Representation theory Lecture Notes: Chapter 5

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

Last update: 20 November 2013

Symmetric functions

Let $$\begin{array}{ccc}P& =& \left\{({\gamma }_1,\dots ,{\gamma }_n)\hspace{0.17em}\right|\hspace{0.17em}{\gamma }_i\in {\mathbb{Z}}_{\ge 0}\},\\ P^{+}& =& \left\{({\gamma }_1\ge \dots \ge {\gamma }_n)\hspace{0.17em}\right|\hspace{0.17em}{\gamma }_i\in {\mathbb{Z}}_{\ge 0}\},\text{and}\\ P^{++}& =& \left\{({\gamma }_1>\dots >{\gamma }_n)\hspace{0.17em}\right|\hspace{0.17em}{\gamma }_i\in {\mathbb{Z}}_{\ge 0}\}\text{.}\end{array}$$ The map $$\begin{array}{ccc}{P}^{++}& \stackrel{1-1}{⟷}& P^{+}\\ \lambda +\delta & ⟼& \lambda \end{array}\text{where}\delta =(n-1,n-2,\dots ,2,1,0),$$ is a bijection. There is an $S_n$ action on $P$ given by $$w\gamma =({\gamma }_{w\left(1\right)},\dots ,{\gamma }_{w\left(n\right)}),\text{for}\hspace{0.17em}w\in S_n,$$ and the elements of $P^{+}$ are representatives of the $S_n$ orbits on $P\text{.}$

The ring of polynomials in $n$ variables $$ℂ[x_1,\dots ,x_n]\text{has basis}\{x^{\gamma }=x_1^{{\gamma }_1}\cdots x_n^{{\gamma }_n}\hspace{0.17em}|\hspace{0.17em}\gamma \in P\}\text{.}$$ and there is an action of $S_n$ on $ℂ[x_1,\dots ,x_n]$ given by $$w{x}^{\gamma }=x^{w\gamma },\text{for all}\hspace{0.17em}w\in S_n,\hspace{0.17em}\gamma \in P\text{.}$$ i.e. $S_n$ acts by permuting the variables. The ring of symmetric polynomials is $${ℂ[x_1,\dots ,x_n]}^{S_n}=\{f\in ℂ[x_1,\dots ,x_n]\hspace{0.17em}|\hspace{0.17em}wf=f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in S_n\}\text{.}$$ The monomial symmetric functions $$m_{\lambda }=\sum _{\mu \in S_n\lambda }{x}^{\mu },\lambda \in P^{+},$$ are a basis of ${ℂ[x_1,\dots ,x_n]}^{S_n}\text{.}$

The vector space of alternating polynomials is $$A=\{f\in ℂ[x_1,\dots ,x_n]\hspace{0.17em}|\hspace{0.17em}wf=\epsilon \left(w\right)f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in S_n\},$$ where $\epsilon \left(w\right)$ is the sign of the permutation $w\text{.}$ If $\gamma \in P$ define $$a_{\gamma }=\sum _{w\in S_n}\epsilon \left(w\right)w{x}^{\gamma }=\text{det}\left(x_i^{{\gamma }_j}\right)\text{.}$$ Then

(a)	$a_{\gamma }$ is an alternating polynomial,
(b)	$a_{\gamma }=0$ is ${\gamma }_i={\gamma }_j$ for some pair $(i,j),$
(c)	$\left\{a_{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in P^{++}\}$ is a basis of $A\text{.}$

Let $f$ be an alternating polynomial and let $\gamma =({\gamma }_1,\dots ,{\gamma }_n)\in {ℂ}^n$ be such that ${\gamma }_i={\gamma }_j\text{.}$ Then $f$ evaluated at $\gamma$ is $0$ since $$f{|}_{{\gamma }_i={\gamma }_j}=\left((i,j)f\right){|}_{{\gamma }_i={\gamma }_j}=-f{|}_{{\gamma }_i={\gamma }_j},$$ where $(i,j)$ is the transposition in $S_n$ which switches $i$ and $j\text{.}$ Since $f{|}_{{\gamma }_i={\gamma }_j}=0$ it follows that $x_i-x_j$ divides the polynomial $f$ in $ℂ[x_1,\dots ,x_n]\text{.}$ Since the polynomials $x_i-x_j$ are coprime in the polynomial ring $ℂ[x_1,\dots ,x_n]$ it follows that the product ${\prod }_{i<j}(x_i-x_j)$ divides $f\text{.}$

In particular, ${\prod }_{i<j}(x_i-x_j)$ divides the alternating polynomial $a_{\delta }\text{.}$ Since $a_{\delta }$ and ${\prod }_{i<j}(x_i-x_j)$ are both homogeneous polynomials of the same degree the quotient must be $1\text{.}$ Thus we have proved the following.

(a)	$a_{\delta }=\prod _{i<j}(x_i-x_j)\text{.}$
(b)	If $f$ is an alternating polynomial the $f$ is divisible by $a_{\delta }\text{.}$
(c)	There is a vector space isomorphism $$\begin{array}{ccc}A& ⟶& {ℂ[x_1,\dots ,x_n]}^{S_n}\\ f& ⟼& \frac{f}{a_{\delta }}\end{array}$$

The quotient $f/a_{\delta }$ is a symmetric polynomial since $w(f/a_{\delta })=\left(wf\right)/\left(w{a}_{\delta }\right)=\left(\epsilon \left(w\right)f\right)/\left(\epsilon \left(w\right)a_{\delta }\right)=f/a_{\delta },$ for all $w\in S_n\text{.}$

Identify $\lambda =({\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n)\in P^{+}$ with the partition which has ${\lambda }_i$ boxes in row $i\text{.}$ A column strict tableau of shape $\lambda$ is a filling of the boxes of $\lambda$ with elements of the set $\{1,2,\dots ,n\}$ such that

(a)	The rows are weakly increasing left to right,
(b)	The columns are strictly increasing top to bottom.

