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 7

Dual vector spaces

Let $𝔽$ be a field, $$\begin{array}{rcl}{\displaystyle {𝔥}^{*}}& =& {\displaystyle \text{span}\{{\omega }_1,\dots ,{\omega }_n\}\hspace{0.17em}\text{a vector space,}}\\ {\displaystyle 𝔥}& =& {\displaystyle \text{Hom}({𝔥}^{*},𝔽)\hspace{0.17em}\text{the dual vector space.}}\end{array}$$ Write $$⟨\mu ,{\lambda }^{\vee }⟩=\mu \left({\lambda }^{\vee }\right),$$ for $\mu \in {𝔥}^{*},$ ${\lambda }^{\vee }\in 𝔥\text{.}$ Let $G=GL\left({𝔥}^{*}\right)\text{.}$ $G$ acts on ${𝔥}^{*}\text{.}$ Define an action of $G$ on $𝔥$ by $$⟨\mu ,g{\lambda }^{\vee }⟩=⟨g^{-1}\mu ,{\lambda }^{\vee }⟩\text{.}$$ Let ${\omega }_1,\dots ,{\omega }_n$ be a basis of ${𝔥}^{*}$ and identify $g$ with its matrix in ${GL}_n\left(𝔽\right)\text{.}$

Let ${\alpha }_1^{\vee },\dots ,{\alpha }_n^{\vee }$ be the dual basis in $𝔥\text{.}$ The matrix of the action of $g$ on $𝔥$ is $$g^{\vee }={\left(g^t\right)}^{-1}\text{.}$$

Reflections

A reflection is $s_{\alpha }\in GL\left({𝔥}^{*}\right)$ such that, in ${GL}_n\left(\overline{𝔽}\right),$ $s_{\alpha }$ is conjugate to $$\left(\begin{array}{c}\xi \\ & 1\\ & & \ddots \\ & & & 1\end{array}\right)$$ with $\xi \in \overline{𝔽},$ $\xi \ne 1\text{.}$ Then $s_{\alpha }^{\vee }\in GL\left(𝔥\right)$ is conjugate to $$\left(\begin{array}{c}{\xi }^{-1}\\ & 1\\ & & \ddots \\ & & & 1\end{array}\right)\text{.}$$ Then $${𝔥}^{*}={𝔥}^{{\alpha }^{\vee }}\oplus ℂ\alpha \text{and}𝔥={𝔥}^{\alpha }\oplus ℂ{\alpha }^{\vee }$$ where $$\begin{array}{rcl}{\displaystyle {𝔥}^{{\alpha }^{\vee }}}& =& {\displaystyle {\left({𝔥}^{*}\right)}^{s_{\alpha }}=\{\mu \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}{s}_{\alpha }\mu =\mu \}\text{(1 eigenspace of}\hspace{0.17em}{s}_{\alpha }\text{),}}\\ {\displaystyle ℂ\alpha }& =& {\displaystyle \left(\xi \text{-eigenspace of}\hspace{0.17em}{s}_{\alpha }\right),}\\ {\displaystyle {𝔥}^{\alpha }}& =& {\displaystyle {𝔥}^{s_{\alpha }^{\vee }}=\{{\lambda }^{\vee }\in 𝔥\hspace{0.17em}|\hspace{0.17em}{s}_{\alpha }^{\vee }{\lambda }^{\vee }={\lambda }^{\vee }\}=\left(1\text{-eigenspace of}\hspace{0.17em}{s}_{\alpha }\right),}\\ {\displaystyle ℂ{\alpha }^{\vee }}& =& {\displaystyle \left({\xi }^{-1}\text{-eigenspace of}\hspace{0.17em}{s}_{\alpha }^{\vee }\right)}\end{array}$$ and $$\begin{array}{rcl}{\displaystyle {𝔥}^{{\alpha }^{\vee }}}& =& {\displaystyle \{\mu \in {𝔥}^{*}\hspace{0.17em}|\hspace{0.17em}⟨\mu ,{\alpha }^{\vee }⟩=0\},}\\ {\displaystyle {𝔥}^{\alpha }}& =& {\displaystyle \{{\lambda }^{\vee }\in 𝔥\hspace{0.17em}|\hspace{0.17em}⟨{\lambda }^{\vee },\alpha ⟩=0\}\text{.}}\end{array}$$ Choose $\alpha$ and ${\alpha }^{\vee }$ so that $⟨\alpha ,{\alpha }^{\vee }⟩=1-\xi =1-\text{det}\left(s_{\alpha }\right)\text{.}$ Then $$s_{\alpha }\mu =\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha \text{and}{s}_{\alpha }^{-1}{\lambda }^{\vee }={\lambda }^{\vee }-⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee }\text{.}$$ Check: $$\begin{array}{rcl}{\displaystyle s_{\alpha }\alpha }& =& {\displaystyle \alpha -⟨\alpha ,{\alpha }^{\vee }⟩\alpha =(1-⟨\alpha ,{\alpha }^{\vee }⟩)\alpha =\xi \alpha ,}\\ {\displaystyle s_{\alpha }^{-1}{\alpha }^{\vee }}& =& {\displaystyle {\alpha }^{\vee }-⟨\alpha ,{\alpha }^{\vee }⟩{\alpha }^{\vee }=(1-⟨\alpha ,{\alpha }^{\vee }⟩){\alpha }^{\vee }=\xi {\alpha }^{\vee },}\end{array}$$ as it should be. $$\begin{array}{rcl}{\displaystyle s_{\alpha }\mu }& =& {\displaystyle \mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha =\mu -0,\text{if}\hspace{0.17em}\mu \in {𝔥}^{{\alpha }^{\vee }},}\\ {\displaystyle s_{\alpha }^{-1}{\lambda }^{\vee }}& =& {\displaystyle {\lambda }^{\vee }-⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee }={\lambda }^{\vee }-0,\text{if}\hspace{0.17em}{\lambda }^{\vee }\in {𝔥}^{\alpha }\text{.}}\end{array}$$

