Crystals from paths and MV polytopes

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

Last update: 8 October 2012

Crystals: The Path Model

Initial data: $({𝔥}_{\mathbb{Z}},W_0)$ where

1. ${𝔥}_{\mathbb{Z}}$ has $\mathbb{Z}$-basis
2. $W_0$ a finite subgroup of $GL\left({𝔥}_{\mathbb{Z}}\right)$ generated by reflections

Let ${𝔥}_{ℝ}=ℝ$-span of the basis and

$$B_{\text{univ}}=\{p:[0,1]⟶{𝔥}_{ℝ}\mid p\text{is continuous, piecewise linear,}p\left(1\right)\in {𝔥}_{\mathbb{Z}},p\left(0\right)=0\}$$

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 ${𝔥}_{ℝ}$

$${𝔥}^{{\alpha }_1},\dots ,{𝔥}^{{\alpha }_n}\text{are the walls of}{C}_0$$ $$𝔥αi p f∼ip pos.side (towardsC0) 𝔥αi p f˜ip=0$$

and

$${\stackrel{\sim }{e}}_{i}q=\{\begin{array}{cc}p,& \text{if}q={\stackrel{\sim }{f}}_{i}p,\\ 0,& \text{otherwise,}\end{array}$$

Example Type ${SL}_3$

$$C0 𝔥α1∨ 𝔥α2∨$$

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

$$p\left([0,1]\right)=\text{im}p\subseteq C_0-p,$$

where

$${𝔥}_{\mathbb{Z}}^{+}={𝔥}_{\mathbb{Z}}\cap {\bar{C}}_0,{𝔥}_{\mathbb{Z}}^{++}={𝔥}_{\mathbb{Z}}\cap C_0\text{and}\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{+}& \stackrel{\sim }{⟶}& {𝔥}_{\mathbb{Z}}^{++}\\ \lambda & ⟼& \rho +\lambda \end{array}\text{as}\mathbb{Z}\text{-modules.}$$

Then

$$p\text{is a highest weight path}\iff {\stackrel{\sim }{e}}_{i}p=0\text{for}i=1,2,\dots ,n\text{.}$$

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

$$B\left(\lambda \right)=\{{\stackrel{\sim }{f}}_{i_1}\dots {\stackrel{\sim }{f}}_{i_n}{p}_{\lambda }\mid \lambda \in {\mathbb{Z}}_{\ge 0},1\le i_1,\dots ,i_d\le n\}$$

The Weyl character

$$s_{\lambda }=\sum _{p\in B\left(\lambda \right)}{X}^{\varphi \left(1\right)}\text{.}$$ $${\stackrel{\sim }{e}}_{i}p=\{\begin{array}{cc}q,& \text{if}{\stackrel{\sim }{f}}_{i}q=p,\\ 0,& \text{otherwise.}\end{array}$$

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

$$B\left(\lambda \right)=\{p_T\mid T\text{is the arabic reading of a column strict tableau of shape}\lambda \text{.}\}$$ $$f∼1 f∼2 f∼1 f∼1 f∼2 f∼2 f∼2 f∼1 f∼2 f∼2 f∼2$$

and

$${\stackrel{\sim }{e}}_{i}q=\{\begin{array}{cc}p,& \text{if}q={\stackrel{\sim }{f}}_{i}q,\\ 0,& \text{otherwise.}\end{array}$$

If

$${𝔥}_{\mathbb{Z}}^{+}={𝔥}_{\mathbb{Z}}\cap {\bar{C}}_0\text{and}{𝔥}_{\mathbb{Z}}^{++}={𝔥}_{\mathbb{Z}}\cap C_0$$

then

$$\begin{array}{ccc}{𝔥}_{\mathbb{Z}}^{+}& \stackrel{\sim }{⟶}& {𝔥}_{\mathbb{Z}}^{++}\\ \lambda & ⟼& \rho +\lambda \end{array}\text{as}{𝔥}_{\mathbb{Z}}^{+}\text{-modules.}$$

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

$$\text{im}p\subseteq C_0-\rho \text{.}$$

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

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

$$B\left(\lambda \right)=\{{\stackrel{\sim }{f}}_{i_1}\dots {\stackrel{\sim }{f}}_{i_d}{p}_{\lambda }\mid d\in {\mathbb{Z}}_{\ge 0},1\le i_1,\dots ,i_d\le n\}\text{.}$$

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

$$s_{\lambda }=\sum _{p\in B\left(\lambda \right)}{X}^{p^{\left(1\right)}}\text{.}$$

MV polytopes

$$\begin{array}{ccc}b=& \[ b_1 \] \[ b_{s_1} \] \[ b_{s_2} \] \[ b_{s_1{s}_2} \] \[ b_{s_2{s}_1} \] \[ b_{s_1{s}_2{s}_1}=b_{s_2{s}_1{s}_2} \]& \begin{array}{c}{\text{per}}_{121}\left(b\right)=({\ell }_1,{\ell }_2,{\ell }_3)\\ {\text{per}}_{212}\left(b\right)=({\ell }_1^{\prime },{\ell }_2^{\prime },{\ell }_3^{\prime })\end{array}\end{array}$$ $$\begin{array}{c}b=\text{Conv}\left(\{b_w\mid w\in W_0\}\right)\text{with}{b}_w\text{the vertices of}b\text{.}\\ b_1=0\text{.}\end{array}$$

Let $w_0$ be the longest element of $W_0$ (Coxeter generators) and $\stackrel{\rightharpoonup }{i}=(i_1,\dots ,i_N)$ with $s_{i_1}\dots s_{i_N}=w_0$ reduced.