If $T$ is a column strict tableau then $$x^T=x_1^{\left(\text{\# of 1s in}\hspace{0.17em}T\right)}{x}_2^{\left(\text{\# of 2s in}\hspace{0.17em}T\right)}\cdots x_n^{\left(\text{\# of}\hspace{0.17em}n\text{s in}\hspace{0.17em}T\right)}\text{.}$$ The Schur function is the generating function $$s_{\lambda }=\sum _T{x}^T,$$ where the sum is over all column strict tableaux of shape $\lambda \text{.}$ In Theorem ??? we will show that, under the isomorphism in (???), $$\begin{array}{ccc}A& ⟶& {ℂ[x_1,\dots ,x_n]}^{S_n}\\ a_{\lambda +\delta }& ⟼& s_{\lambda },\end{array}$$ i.e. that the image of basis $\left\{a_{\lambda +\delta }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ of $A$ is the basis $\left\{s_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ of ${ℂ[x_1,\dots ,x_n]}^{S_n}\text{.}$

Define $e_r\left(x\right),$ $h_r\left(x\right)$ and $p_r\left(x\right)$ by $$\begin{array}{ccc}{e}_r(x_1,\dots ,x_n)& =& \sum _{i_1<\cdots <i_r}{x}_{i_1}\cdots x_{i_r},\\ h_r(x_1,\dots ,x_n)& =& \sum _{i_1<\cdots <i_r}{x}_{i_1}\cdots x_{i_r},\\ p_r(x_1,\dots ,x_n)& =& \sum _i{x}_i^r\text{.}\end{array}$$ Then $h_r\left(x\right)=s_{\left(r\right)}\left(x\right),$ and $e_r\left(x\right)=s_{\left(1^r\right)}\left(x\right)\text{.}$

(a)	$\prod _{i=1}^n\prod _{j=1}^m\frac{1}{1-x_i{y}_j}=\sum _{\lambda }{s}_{\lambda }\left(x\right)s_{\lambda }\left(y\right)=\sum _{\mu }\frac{p_{\mu }\left(x\right)p_{\mu }\left(y\right)}{z_{\mu }}=\sum _{\nu }{h}_{\nu }\left(x\right)m_{\nu }\left(y\right)$
(b)	$\prod _{i=1}^n\prod _{j=1}^m(1+x_i{y}_j)=\sum _{\lambda }{s}_{\lambda }\left(x\right)s_{\lambda \prime }\left(y\right)=\sum _{\nu }{e}_{\nu }\left(x\right)m_{\nu }\left(y\right)$

		Proof.
	(a) The first equality is proved by Schensted insertion. $$\begin{array}{ccc}\prod _{i=1}^n\prod _{j=1}^m\frac{1}{1-x_i{y}_j}& =& \prod _{i,j}\text{exp}\left(\text{log}\left(\frac{1}{1-x_i{y}_j}\right)\right)\\ & =& \text{exp}(\sum _{i,j}-\text{ln}(1-x_i{y}_j))\\ & =& \text{exp}\left(\sum _{i,j}\sum _{r>0}\frac{x_i^r{y}_j^r}{r}\right),\text{since}-\text{ln}(1-x)=\sum _{r>0}\frac{x^r}{r},\\ & =& \text{exp}\left(\sum _{r>0}\frac{p_r\left(x\right)p_r\left(y\right)}{r}\right),\text{since}\sum _{i,j}{x}_i^r{y}_j^r=p_r\left(x\right)p_r\left(y\right),\\ & =& \prod _{r>0}\text{exp}\left(\frac{p_r\left(x\right)p_r\left(y\right)}{r}\right)\\ & =& \prod _{r>0}\left(\sum _{k\ge 0}\frac{p_r{\left(x\right)}^k{p}_r{\left(y\right)}^k}{r^{k}k!}\right)\end{array}$$ and expanding the product gives $$\prod _{i=1}^n\prod _{j=1}^m\frac{1}{1-x_i{y}_j}=\sum _{\mu }\frac{p_{\mu }\left(x\right)p_{\mu }\left(y\right)}{z_{\mu }}\text{.}$$ In order to obtain the third equality in (a) write $$\prod _{i=1}^n\prod _{j=1}^m\frac{1}{1-x_i{y}_j}=\prod _{j=1}^m\left(\sum _{r\ge 0}{h}_r\left(x\right)y_j^r\right)$$ Then, expanding the product gives $$\prod _{i=1}^n\prod _{j=1}^m\frac{1}{1-x_i{y}_j}=\sum _{\nu }{h}_{\nu }\left(x\right)m_{\nu }\left(y\right)\text{.}$$ $\square$

Define $q_r(x_1,\dots ,x_n;t,q)$ by the generating function $$1+(q-t)\sum _{r>0}{q}_r{z}^r=\prod _{i=1}^n\frac{1-t{x}_{i}z}{1-q{x}_{i}z}\text{.}$$ Then