Weyl groups

Let ${𝔥}_{\mathbb{Z}}^{*}$ be a $\mathbb{Z}\text{-vector}$ space. $${𝔥}_{\mathbb{Z}}^{*}=\mathbb{Z}\text{-span}\{{\omega }_1,\dots ,{\omega }_n\},$$ where ${\omega }_1,\dots ,{\omega }_n$ is a $\mathbb{Z}\text{-basis}$ of ${𝔥}_{\mathbb{Z}}^{*}\text{.}$ $$\begin{array}{rcl}{\displaystyle {𝔥}_{\mathbb{Q}}^{*}}& =& {\displaystyle \mathbb{Q}{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*}=\mathbb{Q}\text{-span}\{{\omega }_1,\dots ,{\omega }_n\},}\\ {\displaystyle {𝔥}_{ℝ}^{*}}& =& {\displaystyle ℝ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*}=ℝ\text{-span}\{{\omega }_1,\dots ,{\omega }_n\},}\\ {\displaystyle {𝔥}_{ℂ}^{*}}& =& {\displaystyle ℂ{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*}=ℂ\text{-span}\{{\omega }_1,\dots ,{\omega }_n\},}\\ {\displaystyle {𝔥}_{\bar{\mathbb{Q}}}^{*}}& =& {\displaystyle \bar{\mathbb{Q}}{\otimes }_{\mathbb{Z}}{𝔥}_{\mathbb{Z}}^{*}=\bar{\mathbb{Q}}\text{-span}\{{\omega }_1,\dots ,{\omega }_n\}\text{.}}\end{array}$$ A Weyl group, or crystallographic reflection group, is a finite subgroup $w_0$ of $GL\left({𝔥}_{\mathbb{Z}}^{*}\right)$ generated by reflections.

