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 8

Set up

${𝔥}_{\mathbb{Z}}^{*}$ is a $\mathbb{Z}\text{-vector}$ space,

$W_0\subseteq GL\left({𝔥}_{\mathbb{Z}}^{*}\right)$ a finite group generated by reflections,

$s_{\alpha },$ $\alpha \in R^{+},$ are the reflections in $W_0,$

$s_{\alpha }\mu =\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha ,$

$C$ is a fundamental chamber for $W_0$ acting on ${𝔥}_{ℝ}^{*},$

${𝔥}^{{\alpha }_1^{\vee }},\dots ,{𝔥}^{{\alpha }_n^{\vee }}$ are the walls of $C$ and their reflections,

$s_1,\dots ,s_n$ are the simple reflections, $$W_0\stackrel{1-1}{⟷}\left\{\text{chambers on}\hspace{0.17em}{𝔥}_{ℝ}^{*}\right\}$$

(Type ${SL}_3\text{)}$ ${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\omega }_1,{\omega }_2\}$ $$\begin{array}{c} 1 𝔥α2∨ 𝔥α1∨ s2 s1 s2s1 s1s2 s1s2s1 \end{array}$$

$P^{+}={𝔥}_{\mathbb{Z}}^{*}\cap \bar{C}$ and $P^{++}={𝔥}_{\mathbb{Z}}^{*}\cap C$ are isomorphic semigroups $$\begin{array}{ccc}{P}^{+}& \stackrel{\sim }{⟶}& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}$$

(Type ${GL}_n\text{)}$ ${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\epsilon }_1,\dots ,{\epsilon }_n\}$ and $W_0=S_n,$ which has reflections $$s_{ij}=\begin{array}{c} 1 ⋯ i ⋯ j ⋯ n \end{array}$$ with $${𝔥}^{ij}=\{\mu =({\mu }_1,\dots ,{\mu }_n)\hspace{0.17em}|\hspace{0.17em}{\mu }_i={\mu }_j\}\text{.}$$ Then $$C=\{\mu =({\mu }_1,\dots ,{\mu }_n)\in {ℝ}^n\hspace{0.17em}|\hspace{0.17em}{\mu }_1>{\mu }_2>\cdots >{\mu }_n\}$$ has walls $${𝔥}^{{\alpha }_i^{\vee }}=\{\mu =({\mu }_1,\dots ,{\mu }_n)\hspace{0.17em}|\hspace{0.17em}{\mu }_i={\mu }_{i+1}\}\text{and}{s}_i=\begin{array}{c} 1 2 i i+1 n ⋯ ⋯ \end{array}\text{.}$$ Then $$\begin{array}{rcl}{\displaystyle P^{+}}& =& {\displaystyle \{\lambda =({\lambda }_1,\dots ,{\lambda }_n)\in {\mathbb{Z}}^n\hspace{0.17em}|\hspace{0.17em}{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\},}\\ {\displaystyle P^{++}}& =& {\displaystyle \{\mu =({\mu }_1,\dots ,{\mu }_n)\in {\mathbb{Z}}^n\hspace{0.17em}|\hspace{0.17em}{\mu }_1>\cdots >{\mu }_n\}}\end{array}$$ and $$\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}$$ where $\rho =(n-1,n-2,\dots ,1,0)\text{.}$

$$\begin{array}{rcl}{\displaystyle ℂ\left[X\right]}& =& {\displaystyle \text{span}\left\{X^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*}\}\text{with}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu },}\\ {\displaystyle {ℂ\left[X\right]}^{W_0}}& =& {\displaystyle \{f\in ℂ\left[X\right]\hspace{0.17em}|\hspace{0.17em}wf=f\text{, for all}\hspace{0.17em}w\in W_0\},}\\ {\displaystyle {ℂ\left[X\right]}^{\text{det}}}& =& {\displaystyle \{f\in ℂ\left[X\right]\hspace{0.17em}|\hspace{0.17em}wf=\text{det}\left(w\right)f\text{, for all}\hspace{0.17em}w\in W_0\}\text{.}}\end{array}$$ Then ${ℂ\left[X\right]}^{W_0}$ has basis $$m_{\lambda }=\sum _{\gamma \in W_0\lambda }{X}^{\gamma },\lambda \in P^{+},$$ ${ℂ\left[X\right]}^{\text{det}}$ has basis $$a_{\lambda +\rho }=\sum _{w\in W_0}\text{det}\left(w\right)X^{w(\lambda +\rho )}\text{.}$$