(a)	$q_r={\displaystyle \sum _{i_1<\cdots <i_k\ge \cdots \ge i_r}{(-t)}^{\text{\#}\hspace{0.17em}(i_j<i_{j+1})}{q}^{\text{\#}\hspace{0.17em}(i_j\le i_{j+1})}{x}_{i_1}\cdots x_{i_r}\text{.}}$
(b)	$q_r={\displaystyle \sum _{i_1\ge \cdots \ge i_r}{(q-t)}^{\text{\#}\hspace{0.17em}(i_j>i_{j+1})}{q}^{\text{\#}\hspace{0.17em}(i_j=i_{j+1})}{x}_{i_1}\cdots x_{i_r}\text{.}}$
(c)	$q_r(x;0,1)=h_r\left(x\right)\text{.}$
(d)	$q_r(x;1,0)={(-1)}^{r-1}{e}_r\left(x\right)\text{.}$
(e)	$q_r(x;1,1)=p_r\left(x\right)\text{.}$
(f)	$q_r\left(x\right)s_{\mu }\left(x\right)={\displaystyle \sum _{\lambda }{s}_{\lambda }\left(x\right)\text{wt}(\lambda /\mu )}$ where the sum is over all $\lambda$ such that $|\lambda /\mu |=r$ and $$\text{wt}(\lambda /\mu )=\sum _Q\text{wt}\left(Q\right),$$ where the sum is over all standard tableaux $Q$ of shape $\lambda /\mu$ and $$\text{wt}\left(Q\right)=\prod _{i=1}^{r-1}{d}_i\left(Q\right),$$ where $$d_i\left(Q\right)=\{\begin{array}{cc}q,& \text{if}\hspace{0.17em}i+1\hspace{0.17em}\text{is northeast of}\hspace{0.17em}i,\\ 0,& \text{if}\hspace{0.17em}i+1\hspace{0.17em}\text{is northwest of}\hspace{0.17em}i\hspace{0.17em}\text{and}\hspace{0.17em}i+2\hspace{0.17em}\text{is southwest of}\hspace{0.17em}i+1,\\ -t,& \text{if}\hspace{0.17em}i+1\hspace{0.17em}\text{is southwest of}\hspace{0.17em}i\text{.}\end{array}$$

(a)	$h_r\left(x\right)s_{\lambda }\left(x\right)=\sum _{\lambda }{s}_{\lambda }\left(x\right),$ where the sum is over all $\lambda$ such that $\lambda /\mu$ is a horizontal strip of length $r\text{.}$
(b)	$e_r\left(x\right)s_{\lambda }\left(x\right)=\sum _{\lambda }{s}_{\lambda }\left(x\right),$ where the sum is over all $\lambda$ such that $\lambda /\mu$ is a vertical strip of length $r\text{.}$
(c)	$p_r\left(x\right)s_{\lambda }\left(x\right)=\sum _{\lambda }{(-1)}^{\text{\# rows}{(\lambda /\mu )}^{-1}}{s}_{\lambda }\left(x\right),$ where the sum is over all $\lambda$ such that $\lambda /\mu$ is a border strip of length $r\text{.}$

