Representation Theory

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

Last updated: 2 October 2014

Lecture 6

${𝔰𝔩}_2$ Crystals

Start with $$B\left(a\right)=\{⟶,⟵\}$$ with operators $\stackrel{\sim }{e}$ and $\stackrel{\sim }{f}$ given by $$\begin{array}{ccccc}\begin{array}{rcl}{\displaystyle \stackrel{\sim }{e}(⟵)}& =& {\displaystyle ⟶,}\\ {\displaystyle \stackrel{\sim }{e}(⟶)}& =& {\displaystyle 0,}\end{array}& & \begin{array}{rcl}{\displaystyle \stackrel{\sim }{f}(⟵)}& =& {\displaystyle 0,}\\ {\displaystyle \stackrel{\sim }{f}(⟶)}& =& {\displaystyle ⟵\text{.}}\end{array}& & \stackrel{\sim }{f}\underset{⟵}{\overset{⟶}{⇵}}\stackrel{\sim }{e}\end{array}$$ Tensor products are by concatenation $$\begin{array}{rcl}{\displaystyle B\left(▫\right)\otimes B\left(▫\right)}& =& {\displaystyle \{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\},}\\ {\displaystyle B\left(▫\right)\otimes B\left(▫\right)\otimes B\left(▫\right)}& =& {\displaystyle \{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\text{.}}\end{array}$$ If $B_1$ and $B_2$ are ${𝔰𝔩}_2\text{-crystals}$ then $\stackrel{\sim }{f}$ acts on $B_1\otimes B_2=\{p\otimes q\hspace{0.17em}|\hspace{0.17em}p\in B_1,q\in B_2\}$ by $$\stackrel{\sim }{f}(p\otimes q)=\{\begin{array}{ll}\stackrel{\sim }{f}p\otimes q,& \text{if (last occurrence of) most negative point of}\hspace{0.17em}p\otimes q\hspace{0.17em}\text{is in}\hspace{0.17em}p,\\ p\otimes \stackrel{\sim }{f}q,& \text{if (last occurrence of) most negative point of}\hspace{0.17em}p\otimes q\hspace{0.17em}\text{is in}\hspace{0.17em}q,\end{array}$$ and the action of $\stackrel{\sim }{e}$ is given by $$\stackrel{\sim }{e}b=\{\begin{array}{ll}b\prime ,& \text{if}\hspace{0.17em}b=\stackrel{\sim }{f}b\prime ,\\ 0,& \text{otherwise.}\end{array}$$

Decomposing $B^{\otimes k}$ where $B=B\left(▫\right)$

$$B\left(▫\right)\otimes B\left(▫\right)=\{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}$$ with $$\begin{array}{ccc}\begin{array}{c}0\\ \uparrow \stackrel{\sim }{e}\\ \begin{array}{c} \end{array}\\ \stackrel{\sim }{f}⇵\stackrel{\sim }{e}\\ \begin{array}{c} \end{array}\\ \stackrel{\sim }{f}⇵\stackrel{\sim }{e}\\ \begin{array}{c} \end{array}\\ \stackrel{\sim }{f}\downarrow \\ 0\end{array}& & \begin{array}{c}0\\ \uparrow \stackrel{\sim }{e}\\ \begin{array}{c} \end{array}\\ \stackrel{\sim }{f}\downarrow \\ 0\end{array}\end{array}$$ So $$B\left(▫\right)\otimes B\left(▫\right)=B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(\varnothing \right),$$ where $$B\left(\begin{array}{c}\end{array}\right)=\{\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\}\text{and}B\left(\varnothing \right)=\left\{\begin{array}{c} \end{array}\right\}\text{.}$$ Then $$\begin{array}{rcl}{\displaystyle B^{\otimes 3}}& =& {\displaystyle (B\left(▫\right)\otimes B\left(▫\right))\otimes B\left(▫\right)}\\ & =& {\displaystyle (B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(\varnothing \right))\otimes B\left(▫\right)}\\ & =& {\displaystyle (B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right))\bigsqcup (B\left(\varnothing \right)\otimes B\left(▫\right))}\end{array}$$ $$\begin{array}{ccccc}\begin{array}{c}\begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right)\cong B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(▫\right)\end{array}& & \begin{array}{c}\begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\end{array}& \text{and}& \begin{array}{c}\begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ B\left(\varnothing \right)\otimes B\left(▫\right)\simeq B\left(▫\right)\end{array}\end{array}$$ where $$B\left(\begin{array}{c}\end{array}\right)=\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}\text{.}$$ So far we have $$\begin{array}{c} B(∅) B(∅) B() B() B() B() B⊗0 B⊗1 B⊗2 B⊗3 \end{array}$$

A crystal is a subset of $B^{\otimes \ell }$ closed under the action of $\stackrel{\sim }{e}$ and $\stackrel{\sim }{f}\text{.}$

The crystal graph of a crystal $B$ is the graph with vertices $B$ and edges $p-{\stackrel{\sim }{f}}_p\text{.}$

A crystal is irreducible if its crystal graph is connected.