Let $R^{+}$ be an index set for the reflections in $W_0$ so that $s_{\alpha },$ $\alpha \in R^{+},$ are the reflections in $W_0\text{.}$

WARNING: A Weyl group is really a pair $(W_0,{𝔥}_{\mathbb{Z}}^{*})\text{.}$ $W_0$ cannot exist without ${𝔥}_{\mathbb{Z}}^{*}\text{.}$

Examples (Type ${GL}_n$)

$${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\epsilon }_1,\dots ,{\epsilon }_n\}$$ with $W_0=S_n$ acting by permuting ${\epsilon }_1,\dots ,{\epsilon }_n\text{.}$ The reflections are $$s_{ij}=s_{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }}=\begin{array}{c} 1 ⋯ i ⋯ j ⋯ n \end{array}=\begin{array}{ccccccccccccc}& & & & & i& & & & j& & & \\ & (& 1& & & & & & & & & & & )\\ & & \ddots & & & & & & & & & \\ & & & 1& & & & & & & & \\ i& & & & 0& & & & 1& & & \\ & & & & & 1& & & & & & \\ & & & & & & \ddots & & & & & \\ & & & & & & & 1& & & & \\ j& & & & 1& & & & 0& & & \\ & & & & & & & & & 1& & \\ & & & & & & & & & & \ddots & \\ & & & & & & & & & & & 1\end{array}$$ $$R^{+}=\left\{\left(ij\right)\hspace{0.17em}\right|\hspace{0.17em}1\le i<j\le n\}\text{or}{R}^{+}=\{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }\hspace{0.17em}|\hspace{0.17em}1\le i<j\le n\}$$ and $$\begin{array}{rcl}{\displaystyle {𝔥}^{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }}}& =& {\displaystyle {\left({𝔥}^{*}\right)}^{s_{ij}}}\\ & =& {\displaystyle \{\mu \in {𝔥}_{ℝ}^{*}\hspace{0.17em}|\hspace{0.17em}{s}_{ij}\mu =\mu \}}\\ & =& {\displaystyle \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}⟨\mu ,{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }⟩=0\}}\\ & =& {\displaystyle \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_i={\mu }_j\}\text{.}}\end{array}$$ The arrangement of hyperplanes $${𝔥}^{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }}\text{in}{𝔥}_{ℂ}^{*},1\le i<j\le n$$ is the braid arrangement.

Remark $${\text{Conf}}_n\left({ℂ}^n\right)=({𝔥}_{ℂ}^{*}-\left(\bigcup _{1\le i<j\le n}{𝔥}^{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }}\right))$$ has $${\pi }_1\left({\text{Conf}}_n\left({ℂ}^n\right)\right)=\text{braid group.}$$

(Type ${SL}_3\text{)}$ $${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\omega }_1,{\omega }_2\}\text{and}{W}_0=⟨s_1,s_2\hspace{0.17em}|\hspace{0.17em}{s}_1^2=s_2^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩$$ $$\begin{array}{c} C 𝔥α1∨ 𝔥α2∨ s1C s2C ω1 ω2 s1s2C s2s1C s1s2s1C=s2s1s2C \end{array}$$ where $s_1$ is reflection in ${𝔥}^{{\alpha }_1^{\vee }}$ and $s_2$ is reflection in ${𝔥}^{{\alpha }_2^{\vee }}\text{.}$

Let $C$ be a fundamental chamber for the action of $W_0$ on ${𝔥}_{ℝ}^{*}\text{.}$ $$W_0\stackrel{1-1}{⟷}\left\{\text{chambers in}\hspace{0.17em}{𝔥}_{ℝ}^{*}\right\}\text{.}$$ Let $\bar{C}$ be the closure of $C\text{.}$ The dominant integral weights are $$P^{+}={𝔥}_{\mathbb{Z}}^{*}\cap \bar{C}\text{and}{P}^{++}={𝔥}_{\mathbb{Z}}^{*}\cap C$$ are the strictly dominant integral weights.

