## Crystals from paths and MV polytopes

Last update: 8 October 2012

## Crystals: The Path Model

Initial data: $\left({𝔥}_{ℤ},{W}_{0}\right)$ where

1. ${𝔥}_{ℤ}$ has $ℤ$-basis
2. ${W}_{0}$ a finite subgroup of $GL\left({𝔥}_{ℤ}\right)$ generated by reflections

Let ${𝔥}_{ℝ}=ℝ$-span of the basis and

$Buniv= { p:[0,1]⟶𝔥ℝ ∣pis continuous, piecewise linear, p(1)∈𝔥ℤ, p(0)=0 }$

A crystal is a subset $B$ or ${B}_{\text{univ}}$ closed under the root operators ${\stackrel{\sim }{f}}_{1},\dots ,{\stackrel{\sim }{f}}_{n},{\stackrel{\sim }{e}}_{1},\dots ,{\stackrel{\sim }{e}}_{n}\text{.}$

${C}_{0}$ is a fundamental chamber for ${W}_{0}$ acting on ${𝔥}_{ℝ}$

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

and

$e∼iq= { p, ifq=f∼ip, 0, otherwise,$

Example Type ${SL}_{3}$

$C0 𝔥α1∨ 𝔥α2∨$

A highest weight path is $p\in {B}_{\text{univ}}$ with

$p([0,1])= imp⊆C0-p,$

where

$𝔥ℤ+=𝔥ℤ∩ C‾0, 𝔥ℤ++=𝔥ℤ ∩C0and 𝔥ℤ+ ⟶∼ 𝔥ℤ++ λ⟼ρ+λ asℤ-modules.$

Then

$pis a highest weight path⇔e∼ip=0 fori=1,2,…,n.$

Let $\lambda \in {𝔥}_{ℤ}^{+}$ and ${p}_{\lambda }$ a highest wt. path with ${p}_{\lambda }\left(1\right)=\lambda \text{.}$

$B(λ)= { f∼i1… f∼inpλ ∣λ∈ℤ≥0 ,1≤i1,…,id ≤n }$

The Weyl character

$sλ=∑p∈B(λ) Xϕ(1).$ $e∼ip= { q, iff∼iq=p, 0, otherwise.$

A crystal $B$ is a subset of ${B}_{\text{univ}}$ closed under the action of ${\stackrel{\sim }{e}}_{1},\dots ,{\stackrel{\sim }{e}}_{n}$ and ${\stackrel{\sim }{f}}_{1},\dots ,{\stackrel{\sim }{f}}_{n}\text{.}$

$f∼1 f∼2 f∼2 f∼2 f∼1 f∼2 f∼1 f∼2$

Example Type ${SL}_{3},$ $\lambda ={w}_{1}+{w}_{2}$

$1 1 2 2 1 2 1 1 3 3 1 2 2 1 3 2 2 3 3 1 3 3 2 3 f∼1 f∼2 f∼2 f∼1 f∼2 f∼1 f∼2 f∼1$

## Type ${SL}_{n}$ or ${GL}_{n}$

Let ${p}_{1},{p}_{2},{p}_{3}$ be

$p1 p2 p3$

If ${p}_{\lambda }={p}_{1}\dots {p}_{1}{p}_{2}\dots {p}_{2}\dots$ then

$B(λ)= { pT∣T is the arabic reading of a column strict tableau of shapeλ. }$ $f∼1 f∼2 f∼1 f∼1 f∼2 f∼2 f∼2 f∼1 f∼2 f∼2 f∼2$

and

$e∼iq= { p, ifq=f∼iq, 0, otherwise.$

If

$𝔥ℤ+=𝔥ℤ∩ C‾0and 𝔥ℤ++=𝔥ℤ ∩C0$

then

$𝔥ℤ+ ⟶∼ 𝔥ℤ++ λ ⟼ ρ+λ as𝔥ℤ+-modules.$

A highest weight path is $p\in {B}_{\text{univ}}$ with

$imp⊆C0-ρ.$

$p$ is a highest wt path $⇔{\stackrel{\sim }{e}}_{i}p=0$ for $i=1,\dots ,n\text{.}$

Let $\lambda \in {𝔥}_{ℤ}^{+}$ and ${p}_{\lambda }$ a hw path with ${p}_{\lambda }\left(1\right)=\lambda$