Let $B$ be a crystal. The character of $B$ is $$\text{ch}\left(B\right)=\sum _{p\in B}{x}^{\text{wt}\left(p\right)},$$ where $\text{wt}\left(p\right)$ is the endpoint of $p\text{.}$

A highest weight path is a path which is always $\ge 0\text{.}$

If $b$ is a highest weight path then $\stackrel{\sim }{e}p=0\text{.}$

Our examples

$$B=B\left(▫\right)=\left\{\underset{⟵}{\overset{⟶}{\downarrow \stackrel{\sim }{f}}}\right\}$$ has $\text{char}\left(B\left(▫\right)\right)=x+x^{-1}\text{.}$ Then $$B^{\otimes 2}=B\otimes B=\{\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}\}$$ has character $$\text{char}(B\otimes B)={(x+x^{-1})}^2=x^2+2+x^{-2}\text{.}$$ Next $$B\left(\begin{array}{c}\end{array}\right)=\left\{\begin{array}{c}\begin{array}{c}\end{array}\\ \downarrow \stackrel{\sim }{f}\\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\end{array}\right\}$$ has character $$\text{char}\left(B\left(\begin{array}{c}\end{array}\right)\right)=x^2+1+x^{-2}=x^2+x^0+x^{-2}\text{.}$$ $B\left(\varnothing \right)=\left\{\begin{array}{c}\end{array}\right\}$ has character $\text{char}\left(B\left(\varnothing \right)\right)=x^0=1$ and $$B^{\otimes 2}\simeq B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(\varnothing \right)\text{and}\begin{array}{c}\end{array}\text{and}\begin{array}{c}\end{array}$$ are the highest weight paths in $B\otimes B\text{.}$ $$B\left(\begin{array}{c}\end{array}\right)=\left\{\begin{array}{c}\begin{array}{c}\end{array}\\ \stackrel{\sim }{f}⇵\stackrel{\sim }{e}\\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\end{array}\right\}$$ has character $$\text{char}\left(B\left(\begin{array}{c}\end{array}\right)\right)=x^3+x+x^{-1}+x^{-3}$$ and $$B^{\otimes 3}\simeq B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(▫\right)\bigsqcup B\left(▫\right)$$ has highest weight paths $\begin{array}{c}\end{array},\begin{array}{c}\end{array},\begin{array}{c}\end{array}$ and $$\text{char}\left(B^{\otimes 3}\right)={(x+x^{-1})}^3=(x^3+x+x^{-1}+x^{-3})+(x+x^{-1})+(x+x^{-1})\text{.}$$

Classification of irreducible ${𝔰𝔩}_2\text{-crystals}$

(a)	The irreducible ${𝔰𝔩}_2\text{-crystals}$ are $$B(\underset{\underset{k}{⏟}}{\begin{array}{c}\end{array}})=\left\{\begin{array}{c}\begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\\ \stackrel{\sim }{f}\downarrow \\ ⋮\\ \stackrel{\sim }{f}\downarrow \\ \begin{array}{c}\end{array}\end{array}\right\}$$ with $$\text{char}\left(B(\underset{\underset{k}{⏟}}{\begin{array}{c}\end{array}})\right)=x^k+x^{k-2}+\cdots +x^{-(k-2)}+x^{-k}\text{.}$$
(b)	Every crystal is a disjoint union of irreducible crystals.
(c)	Each irreducible crystal $B$ has a unique highest weight path and $$B\simeq B(\underset{\underset{k}{⏟}}{\begin{array}{c}\end{array}}),$$ if $p$ ends at $k\text{.}$

${𝔰𝔩}_3\text{-crystals}$

For ${𝔰𝔩}_3\text{-crystals}$ the picture $${𝔰𝔩}_2\begin{array}{c} -4 -3 -2 -1 0 1 2 3 4 𝔥α \end{array}$$ with $\stackrel{\sim }{e}$ and $\stackrel{\sim }{f}$ is replaced by $${𝔰𝔩}_3\begin{array}{c} 𝔥α1∨ 𝔥α2∨ ω1 ω2 \end{array}$$ with operators ${\stackrel{\sim }{e}}_1,{\stackrel{\sim }{e}}_2,{\stackrel{\sim }{f}}_1,{\stackrel{\sim }{f}}_2\text{.}$