The unitary group

The unitary group is the real Lie group $$U_n=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1\}\text{.}$$ Note that $U_n$ is not a complex Lie group in a natural way. The maximal torus in $U_n$ is $$\begin{array}{ccc}T& =& \left\{\text{diagonal matrices in}\hspace{0.17em}{U}_n\right\}\\ & =& \left\{\left(\begin{array}{c}{z}_1\\ & z_2\\ & & \ddots \\ & & & z_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}\left|z_i\right|=1\}\\ & =& \left\{\left(\begin{array}{c}{e}^{i{\theta }_1}\\ & e^{i{\theta }_2}\\ & & \ddots \\ & & & e^{i{\theta }_n}\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}0\le {\theta }_i<2\pi \}\end{array}$$ The Weyl group is $$W=N\left(T\right)/T=S_n=\text{the symmetric group,}$$ since, under conjugation, the eigenvalues of diagonal matrices can only get permuted. Let $$\begin{array}{cccc}{X}^{{\epsilon }_i}:& T& ⟶& {ℂ}^{*}\\ & \left(\begin{array}{c}{z}_1\\ & z_2\\ & & \ddots \\ & & & z_n\end{array}\right)& ⟼& z_i\end{array}\text{.}$$ Then the irreducible representations of $T$ are $$X^{\lambda }=X^{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n}={\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}\cdots {\left(X^{{\epsilon }_n}\right)}^{{\lambda }_n},{\lambda }_i\in \mathbb{Z},$$ and so the weight lattice is $$L=\{{\lambda }_1{\epsilon }_1+{\lambda }_2{\epsilon }_2+\cdots +{\lambda }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}\}\cong {\mathbb{Z}}^n\text{.}$$ Then $${𝔲}_n=\text{Lie}\left(U_n\right)=\{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}x+{\bar{x}}^t=0\}$$ which has basis $\{E_{ij}-E_{ji},i{E}_{ij}+i{E}_{ji}\hspace{0.17em}|\hspace{0.17em}i<j\}\cup \left\{i{E}_{ii}\hspace{0.17em}\right|\hspace{0.17em}1\le i\le n\}$ as a vector space over $ℝ\text{.}$ Then $$𝔤=ℂ{\otimes }_{ℝ}{𝔲}_n\oplus i{𝔲}_n={𝔤𝔩}_n\left(ℂ\right)=M_n\left(ℂ\right)$$ which has basis $\left\{E_{ij}\hspace{0.17em}\right|\hspace{0.17em}1\le i,j\le n\}$ as a vector space over $ℂ\text{.}$ Then $$\begin{array}{ccc}𝔱& =& \text{Lie}\left(T\right)=ℝ\text{-span}\left\{i{E}_{ii}\hspace{0.17em}\right|\hspace{0.17em}1\le i\le n\},\text{and}\\ 𝔥& =& 𝔱\oplus i𝔱=ℂ{\otimes }_{ℝ}𝔱=\left\{\left(\begin{array}{c}{h}_1\\ & h_2\\ & & \ddots \\ & & & h_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{h}_i\in ℂ\}\text{.}\end{array}$$ So $$𝔤=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)\text{where}R=\{{\epsilon }_i-{\epsilon }_j\hspace{0.17em}|\hspace{0.17em}1\le i,j\le n,i\ne j\}\text{and}{𝔤}_{{\epsilon }_i-{\epsilon }_j}=ℂE_{ij},$$ since $$[h,E_{ij}]=h{E}_{ij}-E_{ij}h=(x_i-x_j)E_{ij}=({\epsilon }_i-{\epsilon }_j)\left(h\right)E_{ij}\text{.}$$ Then $$X^{{\epsilon }_k-{\epsilon }_{\ell }}\left(t\right)=z_k{z}_{\ell }^{-1}=e^{i{\theta }_k}{e}^{i{\theta }_{\ell }}=e^{i({\theta }_k-{\theta }_{\ell })}\text{.}$$ So $$\begin{array}{ccc}{L}^{+}& =& \{{\lambda }_1{\epsilon }_1+{\lambda }_2{\epsilon }_2+\cdots +{\lambda }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\}\\ & =& \mathbb{Z}({\epsilon }_1+\cdots +{\epsilon }_n)+\{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_{n-1}{\epsilon }_{n-1}\hspace{0.17em}|\hspace{0.17em}{\lambda }_1\ge \cdots \ge {\lambda }_{n-1}\ge 0\}\\ \\ L^{++}& =& \{{\mu }_1{\epsilon }_1+{\mu }_2{\epsilon }_2+\cdots +{\mu }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\mu }_1>{\mu }_2>\cdots >{\mu }_n\}\\ & =& \mathbb{Z}({\epsilon }_1+\cdots +{\epsilon }_n)+\{{\mu }_1{\epsilon }_1+{\mu }_2{\epsilon }_2+\cdots +{\mu }_{n-1}{\epsilon }_{n-1}\hspace{0.17em}|\hspace{0.17em}{\mu }_1>{\mu }_2>\cdots >{\mu }_{n-1}>0\}\end{array}$$ The bijection $$\begin{array}{ccc}{L}^{+}& ⟶& L^{++}\\ k+\lambda & ⟼& k+\left(\frac{n-1}{2}\right)+\lambda +\delta \end{array}$$ restricted to those elements of $L^{+}$ with $k=0$ is the same as the bijection $$\begin{array}{ccc}𝒫& ⟶& {𝒫}^s\\ \lambda & ⟼& \lambda +\delta & =& ({\lambda }_1+n-1,{\lambda }_2+n-2,\dots ,{\lambda }_{n-1}+1)\end{array}$$ where $$\begin{array}{ccc}𝒫& =& \{\text{partitions with length}\hspace{0.17em}\le n-1\}\\ & =& \left\{({\lambda }_1,\dots ,{\lambda }_{n-1})\hspace{0.17em}\right|\hspace{0.17em}{\lambda }_1\ge \cdots \ge {\lambda }_{n-1}\ge 0\}\end{array}\text{and}\begin{array}{ccc}{𝒫}^s& =& \{\text{strict partitions with length}\hspace{0.17em}\le n-1\}\\ & =& \left\{({\mu }_1,\dots ,{\mu }_{n-1})\hspace{0.17em}\right|\hspace{0.17em}{\mu }_1>\cdots >{\mu }_{n-1}>0\}\end{array}$$ The chamber in $𝔱={ℝ}^n$ can be chosen as $$C=\{(x_1,\dots ,x_n)\in {ℝ}^n\hspace{0.17em}|\hspace{0.17em}{x}_1\ge x_2\ge \cdots \ge x_n\}$$ the hyperplanes are $$H_{{\epsilon }_i-{\epsilon }_j}\{(x_1,\dots ,x_n)\in {ℝ}^n\hspace{0.17em}|\hspace{0.17em}{x}_i=x_j\},1\le i<j\le n\text{.}$$ The positive roots are $$R^{+}=\{{\epsilon }_i-{\epsilon }_j\hspace{0.17em}|\hspace{0.17em}1\le i<j\le n\}\text{and}R=R^{+}\cup (-R^{+})\text{.}$$ Then $$R\left(T\right)\cong \mathbb{Z}L=\mathbb{Z}\text{-span}\left\{e^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in L\}=\mathbb{Z}\text{-span}\left\{e^{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n}\hspace{0.17em}\right|\hspace{0.17em}{\lambda }_i\in \mathbb{Z}\}\text{.}$$ We write $$e^{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n}={\left(e^{{\epsilon }_1}\right)}^{{\lambda }_1}\cdots {\left(e^{{\epsilon }_n}\right)}^{{\lambda }_n}=z_1^{{\lambda }_1}\cdots z_n^{{\lambda }_n}\text{.}$$ Then $${\left(\mathbb{Z}P\right)}^W=\text{symmetric polynomials in the}\hspace{0.17em}{z}_i$$ is the set of polynomials such that $p(z_1,\dots ,z_n)=p(z_{w\left(1\right)},\dots ,z_{w\left(n\right)}),$ for all $w\in S_n\text{.}$ $$\begin{array}{ccc}\rho & =& \frac{1}{2}\sum _{\alpha >0}\alpha =\frac{1}{2}\sum _{1\le i<j\le n}({\epsilon }_i-{\epsilon }_j)\\ & =& \left(\frac{n-1}{2}\right){\epsilon }_1+\left(\frac{n-3}{2}\right){\epsilon }_2+\cdots +\left(\frac{-(n-1)}{2}\right){\epsilon }_n\\ & =& -\left(\frac{n-1}{2}\right)({\epsilon }_1+\cdots +{\epsilon }_n)(n-1){\epsilon }_1+(n-2){\epsilon }_2+\cdots +{\epsilon }_{n-1}+0{\epsilon }_n\end{array}$$ and $$\begin{array}{ccc}{a}_{\rho }& =& {\left(z_1\cdots z_n\right)}^{-(n-1)/2}\sum _{w\in S_n}\epsilon \left(w\right)w\left(z_1^{n-1}{z}_2^{n-2}\cdots z_{n-1}\right)\\ & =& \text{det}\left(\begin{array}{ccccc}{z}_1^{n-1}& z_1^{n-2}& \cdots & z_1& 1\\ z_2^{n-1}& z_2^{n-2}& \cdots & z_2& 1\\ ⋮& & \ddots & & ⋮\\ z_n^{n-1}& z_n^{n-2}& \cdots & z_n& 1\end{array}\right)\text{(the Vandermonde determinant!)}\\ & =& {\left(z_1\cdots z_n\right)}^{-(n-1)/2}\prod _{i<j}(z_i-z_j)\\ & =& z_1^{-(n-1)/2}{z}_2^{-(n-3)/2}\cdots z_n^{(n-1)/2}\prod _{i<j}(z_i{z}_j^{-1}-1)\\ & =& X^{-\rho }\left(t\right)\prod _{\alpha >0}(X^{\alpha }\left(t\right)-1)\text{.}\end{array}$$ Then $$\begin{array}{ccc}{\chi }^{\lambda }& =& \frac{a_{\lambda +\rho }}{a_{\rho }}=\frac{{\sum }_{w\in S_n}\epsilon \left(w\right)w\left(z_1^{{\lambda }_1+n-1}{z}_2^{{\lambda }_2+n-2}\cdots z_{n-1}^{{\lambda }_{n-1}+1}{z}_n^{{\lambda }_n}\right)}{{\sum }_{w\in S_n}\epsilon \left(w\right)w\left(z_1^{n-1}{z}_2^{n-2}\cdots z_{n-1}^1{z}_n^0\right)}\\ & =& \frac{\text{det}\left(z_i^{{\lambda }_j+n-j}\right)}{\text{det}\left(z_i^{n-j}\right)}\text{(the Schur function!).}\end{array}$$