$B(λ)= { f∼i1… f∼idpλ ∣d∈ℤ≥0 ,1≤i1,…, id≤n } .$

(Littelmann) $B\left(\lambda \right)$ is a crystal and

$sλ=∑p∈B(λ) Xp(1).$

## MV polytopes

$b= b1 bs1 bs2 bs1s2 bs2s1 bs1s2s1=bs2s1s2 per121(b) =(ℓ1,ℓ2,ℓ3) per212(b)= ( ℓ1′, ℓ2′, ℓ3′ )$ $b=Conv ({bw∣w∈W0}) withbwthe vertices ofb. b1=0.$

Let ${w}_{0}$ be the longest element of ${W}_{0}$ (Coxeter generators) and $\stackrel{⇀}{i}=\left({i}_{1},\dots ,{i}_{N}\right)$ with ${s}_{{i}_{1}}\dots {s}_{{i}_{N}}={w}_{0}$ reduced.

The $\stackrel{⇀}{i}$-perimeter of $b$ is

$peri(b)= (ℓ1,…,ℓN) withℓkβ= bsi1…sik - bsi1…sik-1$

so that $\ell$ is the distance from ${b}_{{s}_{{i}_{1}}\dots {s}_{{i}_{k-1}}}$ to ${b}_{{s}_{{i}_{1}}\dots {s}_{{i}_{k}}}$

?????????????????

Any other ${\text{per}}_{j}\left(b\right)$ is obtained from ${\text{per}}_{i}\left(0b\right)$ by a sequence of transformations

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

The crystal operator ${\stackrel{\sim }{f}}_{{i}_{1}}b$ is given by

$peri⇀ ( f∼i1b ) = ( ℓ1+1,ℓ2, ℓ3,…,ℓN ) ifperi⇀ (b)= ( ℓ1,ℓ2, …,ℓN )$

Let ${b}_{+}$ be given by ${\text{per}}_{\stackrel{⇀}{i}}\left({b}_{+}\right)=\left(0,0,\dots ,0\right)\text{.}$

$B(∞)= { f∼i1… f∼idb+ ∣d∈ℤ>0 ,1≤i1,…, id≤n }$

For $\lambda \in {𝔥}_{ℤ}^{+}$ let

$B(λ)= { b∈B(∞)∣ λ+b⊆Conv (W0λ) } .$ $λ s1s2s1λ s2λ s1λ s2s1λ s1s2λ$

(Anderson-Kannitzer)

$sλ=∑b∈B(λ) Xλ+bw0.$

Let $\lambda \in {𝔥}_{ℤ}$ with $\lambda \in {\stackrel{‾}{C}}_{0}$ and

$B(λ)= { MV polytopesbwith b⊆Conv(W0λ) } .$

The Weyl character is

$sλ=∑b∈B(λ) Xλ-bw0$ $f1˜ f1˜ f1˜ f2˜ f1˜ f2˜ f2˜$

## MV-cycles

$ℂ((t)) = { aℓt-ℓ+ aℓ+1 t-ℓ+1+… ∣ai ∈ℂ,ℓ∈ℤ } ⊆ ℂ[[t]] = { a0+a1t+ a2t2+… ∣ai∈ℂ } ⟶t=0ℂ.$

$G\left(ℂ\right)$ is a complex reductive algebraic group, say $G={SL}_{3}\text{.}$

$G= SL3 (ℂ((t))) ⊆ K= SL3 (ℂ[[t]]) ⟶t=0 SL3(ℂ)$

$G}{K}$ is the loop Grassmannian.

$G=⨆λ∈𝔥ℤ+ KtλKand G=⨆μ∈𝔥ℤ U-tμK$

where

$tλ=hλ(t) andU-= { ( 10 ⋱ ✶1 ) ∣✶∈ ℂ((t)) } .$

The Mirkovic-Vilonen intersections are

$KtλK∩U- tμK.$

The MV cycles are the irreducible components

$zbin IW ( KtλK∩ U-tμK ‾ ) .$

Then the Weyl character is

$sλ=∑μ∈𝔥ℤ Card ( IW ( KtλK∩ U-tμK ‾ ) ) Xμ.$

Define ${f}_{i}b$ 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.