Some examples:

$$B\left(▫\right)=\left\{\begin{array}{ccc}& & \nearrow \\ & {\stackrel{\sim }{f}}_1\curvearrowleft & \\ ⟵\\ & {\stackrel{\sim }{f}}_2⤹& \\ & & \searrow \\ \end{array}\right\}$$ has character $\text{char}\left(B\left(▫\right)\right)=x_1+x_2+x_3\text{.}$ Let $B=B\left(▫\right)\text{.}$ Then $$\begin{array}{rcl}{\displaystyle B^{\otimes 2}}& =& {\displaystyle B\left(▫\right)\otimes B\left(▫\right)}\\ & =& {\displaystyle \begin{array}{c}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \\ \begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c}\end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\downarrow \\ \begin{array}{c} \end{array}\end{array}\begin{array}{c}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\swarrow \\ \begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \\ \begin{array}{c} \end{array}\end{array}}\\ & =& {\displaystyle B\left(\begin{array}{c}\end{array}\right)\bigsqcup B\left(\begin{array}{c}\end{array}\right)\text{.}}\end{array}$$ The points of the positive/dominant chamber $$P^{+}=\{k{\omega }_1^{\pm }+\ell {\omega }_2\hspace{0.17em}|\hspace{0.17em}k,\ell \in {\mathbb{Z}}_{\ge 0}\}$$ are in bijection with partitions with $\le 2$ rows. $$\begin{array}{ccc}{P}^{+}& \stackrel{1-1}{⟷}& \{\text{partitions with}\hspace{0.17em}\le 2\hspace{0.17em}\text{rows}\}\\ k{\omega }_1+\ell {\omega }_2& ⟼& \begin{array}{c} ⏟ k ⏟ ℓ \end{array}\end{array}$$ We have $$\begin{array}{rcl}{\displaystyle \text{char}\left(B^{\otimes 2}\right)}& =& {\displaystyle {(x_1+x_2+x_3)}^2}\\ & =& {\displaystyle (x_1^2+x_1{x}_2+x_3{x}_1+x_2^2+x_3{x}_2+x_3^2)+(x_1{x}_2+x_1{x}_3+x_2{x}_3),}\end{array}$$ with $$\begin{array}{rcl}{\displaystyle \text{char}\left(B\left(\begin{array}{c}\end{array}\right)\right)}& =& {\displaystyle x_1^2+x_1{x}_2+x_1{x}_3+x_2^2+x_2{x}_3+x_3^2}\\ & =& {\displaystyle \sum _{1\le i\le j\le 3}{x}_i{x}_j,}\end{array}$$ and $$\begin{array}{ccc}{\displaystyle \text{char}\left(B\left(\begin{array}{c}\end{array}\right)\right)}& =& {\displaystyle x_1{x}_2+x_1{x}_3+x_2{x}_3}\\ & =& {\displaystyle \sum _{1\le i<j\le 3}{x}_i{x}_j\text{.}}\end{array}$$ Now compute $$\begin{array}{rcl}{\displaystyle B^{\otimes 3}}& =& {\displaystyle B\left(▫\right)\otimes B\left(▫\right)\otimes B\left(▫\right)}\\ & =& {\displaystyle B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right)\bigsqcup B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right)\text{.}}\end{array}$$

Three realizations of $B\left(\begin{array}{c}\end{array}\right)$

Inside $B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right)\text{:}$ $$\begin{array}{c}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ \begin{array}{c} f∼1 f∼2 \end{array}\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\end{array}$$ Inside $B\left(\begin{array}{c}\end{array}\right)\otimes B\left(▫\right)\text{:}$ $$\begin{array}{c}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ \begin{array}{c} f∼1 f∼2 \end{array}\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\end{array}$$ With the straight line path as highest weight path: $$\begin{array}{c}\uparrow \\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \nwarrow \nearrow \\ {\stackrel{\sim }{f}}_1\searrow \swarrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ \begin{array}{c} f∼1 f∼2 \end{array}\\ \swarrow \searrow \\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \downarrow \end{array}$$