(Kostka 18??) If $({\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n)\in L^{+}$ then $$s_{\lambda }(z_1,\dots ,z_n)={\left(z_1\cdots z_n\right)}^{\lambda }\sum _T{z}^T,$$ where the sum is over all column strict tableaux of shape $\lambda \text{.}$

The Littelmann theorem is the generalization of this theorem to all compact Lie groups.

Think about the derivation of the Weyl dimension formula in the context of Schur functions. $$\begin{array}{ccc}{s}_{\lambda }\left(e^{h\delta }\right)& =& s_{\lambda }(e^{h(n-1)},e^{h(n-2)},\dots ,e^h,1)=s_{\lambda }(q^{(n-1)},q^{(n-2)},\dots ,q,1)\\ & =& \frac{\text{det}\left({\left(q^{n-i}\right)}^{{\lambda }_j+n-j}\right)}{\text{det}\left({\left(q^{n-i}\right)}^{n-j}\right)}=\frac{\text{det}\left({\left(q^{{\lambda }_j+n-j}\right)}^{n-i}\right)}{\text{det}\left({\left(q^{n-i}\right)}^{n-j}\right)}=\frac{a_{\delta }(q^{{\lambda }_1+n-1},\dots ,q^{{\lambda }_n+n-n})}{a_{\delta }(q^{n-1},\dots ,q^{n-n})}\\ & =& \prod _{i<j}\frac{q^{{\lambda }_j+n-j}-q^{{\lambda }_i+n-i}}{q^{n-j}-q^{n-i}}=q^{\sum (i-1){\lambda }_i}\left(\prod _{i<j}\frac{1-q^{({\lambda }_j+n-j)-({\lambda }_i+n-i)}}{q^{(n-j)-(n-i)}}\right)\\ & =& q^{n\left(\lambda \right)}\frac{{\prod }_{i\ge 1}{\prod }_{k=1}^{{\lambda }_i+n-i}(1-q^k)}{({\prod }_{b\in \lambda }1-q^{h\left(b\right)}){\prod }_{i<j}(1-q^{i-j})}=q^{n\left(\lambda \right)}\prod _{b\in \lambda }\frac{1-q^{n+c\left(b\right)}}{1-q^{h\left(b\right)}}\end{array}$$ For example, if $\lambda =5{\epsilon }_1+4{\epsilon }_2+4{\epsilon }_3+3{\epsilon }_4+2{\epsilon }_5,$ $$\begin{array}{c} 2 3 5 7 9 8 6 4 1 λ1+n-1=9 1 3 5 7 6 4 2 λ2+n-2=7 2 4 6 5 3 1 λ3+n-3=6 2 4 3 1 λ4+n-4=4 2 1 λ5+n-5=2 \end{array}$$ Then $$\begin{array}{c} 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 1 2 3 4 5 6 1 2 3 4 1 2 \end{array}$$ So $$\text{dim}\left(U^{\lambda }\right)=\frac{5·6·7·8·9·4·5·6·7·3·4·5·6·2·3·4·1·2}{9·8·6·4·1·7·6·4·2·5·4·3·1·4·3·1·2·1}=\frac{5·7·5·6}{6}=5·7·5=175\text{.}$$