As $ℂ{\left[X\right]}^{W_0}\text{-modules}$ $$\begin{array}{cccc}\mathrm{\Phi }:& ℂ{\left[X\right]}^{W_0}& \stackrel{\sim }{⟶}& ℂ{\left[X\right]}^{\text{det}}\\ & f& ⟼& a_{\rho }f\text{.}\end{array}$$

		Proof.
	(a) $\mathrm{\Phi }$ is a $ℂ{\left[X\right]}^{W_0}\text{-homomorphism.}$ If $g\in ℂ{\left[X\right]}^{W_0}$ then $$\mathrm{\Phi }\left(gf\right)=a_{\rho }gf=g{a}_{\rho }f=g\mathrm{\Phi }\left(f\right)\text{.}$$ (b) $\mathrm{\Phi }$ is well defined. If $w\in W_0$ then $$\begin{array}{rcl}{\displaystyle w\mathrm{\Phi }\left(f\right)}& =& {\displaystyle w{a}_{\rho }f}\\ & =& {\displaystyle \left(w{a}_{\rho }\right)\left(wf\right)}\\ & =& {\displaystyle \text{det}\left(w\right)a_{\rho }f}\\ & =& {\displaystyle \text{det}\left(w\right)\mathrm{\Phi }\left(f\right)\text{.}}\end{array}$$ (c) $\mathrm{\Phi }$ is invertible. Let $g\in ℂ{\left[X\right]}^{\text{det}},$ $g=\sum _{\mu \in {𝔥}_{\mathbb{Z}}^{*}}{g}_{\mu }{X}^{\mu }\text{.}$ Let $s_{\alpha }$ be a reflection in $W_0,$ $$s_{\alpha }\mu =\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha ,\text{with}\hspace{0.17em}⟨\mu ,{\alpha }^{\vee }⟩\in \mathbb{Z}\text{.}$$ Then $$s_{\alpha }g=\text{det}\left(s_{\alpha }\right)g=-g,$$ so that $g=\frac{1}{2}(g-s_{\alpha }g)\text{.}$ So $$\begin{array}{rcl}{\displaystyle g}& =& {\displaystyle \frac{1}{2}(1-s_{\alpha })g}\\ & =& {\displaystyle \frac{1}{2}\sum _{\mu \in {𝔥}_{\mathbb{Z}}^{*}}{g}_{\mu }(X^{\mu }-X^{s_{\alpha }\mu })}\\ & =& {\displaystyle \frac{1}{2}\sum _{\mu \in {𝔥}_{\mathbb{Z}}^{*}}{g}_{\mu }{X}^{\mu }(1-X^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha })\text{.}}\end{array}$$ Note, for example, $$\begin{array}{rcl}{\displaystyle 1-X^{-5\alpha }}& =& {\displaystyle (1-X^{-\alpha })(1+X^{\alpha }+X^{2\alpha }+X^{3\alpha }+X^{4\alpha }),}\\ {\displaystyle 1-X^{5\alpha }}& =& {\displaystyle X^{5\alpha }(1-X^{-\alpha })(1+X^{\alpha }+X^{2\alpha }+X^{3\alpha }+X^{4\alpha })(-1)\text{.}}\end{array}$$ In any case, $g$ is divisible by $1-X^{-\alpha }\text{.}$ Since $1-X^{-\alpha },$ $\alpha \in R^{+},$ are relatively prime, $g$ is divisible by $\prod _{\alpha \in R^{+}}(1-X^{-\alpha })\text{.}$ $\square$