There is a bijection $$\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}$$ where $\rho$ is the point of $P^{++}$ closest to $0\text{.}$

Symmetric functions

Let $$X=\left\{X^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*}\}\text{with}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu }\text{.}$$ $X$ is the same group as ${𝔥}_{\mathbb{Z}}^{*},$ except written multiplicatively. $W_0$ acts on $X$ by $$w{X}^{\mu }=X^{w\mu },\text{for}\hspace{0.17em}w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*}\text{.}$$ Two one-dimensional representations of $W_0$ are $$\begin{array}{ccc}{W}_0& ⟶& ℂ\\ w& ⟼& 1\end{array}\text{and}\begin{array}{ccc}{W}_0& ⟶& ℂ\\ w& ⟼& \text{det}\left(w\right)\text{.}\end{array}$$ The ring of symmetric functions is $$ℂ{\left[X\right]}^{W_0}=\{f\in ℂ\left[X\right]\hspace{0.17em}|\hspace{0.17em}wf=f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W_0\}\text{.}$$ The vector space of determinant symmetric functions is $$ℂ{\left[x\right]}^{\text{det}}=\{f\in ℂ\left[X\right]\hspace{0.17em}|\hspace{0.17em}wf=\text{det}\left(w\right)f,\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W_0\}\text{.}$$

(Type ${GL}_3\text{)}$ $${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\epsilon }_1,{\epsilon }_2,{\epsilon }_3\}\text{and}X=\left\{X^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*}\}$$ where, for ${\mu }_1,{\mu }_2,{\mu }_3\in \mathbb{Z},$ $$\begin{array}{rcl}{\displaystyle X^{\mu }}& =& {\displaystyle X^{{\mu }_1{\epsilon }_1+{\mu }_2{\epsilon }_2+{\mu }_3{\epsilon }_3}}\\ & =& {\displaystyle {\left(X^{{\epsilon }_1}\right)}^{{\mu }_1}{\left(X^{{\epsilon }_2}\right)}^{{\mu }_2}{\left(X^{{\epsilon }_3}\right)}^{{\mu }_3}}\\ & =& {\displaystyle X_1^{{\mu }_1}{X}_2^{{\mu }_2}{X}_3^{{\mu }_3},\text{where}\hspace{0.17em}{X}_i=X^{{\epsilon }_i}\text{.}}\end{array}$$ Then $$m_{(1,1,-2)}=X_1{X}_2{X}_3^{-2}+X_1{X}_2^{-2}{X}_3+X_1^{-2}{X}_2{X}_3$$ is symmetric, and $$a_{(1,1,-2)}=X_1{X}_2{X}_3^{-2}-X_2{X}_1{X}_3^{-2}-X_1^{-2}{X}_2{X}_3-X_1{X}_2^{-2}{X}_3+X_2{X}_3{X}_1^{-2}+X_3{X}_1{X}_2^{-2}$$ is determinant symmetric since $\text{det}\left(s_1\right)=-1$ and $\text{det}\left(s_2\right)=-1\text{.}$

(Type ${SL}_3\text{)}$ $$\begin{array}{c} C 𝔥α1∨ 𝔥α2∨ ω1 ω2 ρ s1ρ s2ρ \end{array}$$ $$\begin{array}{rcl}{\displaystyle m_{\rho }}& =& {\displaystyle X^{\rho }+X^{s_1\rho }+X^{s_2\rho }+X^{s_1{s}_2\rho }+X^{s_2{s}_1\rho }+X^{s_1{s}_2{s}_1\rho }\text{and}}\\ {\displaystyle m_{2{\omega }_1}}& =& {\displaystyle X^{2{\omega }_1}+X^{2{\omega }_2-2{\omega }_1}+X^{-2{\omega }_2}\text{are symmetric and}}\\ {\displaystyle a_{\rho }}& =& {\displaystyle X^{\rho }-X^{s_1\rho }-X^{s_2\rho }+X^{s_1{s}_2\rho }+X^{s_2{s}_1\rho }-X^{s_1{s}_2{s}_1\rho }\text{and}}\\ {\displaystyle a_{2{\omega }_1}}& =& {\displaystyle X^{2{\omega }_1}-X^{s_{1}2{\omega }_1}-X^{s_{2}2{\omega }_1}+X^{s_1{s}_{2}2{\omega }_1}+X^{s_2{s}_{1}2{\omega }_1}-X^{s_1{s}_2{s}_{1}2{\omega }_1}}\\ & =& {\displaystyle X^{2{\omega }_1}-X^{2{\omega }_1}-X^{2{\omega }_2-2{\omega }_1}+X^{-2{\omega }_2}+X^{2{\omega }_2-2{\omega }_1}-X^{-2{\omega }_2}}\end{array}$$ are determinant symmetric.