The general linear group ${GL}_n\left(ℝ\right)$

$${GL}_n\left(ℝ\right)=\left\{\text{invertible matrices in}\hspace{0.17em}{M}_n\left(ℝ\right)\right\}\text{.}$$ Note that ${GL}_n\left(ℝ\right)$ is dense in $M_nℝ\text{.}$ Writing $g={\left(g_{k\ell }\right)}_{1\le k,\ell \le n}$ for matrices in ${GL}_n\left(ℝ\right)$ let $f_{k\ell }$ be the coordinate functions on ${GL}_n\left(ℝ\right),$ $$\begin{array}{cc}\begin{array}{cccc}{f}_{k\ell }:& {GL}_n\left(ℝ\right)& ⟶& ℝ\\ & g& ⟼& g_{k\ell }\end{array}\text{and}{X}_{k\ell }=\left(\frac{\partial }{{\partial }_{f_{k\ell }}}{|}_{g=1}\right),& \text{(3.1)}\end{array}$$ the corresponding basis of tangent vectors (derivatives on $G\text{)}$ in the tangent space $T_1\left(G\right)\text{.}$ Let $E_{ij}$ denote the matrix with a $1$ in the $(i,j)\text{th}$ entry and $0$ everywhere else. Then one parameter subgroups corresponding to the tangent vectors in ??? are $$\begin{array}{cccc}{\gamma }_{ij}:& ℝ& ⟶& {GL}_n\left(ℝ\right)\\ & t& ⟼& (\text{Id}+t{E}_{ij})& =& \left(\begin{array}{c}1\\ & \ddots & t\\ & & \ddots \\ & & & \ddots \\ & & & & 1\end{array}\right)\end{array}\text{and}$$ $$\begin{array}{cccc}{\gamma }_{ii}:& ℝ& ⟶& {GL}_n\left(ℝ\right)\\ & t& ⟼& \left(\begin{array}{c}1\\ & \ddots \\ & & 1\\ & & & e^t\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\end{array}\right)\end{array}$$ since $$\left(\begin{array}{c}1\\ & \ddots & s\\ & & \ddots \\ & & & \ddots \\ & & & & 1\end{array}\right)\left(\begin{array}{c}1\\ & \ddots & t\\ & & \ddots \\ & & & \ddots \\ & & & & 1\end{array}\right)=\left(\begin{array}{c}1\\ & \ddots & s+t\\ & & \ddots \\ & & & \ddots \\ & & & & 1\end{array}\right)$$ and $$\left(\begin{array}{c}1\\ & \ddots \\ & & 1\\ & & & e^s\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\end{array}\right)\left(\begin{array}{c}1\\ & \ddots \\ & & 1\\ & & & e^t\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\end{array}\right)=\left(\begin{array}{c}1\\ & \ddots \\ & & 1\\ & & & e^{s+t}\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\end{array}\right)\text{.}$$ In general, the vector field given by $${\delta }_X=\sum _{i,j=1}^n{x}_{ij}\frac{\partial }{\partial f_{ij}},x_{ij}\in ℝ,$$ corresponds to the one parameter subgroup $$\begin{array}{cccc}{\gamma }_X:& ℝ& ⟶& {GL}_n\left(ℝ\right)\\ & t& ⟼& e^{tX}=\sum _{k\ge 0}{t}^k\frac{X^k}{k!},\end{array}$$ where $X=\left(x_{ij}\right)$ is the corresponding matrix in $M_n\left(ℝ\right)\text{.}$ In particular, $$e^{t{E}_{ij}}=\left(\begin{array}{c}1\\ & \ddots \\ & & \ddots \\ & & & 1\end{array}\right)+t\left(\begin{array}{c}1\\ & \ddots & 1\\ & & \ddots \\ & & & 1\end{array}\right)+\frac{t^2}{2!}{\left(\begin{array}{c}1\\ & \ddots & 1\\ & & \ddots \\ & & & 1\end{array}\right)}^2=\left(\begin{array}{c}1\\ & \ddots & t\\ & & \ddots \\ & & & 1\end{array}\right)$$ and $$e^{t{E}_{ii}}=\left(\begin{array}{c}1\\ & \ddots \\ & & 1\\ & & & e^t\\ & & & & 1\\ & & & & & \ddots \\ & & & & & & 1\end{array}\right)$$ as they should. If $X=\left(x_{ij}\right)$ is a matrix in $M_n\left(ℝ\right)$ and $d_X$ is the corresponding tangent vector $d_X\in T_1\left(G\right)$ then $$\begin{array}{cc}{d}_X{f}_{k\ell }=\sum _{i,j=1}^n{x}_{ij}\frac{\partial }{\partial f_{ij}}{f}_{k\ell }{|}_{g=1}=x_{k\ell }\text{.}& \text{(3.2)}\end{array}$$ If ${\partial }_X$ is the corresponding left invariant vector field $$\begin{array}{cc}\begin{array}{ccc}\left({\partial }_X{f}_{k\ell }\right)\left(g\right)& =& \left(L_{g^{-1}}{\partial }_X{f}_{k\ell }\right)\left(1\right)=\left({\partial }_X{L}_{g^{-1}}{f}_{k\ell }\right)\left(1\right)\\ & =& d_X\left(L_{g^{-1}}{f}_{k\ell }\right)=d_X\left(\sum _{j=1}^n{g}_{kj}{f}_{j\ell }\right)=\sum _{j=1}^n{g}_{kj}{x}_{j\ell }=\sum _{j=1}^n{x}_{j\ell }{f}_{kj}\left(g\right)\end{array}& \text{(3.3)}\end{array}$$ since $$\begin{array}{cc}\left(L_{g^{-1}}{f}_{k\ell }\right)\left(h\right)=f_{k\ell }\left(gh\right)={\left(gh\right)}_{k\ell }=\sum _{j=1}^n{g}_{kj}{h}_{j\ell }=\sum _{j=1}^n{g}_{kj}{f}_{j\ell }\left(h\right)& \text{(3.4)}\end{array}$$ for all $h\in G\text{.}$ We can express formula ??? more concisely as $${\partial }_{X}f=fX,\text{where}\hspace{0.17em}f=\left(f_{k\ell }\right)\hspace{0.17em}\text{and}\hspace{0.17em}X=\left(x_{ij}\right)\in M_n\left(ℝ\right)\text{.}$$ It follows that $[{\partial }_X,{\partial }_Y]f=({\partial }_X{\partial }_Y-{\partial }_Y{\partial }_X)f=f(XY-YX)$ and so we may identify $𝔤=\text{Lie}\left({GL}_n\left(ℝ\right)\right)$ with the Lie algebra $$M_n\left(ℝ\right)\text{with bracket}[X,Y]=XY-YX,\text{for}\hspace{0.17em}X,Y\in M_n\left(ℝ\right)\text{.}$$ If $g\in G$ and $X\in M_n\left(ℝ\right)$ then $$\left({\text{Ad}}_g{\partial }_X\right)f=R_g{\partial }_X{R}_{g^{-1}}f=fgX{g}^{-1},\text{where}\hspace{0.17em}f=\left(f_{k\ell }\right)\text{.}$$ Thus, for $G={GL}_n\left(ℝ\right)$ with $𝔤=\text{Lie}\left(G\right)$ identified with $M_n\left(ℝ\right),$ $$\begin{array}{cc}\begin{array}{cccc}{\text{Ad}}_g:& M_n\left(ℝ\right)& ⟶& M_n\left(ℝ\right)\\ & X& ⟼& gX{g}^{-1},\end{array}& \text{for all}\hspace{0.17em}g\in {GL}_n\left(ℝ\right),\text{and}\\ \begin{array}{cccc}{\text{ad}}_X:& M_n\left(ℝ\right)& ⟶& M_n\left(ℝ\right)\\ & Y& ⟼& [X,Y]& =& XY-YX,\end{array}& \text{for all}\hspace{0.17em}X,Y\in M_n\left(ℝ\right),\end{array}$$