Note: $$\begin{array}{rcl}{\displaystyle a_{\rho }}& =& {\displaystyle \left(\prod _{\alpha \in R^{+}}{X}^{\alpha /2}\right)\prod _{\alpha \in R^{+}}(1-X^{-\alpha })}\\ & =& {\displaystyle X^{\rho }\prod _{\alpha \in R^{+}}(1-X^{-\alpha })}\end{array}$$ since $$a_{\rho }=X^{\rho }+\cdots +X^{W_0\rho }\text{and}\rho =\frac{1}{2}\sum _{\alpha \in R^{+}}\alpha$$ because

(a)	$s_i$ permutes $R^{+}-\left\{{\alpha }_i\right\}$ and $s_i{\alpha }_i=-{\alpha }_i\text{.}$
(b)	$W_0$ sends $R^{+}=\left\{\alpha \right\}$ to $R^{-}=\{-\alpha \hspace{0.17em}|\hspace{0.17em}\alpha \in R^{+}\}\text{.}$

The Weyl character, or Schur function is $$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }}\text{so that}\begin{array}{ccc}ℂ{\left[X\right]}^{W_0}& ⟶& ℂ{\left[X\right]}^{\text{det}}\\ s_{\lambda }& ⟼& a_{\lambda +\rho }\end{array}$$

Crystals

A path is $p:[0,1]\to {𝔥}_{ℝ}^{*}$ (piecewise linear) with $p\left(0\right)=0$ and $p\left(1\right)\in {𝔥}_{\mathbb{Z}}^{*}\text{.}$

A crystal is a set of paths $B,$ closed under the action of the root operators ${\stackrel{\sim }{e}}_i,{\stackrel{\sim }{f}}_i\text{.}$ $$\begin{array}{c} 𝔥αi∨ p f∼ip 𝔥αi∨ p f˜ip=0 \end{array}$$ and ${\stackrel{\sim }{e}}_i{\stackrel{\sim }{f}}_i=p,$ if ${\stackrel{\sim }{f}}_{i}p\ne 0$ and ${\stackrel{\sim }{f}}_i{\stackrel{\sim }{e}}_{i}p=p,$ if ${\stackrel{\sim }{e}}_{i}p\ne 0\text{.}$

The character of $B$ is $$\text{char}\left(B\right)=\sum _{p\in B}{X}^{\text{wt}\left(p\right)}\text{.}$$

Favourite example

$$\begin{array}{c} 𝔥α2∨ 𝔥α1∨ \end{array}\begin{array}{c}\uparrow \\ {\stackrel{\sim }{f}}_2\swarrow \searrow {\stackrel{\sim }{f}}_1\\ \nwarrow \nearrow \\ {\stackrel{\sim }{f}}_1\searrow \swarrow {\stackrel{\sim }{f}}_2\\ ⤣⤤\\ \begin{array}{c} f∼2 f∼1 \end{array}\\ \swarrow \searrow \\ {\stackrel{\sim }{f}}_1\searrow \swarrow {\stackrel{\sim }{f}}_2\\ \downarrow \end{array}$$ Let $B$ be a crystal and let $p\in B\text{.}$ The $i\text{-string}$ of $p$ is $${\stackrel{\sim }{f}}_i^{d^{-}}-\cdots -{\stackrel{\sim }{f}}_{i}p-p-{\stackrel{\sim }{e}}_{i}p-{\stackrel{\sim }{e}}_i^{2}p-\cdots -{\stackrel{\sim }{e}}_i^{d^{+}}p$$ where ${\stackrel{\sim }{e}}_i^{d^{+}+1}p=0$ and ${\stackrel{\sim }{f}}_i^{d^{-}+1}p=0\text{.}$ If $h={\stackrel{\sim }{e}}_i^{d^{+}}p$ then the paths in the $i\text{-string}$ $${\stackrel{\sim }{f}}_i^{⟨\mu ,{\alpha }_i^{\vee }⟩}h-\cdots -{\stackrel{\sim }{f}}_i^{2}h-{\stackrel{\sim }{f}}_i-h$$ have weights $$\mu -⟨\mu ,{\alpha }_i^{\vee }⟩{\alpha }_i,\dots ,\mu -2{\alpha }_i,\mu -{\alpha }_i,\mu$$ with $$s_i\mu =\mu -⟨\mu ,{\alpha }_i^{\vee }⟩{\alpha }_i\text{.}$$ Define an action of $W_0$ on $B$ by setting $s_{i}p$ to be the opposite of $p$ in its $i\text{-string.}$ $${\stackrel{\sim }{f}}_i^{⟨\mu ,{\alpha }_i^{\vee }⟩}h-{\stackrel{\sim }{f}}_i^{⟨\mu ,{\alpha }_i^{\vee }⟩-1}h-\cdots -{\stackrel{\sim }{f}}_i^{2}h-{\stackrel{\sim }{f}}_{i}h-h$$ Then $$\text{wt}\left(s_{i}p\right)=s_i\text{wt}\left(p\right)\text{.}$$ So $$\text{char}\left(B\right)=s_i\text{char}\left(B\right)$$ for $i=1,\dots ,n\text{.}$ Since $s_1,\dots ,s_n$ generate $W_0$ it follows that $$\text{char}\left(B\right)\in ℂ{\left[x\right]}^{W_0}\text{.}$$

