1 f2 f2 f2 f1 f2 f1 f2

Example Type SL3, λ=w1+w2

1 1 2 2 1 2 1 1 3 3 1 2 2 1 3 2 2 3 3 1 3 3 2 3 f1 f2 f2 f1 f2 f1 f2 f1

Type SLn or GLn

Let p1,p2,p3 be

p1 p2 p3

If pλ=p1p1p2p2 then

B(λ)= { pTT is the arabic reading of a column strict tableau of shapeλ. } f1 f2 f1 f1 f2 f2 f2 f1 f2 f2 f2


eiq= { p, ifq=fiq, 0, otherwise.


𝔥+=𝔥 C0and 𝔥++=𝔥 C0


𝔥+ 𝔥++ λ ρ+λ as𝔥+-modules.

A highest weight path is pBuniv with


p is a highest wt path eip=0 for i=1,,n.

Let λ𝔥+ and pλ a hw path with pλ(1)=λ

B(λ)= { fi1 fidpλ d0 ,1i1,, idn } .

(Littelmann) B(λ) is a crystal and

sλ=pB(λ) Xp(1).

MV polytopes

b= b1 bs1 bs2 bs1s2 bs2s1 bs1s2s1=bs2s1s2 per121(b) =(1,2,3) per212(b)= ( 1, 2, 3 ) b=Conv ({bwwW0}) withbwthe vertices ofb. b1=0.

Let w0 be the longest element of W0 (Coxeter generators) and i=(i1,,iN) with si1siN=w0 reduced.

The i-perimeter of b is

peri(b)= (1,,N) withkβ= bsi1sik - bsi1sik-1

so that is the distance from bsi1sik-1 to bsi1sik


Any other perj(b) is obtained from peri(0b) by a sequence of transformations

Rijji (k,k+1)= (k+1,k), ifsisj= sjsi Rijijij ( k,k+1 ,k+2 ) = ( k+k+1- min(k,k+2) ,min(k,k+2) ,k+1+k+2- min(k,k+2) ) , ifsisjsi=sjsisj.

The crystal operator fi1b is given by

peri ( fi1b ) = ( 1+1,2, 3,,N ) ifperi (b)= ( 1,2, ,N )

Let b+ be given by peri(b+) =(0,0,,0).

B()= { fi1 fidb+ d>0 ,1i1,, idn }

For λ𝔥+ let

B(λ)= { bB() λ+bConv (W0λ) } . λ s1s2s1λ s2λ s1λ s2s1λ s1s2λ


sλ=bB(λ) Xλ+bw0.

Let λ𝔥 with λC0 and

B(λ)= { MV polytopesbwith bConv(W0λ) } .

The Weyl character is

sλ=bB(λ) Xλ-bw0 f1˜ f1˜ f1˜ f2˜ f1˜ f2˜ f2˜


((t)) = { at-+ a+1 t-+1+ ai , } [[t]] = { a0+a1t+ a2t2+ ai } t=0.

G() is a complex reductive algebraic group, say G=SL3.

G= SL3 (((t))) K= SL3 ([[t]]) t=0 SL3()

GK is the loop Grassmannian.

G=λ𝔥+ KtλKand G=μ𝔥 U-tμK


tλ=hλ(t) andU-= { ( 10 1 ) ((t)) } .

The Mirkovic-Vilonen intersections are

KtλKU- tμK.

The MV cycles are the irreducible components

zbin IW ( KtλK U-tμK ) .

Then the Weyl character is

sλ=μ𝔥 Card ( IW ( KtλK U-tμK ) ) Xμ.

Define fib by

yi(ctk) zbhas zfib as a dense open subset.

Notes and References

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

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