The special unitary group

The special unitary group is the real Lie group $$S{U}_n=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)=1,g{\bar{g}}^t=1\}\text{.}$$ A maximal torus in $S{U}_n$ is $$T=\left\{\text{diagonal matrices in}\hspace{0.17em}S{U}_n\right\}\text{and}W=N\left(T\right)/T=S_n=\text{the symmetric group,}$$ is the Weyl group. Then $$\begin{array}{ccc}𝔰{𝔲}_n& =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}x\in {\bar{x}}^t=0,\hspace{0.17em}\text{tr}\left(x\right)=0\}\text{and}\\ {𝔰𝔩}_n\left(ℂ\right)& =& 𝔰{𝔲}_n\oplus i𝔰{𝔲}_n=\{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{tr}\left(x\right)=0\}\text{.}\end{array}$$ Then $𝔤=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)$ where $$\begin{array}{ccc}R& =& \{{\epsilon }_i-{\epsilon }_j\hspace{0.17em}|\hspace{0.17em}1\le i,j\le n,i\ne j\},\\ {𝔤}_{{\epsilon }_i-{\epsilon }_j}& =& ℂE_{ij},\end{array}$$ and $$𝔥=𝔱\oplus i𝔱=ℂ{\otimes }_{ℝ}𝔱=\left\{\left(\begin{array}{c}{h}_1\\ & h_2\\ & & \ddots \\ & & & h_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{h}_i\in ℂ,\hspace{0.17em}\sum _{i=1}^n{h}_i=0\}\cong {ℝ}^n,$$ The lattice of representations of $T$ is $$P=\{\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\lambda }_1+\dots +{\lambda }_n=0,{\lambda }_i\in \mathbb{Z}\}\cong {\mathbb{Z}}^{n-1}\text{.}$$

The special orthogonal groups

Let $$SO(2r+1,ℝ)=\{g\in M_{2r+1}\left(ℝ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1,\text{det}\left(g\right)=1\}\text{.}$$ A maximal torus of $SO(2r+1,ℝ)$ is $$T=\left\{\left(\begin{array}{c}\left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_1& \text{sin}\hspace{0.17em}{\theta }_1\\ -\text{sin}\hspace{0.17em}{\theta }_1& \text{cos}\hspace{0.17em}{\theta }_1\end{array}\right)\\ & \ddots \\ & & \left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_r& \text{sin}\hspace{0.17em}{\theta }_r\\ -\text{sin}\hspace{0.17em}{\theta }_r& \text{cos}\hspace{0.17em}{\theta }_r\end{array}\right)\\ & & & 1\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}0\le {\theta }_i<2\pi \}\text{.}$$ The Weyl group $$W=N\left(T\right)/T=W{B}_r=\left\{\genfrac{}{}{0ex}{}{r\times r\hspace{0.17em}\text{matrices with exactly one nonzero entry in each}}{\text{row and each column, and nonzero entries}\hspace{0.17em}\pm 1}\right\}$$ and is generated by permuting the blocks of a matrix in $T$ and the transformations $$\left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_j& \text{sin}\hspace{0.17em}{\theta }_j\\ -\text{sin}\hspace{0.17em}{\theta }_j& \text{cos}\hspace{0.17em}{\theta }_j\end{array}\right)⟶\left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_j& -\text{sin}\hspace{0.17em}{\theta }_j\\ \text{sin}\hspace{0.17em}{\theta }_j& \text{cos}\hspace{0.17em}{\theta }_j\end{array}\right)$$ which come from conjugation by the block $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\text{.}$ The Lie algebra $$\begin{array}{ccc}𝔲=𝔰𝔬(2r+1,ℝ)& =& \{x\in M_{2r+1}\left(ℝ\right)\hspace{0.17em}|\hspace{0.17em}x+x^t=0\hspace{0.17em}\text{and}\hspace{0.17em}\text{tr}\left(x\right)=0\}\\ & =& \{x\in M_{2r+1}\left(ℝ\right)\hspace{0.17em}|\hspace{0.17em}x+x^t=0\}\end{array}$$ has basis $\{E_{ij}-E_{ji}\hspace{0.17em}|\hspace{0.17em}1\le i<j\le r\}$ and $$𝔥=ℝ\text{-span}\{E_{2i-1,2i}-E_{2i,2i-1}\hspace{0.17em}|\hspace{0.17em}1\le i\le r\}\text{.}$$ Note that $$e^{\left(\begin{array}{cc}0& \theta \\ -\theta & 0\end{array}\right)}=\left(\begin{array}{cc}\text{cos}\hspace{0.17em}\theta & \text{sin}\hspace{0.17em}\theta \\ -\text{sin}\hspace{0.17em}\theta & \text{cos}\hspace{0.17em}\theta \end{array}\right)\text{.}$$ Then $$\begin{array}{ccc}G& =& {SO}_{2r+1}\left(ℂ\right)=\{A\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}A{A}^t=1,\text{det}\left(A\right)=1\},\\ \\ 𝔤& =& {𝔰𝔬}_{2r+1}\left(ℂ\right)=\{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}A{A}^t=1,\text{det}\left(A\right)=1\},\\ & =& 𝔲\oplus i𝔲=ℂ{\otimes }_{ℝ}𝔲=ℂ\text{-span}\{E_{ij}-E_{ji}\hspace{0.17em}|\hspace{0.17em}1\le i<j\le n\},\\ \\ 𝔥& =& 𝔱\oplus i𝔱=ℂ\text{-span}\{E_{2i-1,2i}-E_{2i,2i-1}\hspace{0.17em}|\hspace{0.17em}1\le i\le r\}\text{.}\end{array}$$ Let $$\begin{array}{ccc}{X}_{{\epsilon }_i-{\epsilon }_j}=\left(\begin{array}{cc}0& \left(\begin{array}{cc}1& i\\ -i& 1\end{array}\right)\\ \left(\begin{array}{cc}-1& i\\ -i& -1\end{array}\right)& 0\end{array}\right)& & X_{-{\epsilon }_i+{\epsilon }_j}=\left(\begin{array}{cc}0& \left(\begin{array}{cc}1& -i\\ -i& 1\end{array}\right)\\ \left(\begin{array}{cc}-1& -i\\ -i& -1\end{array}\right)& 0\end{array}\right)\\ X_{{\epsilon }_i+{\epsilon }_j}=\left(\begin{array}{cc}0& \left(\begin{array}{cc}1& -i\\ -i& -1\end{array}\right)\\ \left(\begin{array}{cc}-1& i\\ i& 1\end{array}\right)& 0\end{array}\right)& & X_{-{\epsilon }_i-{\epsilon }_j}=\left(\begin{array}{cc}0& \left(\begin{array}{cc}1& i\\ i& -1\end{array}\right)\\ \left(\begin{array}{cc}-1& -i\\ -i& 1\end{array}\right)& 0\end{array}\right)\\ X_{{\epsilon }_i}=\left(\begin{array}{cc}0& \left(\begin{array}{c}1\\ -i\end{array}\right)\\ \left(\begin{array}{cc}-1& i\end{array}\right)& 0\end{array}\right)& & X_{-{\epsilon }_i}=\left(\begin{array}{cc}0& \left(\begin{array}{c}1\\ i\end{array}\right)\\ \left(\begin{array}{cc}-1& -i\end{array}\right)& 0\end{array}\right)\end{array}$$ where, in the first four matrices, the $i\text{th}$ and $j\text{th}$ blocks are shown and, in the last two, the $i\text{th}$ block and the $2r+1$ row are shown. This is yucky, typeset this into the matrices as $i\text{th}$ block, $j\text{th}$ block $\dots$ Then $$𝔤=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)\text{where}{𝔤}_{\alpha }=ℂX_{\alpha }\text{and}R=\{\pm ({\epsilon }_i-{\epsilon }_j),\pm ({\epsilon }_i+{\epsilon }_j),\pm {\epsilon }_i\}\text{.}$$