A highest weight path is $p\subseteq C-\rho$ $$\begin{array}{c} C \end{array}\begin{array}{c} C-ρ \end{array}$$ A path $p$ is highest weight, if and only if ${\stackrel{\sim }{e}}_{i}p=0$ for all $i=1,\dots ,n\text{.}$

Let $B$ be a crystal. Then $$\text{char}\left(B\right)=\sum _{\underset{p\subseteq C-\rho }{p\in B}}{s}_{\text{wt}\left(p\right)}\text{.}$$

		Proof.
	Define $$s_{\mu }=\frac{a_{\mu +\rho }}{a_{\rho }},\text{for}\hspace{0.17em}\mu \in P\text{.}$$ Then define a new action of $W_0$ on $P$ by $$w\circ \mu =w(\mu +\rho )-\rho ,\text{for}\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\text{.}$$ $$\begin{array}{c} 0 -ρ s1∘0 s2∘0 s1s2∘0 s2s1∘0 s1s2s1∘0 \end{array}$$ This is the dot action of $W_0\text{.}$ We have $$\begin{array}{cc}\begin{array}{rcl}{\displaystyle s_{w\circ \mu }}& =& {\displaystyle s_{w(\mu +\rho )-\rho }=\frac{a_{w(\mu +\rho )-\rho +\rho }}{a_{\rho }}=\frac{a_{w(\mu +\rho )}}{a_{\rho }}}\\ & =& {\displaystyle \frac{\text{det}\left(w\right)a_{\mu +\rho }}{a_{\rho }}=\text{det}\left(w\right)s_{\mu }\text{.}}\end{array}& (*)\end{array}$$ Now let $$\epsilon =\sum _{w\in W_0}\text{det}\left(w\right)w$$ so that $\epsilon \left(X^{\mu }\right)=a_{\mu }\text{.}$ then $$\begin{array}{rcl}{\displaystyle \text{char}\left(B\right)}& =& {\displaystyle \frac{1}{a_{\rho }}\text{char}\left(B\right)a_{\rho }}\\ & =& {\displaystyle \frac{1}{a_{\rho }}\text{char}\left(B\right)\epsilon \left(X^{\rho }\right)}\\ & =& {\displaystyle \frac{1}{a_{\rho }}\epsilon \left(\text{char}\left(B\right)X^{\rho }\right)}\\ & =& {\displaystyle \frac{1}{a_{\rho }}\sum _{p\in B}\epsilon \left(X^{\text{wt}\left(\rho \right)+\rho }\right)}\\ & =& {\displaystyle \sum _{p\in B}{s}_{\text{wt}\left(p\right)}\text{.}}\end{array}$$ The equation $(*)$ can cause some cancellation in this sum. Let $p\in B$ such that $p$ is not highest weight. Let $i$ be minimal such that $p$ leaves $C-p$ by crossing ${𝔥}^{{\alpha }_i+\delta }\text{.}$ Define $s_i\circ p$ to be the element of the $i\text{-string}$ of $p$ such that $$\text{wt}(s_i\circ p)=s_i\circ p$$ $$t-{\stackrel{\sim }{e}}_{i}t-{\stackrel{\sim }{e}}_i^{2}t-\cdots -{\stackrel{\sim }{f}}_i^{2}h-{\stackrel{\sim }{f}}_{i}h-h$$ Then $$s_{\text{wt}(s_i\circ p)}=\text{det}\left(s_i\right)s_{\text{wt}\left(p\right)}=-s_{\text{wt}\left(p\right)}$$ and $$s_{\text{wt}(s_i\circ p)}+s_{\text{wt}\left(p\right)}=0\text{.}$$ Note that $s_i\circ p$ leaves $C-\rho$ at the same place that $p$ leaves $C-\rho \text{.}$ thus $$\text{char}\left(B\right)=\sum _{\underset{p\subseteq C-\rho }{p\in B}}{s}_{\text{wt}\left(p\right)}\text{.}$$ $\square$