Let $$m_{\mu }=\sum _{\gamma \in W_0\mu }{X}^{\gamma },\text{for}\hspace{0.17em}\mu \in P\text{.}$$ Then $m_{w\mu }=m_{\mu },$ for $w\in W_0$ and the orbit sums, or monomial symmetric functions, $$m_{\lambda }=\sum _{\gamma \in W_0\lambda }{X}^{\gamma },\lambda \in P^{+}$$ form a basis of $ℂ{\left[X\right]}^{W_0}\text{.}$

Let $$a_{\mu }=\sum _{w\in W_0}\text{det}\left(w^{-1}\right)X^{w\mu },\text{for}\hspace{0.17em}\mu \in P\text{.}$$ Then $$\begin{array}{rcl}{\displaystyle a_{v\mu }}& =& {\displaystyle \sum _{w\in W_0}\text{det}\left(w^{-1}\right)X^{wv\mu }}\\ & =& {\displaystyle \sum _{w\in W_0}\text{det}\left(v\right)\text{det}\left({\left(wv\right)}^{-1}\right)X^{wv\mu }}\\ & =& {\displaystyle \text{det}\left(v\right)a_{\mu },}\end{array}$$ for $\mu \in P$, $v\in W_0\text{.}$

If $\mu \in {𝔥}^{{\alpha }^{\vee }}$ so that $s_{\alpha }\mu =\mu$ then $$a_{\mu }=a_{s_{\alpha }\mu }=\text{det}\left(s_{\alpha }\right)a_{\mu }$$ implies $a_{\mu }=0,$ since $\text{det}\left(s_{\alpha }\right)\ne 1\text{.}$ Thus $$a_{\mu }=\sum _{w\in W_0}\text{det}\left(w^{-1}\right)X^{\mu },$$ $\mu \in P^{++},$ form a basis of $ℂ{\left[X\right]}^{\text{det}}\text{.}$

Boson-Fermion correspondence

$$\begin{array}{ccc}ℂ{\left[X\right]}^{W_0}& ⟶& ℂ{\left[X\right]}^{\text{det}}\\ f& ⟼& a_{\rho }f\end{array}$$ is well defined since, if $f\in ℂ{\left[X\right]}^{W_0}$ then $$w\left(a_{\rho }f\right)=\left(w{a}_{\rho }\right)\left(wf\right)=\text{det}\left(w\right)a_{\rho }f,$$ for $w\in W_0\text{.}$ In fact, this map is invertible!