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

$${\text{per}}_i\left(b\right)=({\ell }_1,\dots ,{\ell }_N)\text{with}{\ell }_k\beta =b_{s_{i_1}\dots s_{i_k}}-b_{s_{i_1}\dots s_{i_{k-1}}}$$

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 ${\text{per}}_i(0b)$ by a sequence of transformations

$$R_{ij}^{ji}({\ell }_k,{\ell }_{k+1})=({\ell }_{k+1},{\ell }_k),\text{if}{s}_i{s}_j=s_j{s}_i$$ $$R_{iji}^{jij}({\ell }_k,{\ell }_{k+1},{\ell }_{k+2})=({\ell }_k+{\ell }_{k+1}-\text{min}({\ell }_k,{\ell }_{k+2}),\text{min}({\ell }_k,{\ell }_{k+2}),{\ell }_{k+1}+{\ell }_{k+2}-\text{min}({\ell }_k,{\ell }_{k+2})),\text{if}{s}_i{s}_j{s}_i=s_j{s}_i{s}_j\text{.}$$

The crystal operator ${\stackrel{\sim }{f}}_{i_1}b$ is given by

$${\text{per}}_{\stackrel{\rightharpoonup }{i}}\left({\stackrel{\sim }{f}}_{i_1}b\right)=({\ell }_1+1,{\ell }_2,{\ell }_3,\dots ,{\ell }_N)\text{if}{\text{per}}_{\stackrel{\rightharpoonup }{i}}\left(b\right)=({\ell }_1,{\ell }_2,\dots ,{\ell }_N)$$

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

$$B\left(\infty \right)=\{{\stackrel{\sim }{f}}_{i_1}\dots {\stackrel{\sim }{f}}_{i_d}{b}_{+}\mid d\in {\mathbb{Z}}_{>0},1\le i_1,\dots ,i_d\le n\}$$

For $\lambda \in {𝔥}_{\mathbb{Z}}^{+}$ let

$$B\left(\lambda \right)=\{b\in B\left(\infty \right)\mid \lambda +b\subseteq \text{Conv}\left(W_0\lambda \right)\}\text{.}$$ $$λ s1s2s1λ s2λ s1λ s2s1λ s1s2λ$$

(Anderson-Kannitzer)

$$s_{\lambda }=\sum _{b\in B\left(\lambda \right)}{X}^{\lambda +b_{w_0}}\text{.}$$

Let $\lambda \in {𝔥}_{\mathbb{Z}}$ with $\lambda \in {\bar{C}}_0$ and

$$B\left(\lambda \right)=\{\text{MV polytopes}b\text{with}b\subseteq \text{Conv}\left(W_0\lambda \right)\}\text{.}$$

The Weyl character is

$$s_{\lambda }=\sum _{b\in B\left(\lambda \right)}{X}^{\lambda -b_{w_0}}$$ $$\[ \widetilde{f_1} \] \[ \widetilde{f_1} \] \[ \widetilde{f_1} \] \[ \widetilde{f_2} \] \[ \widetilde{f_1} \] \[ \widetilde{f_2} \] \[ \widetilde{f_2} \]$$

MV-cycles

$$\begin{array}{ccc}ℂ\left(\left(t\right)\right)& =& \{a_{\ell }{t}^{-\ell }+a_{\ell +1}{t}^{-\ell +1}+\dots \mid a_i\in ℂ,\ell \in \mathbb{Z}\}\\ ⊆ \\ ℂ\left[\left[t\right]\right]& =& \{a_0+a_{1}t+a_2{t}^2+\dots \mid a_i\in ℂ\}\stackrel{t=0}{⟶}ℂ\text{.}\end{array}$$

$G\left(ℂ\right)$ is a complex reductive algebraic group, say $G={SL}_3\text{.}$

$$\begin{array}{ccc}G& =& {SL}_3\left(ℂ\left(\left(t\right)\right)\right)\\ & & ⊆ \\ K& =& {SL}_3\left(ℂ\left[\left[t\right]\right]\right)& \stackrel{t=0}{⟶}& {SL}_3\left(ℂ\right)\end{array}$$

$\raisebox{1ex}{$G$}\!\left/ \!\raisebox{-1ex}{$K$}\right.$ is the loop Grassmannian.

$$G=\underset{\lambda \in {𝔥}_{\mathbb{Z}}^{+}}{⨆}K{t}_{\lambda }K\text{and}G=\underset{\mu \in {𝔥}_{\mathbb{Z}}}{⨆}{U}^{-}{t}_{\mu }K$$

where

$$t_{\lambda }=h_{\lambda }\left(t\right)\text{and}{U}^{-}=\{\left(\begin{array}{ccc}1& & 0\\ & \ddots & \\ ✶& & 1\end{array}\right)\mid ✶\in ℂ\left(\left(t\right)\right)\}\text{.}$$

The Mirkovic-Vilonen intersections are

$$K{t}_{\lambda }K\cap U^{-}{t}_{\mu }K\text{.}$$

The MV cycles are the irreducible components

$$z_b\text{in}{I}_W\left(\overline{K{t}_{\lambda }K\cap U^{-}{t}_{\mu }K}\right)\text{.}$$

Then the Weyl character is

$$s_{\lambda }=\sum _{\mu \in {𝔥}_{\mathbb{Z}}}\text{Card}\left(I_W\left(\overline{K{t}_{\lambda }K\cap U^{-}{t}_{\mu }K}\right)\right)X^{\mu }\text{.}$$

Define $f_{i}b$ by

$$y_i\left(c{t}^k\right)z_b\text{has}{z}_{f_{i}b}\text{as a dense open subset.}$$

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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