Definitions

A ${𝔰𝔩}_3\text{-crystal}$ is a collection of paths closed under the root operators ${\stackrel{\sim }{e}}_1,{\stackrel{\sim }{e}}_2,{\stackrel{\sim }{f}}_1,{\stackrel{\sim }{f}}_2\text{.}$ The root operators ${\stackrel{\sim }{e}}_1,{\stackrel{\sim }{f}}_1$ act like the ${𝔰𝔩}_2\text{-crystal}$ operators $\stackrel{\sim }{e},\stackrel{\sim }{f}$ in the ${\left({𝔥}^{{\alpha }_1}\right)}^{\perp }$ projection $$\begin{array}{c} 𝔥α1 p f∼1p \end{array}$$ and ${\stackrel{\sim }{e}}_2,{\stackrel{\sim }{f}}_2$ act like the ${𝔰𝔩}_2\text{-crystal}$ operators $\stackrel{\sim }{e},\stackrel{\sim }{f}$ in the ${\left({𝔥}^{{\alpha }_2}\right)}^{\perp }$ projection.

A highest weight path is a path $p$ contained in $$C=\bigcap \left(\text{positive halfspaces}\right)=\begin{array}{c} \end{array}$$ A path $p$ is highest weight if and only if ${\stackrel{\sim }{e}}_{1}p=0$ and ${\stackrel{\sim }{e}}_{2}p=0\text{.}$

The crystal graph has edges labeled ${\stackrel{\sim }{f}}_1$ and ${\stackrel{\sim }{f}}_2\text{.}$

The crystal graph is irreducible if the crystal graph is connected.

(a)	The irreducible ${𝔰𝔩}_3\text{-crystals}$ are indexed by the points in $$P^{+}=\{k{\omega }_1+\ell {\omega }_2\hspace{0.17em}|\hspace{0.17em}k,\ell \in {\mathbb{Z}}_{\ge 0}\}\begin{array}{c} \end{array}$$
(b)	Every ${𝔰𝔩}_3$ crystal is a disjoint union of irreducible crystals.
(c)	Each irreducible crystal $B$ has a unique highest weight path $p$ and $$B\simeq B\left(\begin{array}{c} ⏟ k ⏟ ℓ \end{array}\right)$$ if $p$ ends at $k{\omega }_1+\ell {\omega }_2\text{.}$

A column strict tableau of shape $$\lambda =\begin{array}{c} ⏟ k ⏟ ℓ \end{array}$$ is a filling of the boxes of $\lambda$ from $\{1,2,3\}$ such that

(a)	the rows weakly increase (left to right)
(b)	the columns strictly increase (top to bottom).

$$\begin{array}{c} 1 1 1 2 2 2 3 2 2 3 3 \end{array}$$ Let $p_1=\nearrow ,p_2=⟵,p_3=\searrow \text{.}$ Let $$\lambda =k{\omega }_1+\ell {\omega }_2\text{and}p=\underset{\underset{k+\ell }{⏟}}{p_1{p}_1\cdots p_1}\underset{\underset{k}{⏟}}{p_2{p}_2{p}_2\cdots p_2}\text{.}$$ There is a bijection from $$B=\left(\text{the irreducible crystal with highest weight path}\hspace{0.17em}p\right)$$ to $$B\left(\lambda \right)=\left\{\text{column strict tableaux of shape}\hspace{0.17em}\lambda \hspace{0.17em}\text{filled from}\hspace{0.17em}\{1,2,3\}\right\}$$ given by reading the tableau in arabic reading order and taking the corresponding word in $p_1,p_2,p_3\text{.}$

$$\begin{array}{c}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ \begin{array}{c} f∼1 f∼2 \end{array}\\ \begin{array}{c} \end{array}\begin{array}{c} \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} \end{array}\end{array}\begin{array}{c}\begin{array}{c} 1 1 2 \end{array}\\ {\stackrel{\sim }{f}}_1\swarrow \searrow {\stackrel{\sim }{f}}_2\\ \begin{array}{c} 1 2 2 \end{array}\begin{array}{c} 1 1 3 \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} 1 3 2 \end{array}\begin{array}{c} 1 2 3 \end{array}\\ \begin{array}{c} f∼1 f∼2 \end{array}\\ \begin{array}{c} 2 2 3 \end{array}\begin{array}{c} 1 3 3 \end{array}\\ {\stackrel{\sim }{f}}_2\searrow \swarrow {\stackrel{\sim }{f}}_1\\ \begin{array}{c} 2 3 3 \end{array}\end{array}$$

Notes and References

These are a typed copy of Lecture 6 from a series of handwritten lecture notes for the class Representation Theory given on September 2, 2008.

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