Let $g\in ℂ{\left[X\right]}^{\text{det}},$ $$g=\sum _{\mu \in P}{g}_{\mu }{X}^{\mu },$$ with $g_{\mu }\in ℂ\text{.}$ Let $s_{\alpha }$ be a reflection in $W_0\text{.}$ Then $$\frac{1}{2}(g-s_{\alpha }g)=\frac{1}{2}(g-\text{det}\left(s_{\alpha }\right)g)=\frac{1}{2}(g+g)=g$$ and $$\begin{array}{rcl}{\displaystyle X^{\mu }-X^{s_{\alpha }\mu }}& =& {\displaystyle X^{\mu }-X^{\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha }}\\ & =& {\displaystyle X^{\mu }(1-X^{-⟨\mu ,{\alpha }^{\vee }⟩\alpha })}\\ & =& {\displaystyle X^{\mu }(1-X^{-\alpha })(1+X^{-\alpha }+X^{-2\alpha }+\cdots +X^{-(⟨\mu ,{\alpha }^{\vee }⟩-1)\alpha }\text{.})}\end{array}$$ Hence $$\frac{X^{\mu }-X^{s_{\alpha }\mu }}{1-X^{-\alpha }}=X^{\mu }(1+X^{-\alpha }+X^{-2\alpha }+\cdots +X^{-(⟨\mu ,{\alpha }^{\vee }⟩-1)\alpha })$$ and $$g=\frac{1}{2}(g-s_{\alpha }g)$$ is divisible by $1-X^{-\alpha }\text{.}$ It follows that if $g\in ℂ{\left[X\right]}^{\text{det}}$ then $g$ is divisible by $\prod _{\alpha \in R^{+}}(1-X^{-\alpha })\text{.}$

Claim: $$\begin{array}{rcl}{\displaystyle a_{\rho }}& =& {\displaystyle X^{\rho }+\text{lower stuff}}\\ & =& {\displaystyle X^{\rho }\prod _{\alpha \in R^{+}}(1-X^{-\alpha })\text{.}}\end{array}$$

(Type ${GL}_n\text{)}$ $$X_i=s_{\mu }{X}^{{\epsilon }_i},$$ where ${𝔥}_{\mathbb{Z}}^{*}=\text{span}\{{\epsilon }_1,\dots ,{\epsilon }_n\}$ and $W_0=S_n\text{.}$ Then $$\begin{array}{rcl}{\displaystyle a_{\mu }}& =& {\displaystyle \sum _{w\in S_n}\text{det}\left(w^{-1}\right)X^{w\mu }}\\ & =& {\displaystyle \sum _{w\in S_n}\text{det}\left(w^{-1}\right)X_{w\left(1\right)}^{{\mu }_1}\cdots X_{w\left(n\right)}^{{\mu }_n}}\\ & =& {\displaystyle \text{det}\left(X_i^{{\mu }_j}\right)\text{.}}\end{array}$$ In this case $$C=\{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_i>{\mu }_j\hspace{0.17em}\text{for}\hspace{0.17em}1\le i<j\le n\}$$ since ${𝔥}^{{\epsilon }_i^{\vee }-{\epsilon }_j^{\vee }}=\{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_u\hspace{0.17em}|\hspace{0.17em}{\mu }_i={\mu }_j\}\text{.}$ Hence $$\begin{array}{rcl}{\displaystyle P^{+}}& =& {\displaystyle \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_i\in \mathbb{Z},{\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_n\},}\\ {\displaystyle P^{++}}& =& {\displaystyle \{\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_i\in \mathbb{Z},{\mu }_1>{\mu }_2>\cdots >{\mu }_n\}}\end{array}$$ and $$\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \mu & ⟼& \mu +\rho \end{array}$$ where $\rho =(n-1){\epsilon }_1+(n-2){\epsilon }_2+\cdots +{\epsilon }_{n-1}\text{.}$ Hence $$a_{\rho }=\text{det}\left(x_i^{n-j}\right)=\left(\begin{array}{ccccc}{x}_1^{n-1}& x_1^{n-2}& \cdots & x_1& 1\\ x_2^{n-1}& x_2^{n-2}& \cdots & x_2& 1\\ & ⋮\\ x_n^{n-1}& x_n^{n-2}& \cdots & x_n& 1\end{array}\right)=\prod _{i<j}(x_i-x_j)\text{.}$$

Notes and References

These are a typed copy of Lecture 7 from a series of handwritten lecture notes for the class Representation Theory given on September 9, 2008.

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