Let $p_{\lambda }^{+}$ be a path, $p_{\lambda }^{+}\subseteq C-\rho$ with $\text{wt}\left(p_{\lambda }^{+}\right)=\lambda \text{.}$ Let $B\left(\lambda \right)$ be the crystal generated by $p_{\lambda }^{+}\text{.}$ Then $$\text{char}\left(B\left(\lambda \right)\right)=s_{\lambda }\text{.}$$

Let $B$ be a crystal. Let $J\subseteq \{1,\dots ,n\}\text{.}$ By ignoring the action of ${\stackrel{\sim }{e}}_i,{\stackrel{\sim }{f}}_i$ for $i\notin J,$ $B$ is a $(W_J,{𝔥}_{\mathbb{Z}}^{*})\text{-crystal,}$ where $W_J=⟨s_j\hspace{0.17em}|\hspace{0.17em}j\in J⟩\text{.}$ Let $C_J-{\rho }_J$ be the region on the positive side of ${𝔥}^{{\alpha }_j^{\vee }+{\delta }_j}$ for $j\in J\text{.}$ Then $$\text{char}\left(B\right)=\sum _{\underset{p\subseteq C_J-{\rho }_J}{p\in B}}{s}_{\text{wt}\left(p\right)}^J\text{.}$$

Let $\lambda ,\mu \in P^{+}\text{.}$ Let $B\left(\lambda \right)$ and $B\left(\mu \right)$ be the irreducible crystals of highest weights $\lambda$ and $\mu ,$ respectively. Then $$B\left(\lambda \right)\otimes B\left(\mu \right)=\{p\otimes q\hspace{0.17em}|\hspace{0.17em}p\in B\left(\lambda \right),q\in B\left(\mu \right)\}$$ is a crystal $\text{(}p\otimes q$ is the concatenation of $p$ and $q\text{).}$ Then $$\text{char}(B\left(\lambda \right)\otimes B\left(\mu \right))=\sum _{\underset{p_{\lambda }^{+}\otimes q\subseteq C-\rho }{q\in B\left(\mu \right)}}{s}_{\text{wt}\left(q\right)+\lambda }\text{.}$$

		Proof.
	$p\otimes q\subseteq C-\rho$ only if $p=p_{\lambda }^{+},$ in which case $$s_{\text{wt}(p_{\lambda }^{+}\otimes q)}=s_{\lambda +\text{wt}\left(q\right)}\text{.}$$ $\square$

Notes and References

These are a typed copy of Lecture 8 from a series of handwritten lecture notes for the class Representation Theory given on September 16, 2008.

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