Let $$SO(2r,ℝ)=\{g\in M_{2r+1}\left(ℝ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1,\text{det}\left(g\right)=1\}\text{.}$$ A maximal torus of $SO(2r+1,ℝ)$ is $$T=\left\{\left(\begin{array}{c}\left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_1& \text{sin}\hspace{0.17em}{\theta }_1\\ -\text{sin}\hspace{0.17em}{\theta }_1& \text{cos}\hspace{0.17em}{\theta }_1\end{array}\right)\\ & \ddots \\ & & \left(\begin{array}{cc}\text{cos}\hspace{0.17em}{\theta }_r& \text{sin}\hspace{0.17em}{\theta }_r\\ -\text{sin}\hspace{0.17em}{\theta }_r& \text{cos}\hspace{0.17em}{\theta }_r\end{array}\right)\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}0\le {\theta }_i<2\pi \}\text{.}$$ The Weyl group $$W=N\left(T\right)/T=W{D}_r=\left\{\begin{array}{c}r\times r\hspace{0.17em}\text{matrices with exactly one nonzero entry in each}\\ \text{row and each column, nonzero entries}\hspace{0.17em}\pm 1,\\ \text{and an even number of}\hspace{0.17em}-1\text{s}\end{array}\right\}$$ and is generated by permuting the blocks of a matrix in $T$ and conjugation by the matrices $$\left(\begin{array}{c}\begin{array}{cc}0& 1\\ 1& 0\end{array}\\ & \begin{array}{cc}0& 1\\ 1& 0\end{array}\end{array}\right)\text{.}$$ The rest of the story is exactly as in the $SO(2r+1,ℝ)$ case except that $$𝔤=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)\text{where}{𝔤}_{\alpha }=ℂX_{\alpha }\text{and}R=\{\pm ({\epsilon }_i-{\epsilon }_j),\pm ({\epsilon }_i+{\epsilon }_j)\}\text{.}$$

The symplectic group ${Sp}_{2r}$

Let $$\begin{array}{ccc}U={Sp}_{2r}& =& \{g\in U_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{g}^{t}Jg=J\}\\ & =& \{g\in U_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}J\bar{g}=aJ\}\\ & =& \{g\in U_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g=\left(\begin{array}{cc}x& y\\ -\bar{y}& \bar{x}\end{array}\right)\}\end{array}\text{where}J=\left(\begin{array}{cccc}& & & 1\\ & 0& & & \ddots \\ & & & & & 1\\ -1\\ & \ddots & & & 0\\ & & -1\end{array}\right)\text{.}$$ A maximal torus in ${Sp}_{2r}$ is $$T=\left\{\left(\begin{array}{c}{z}_1\\ & \ddots & & & 0\\ & & z_r\\ & & & {\bar{z}}_1\\ & 0& & & \ddots \\ & & & & & {\bar{z}}_r\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{z}_i\in {ℂ}^{*},\left|z_i\right|=1\}$$ and the Weyl group is $$W=N\left(T\right)/T=W{C}_r=\left\{\genfrac{}{}{0ex}{}{r\times r\hspace{0.17em}\text{matrices with exactly one nonzero entry in each}}{\text{row and each column, and nonzero entries}\hspace{0.17em}\pm 1}\right\}$$ and is generated by permuting the $z_i$ and the transformations $$\left(\begin{array}{c}{z}_1\\ & \ddots & & & 0\\ & & z_r\\ & & & {\bar{z}}_1\\ & 0& & & \ddots \\ & & & & & {\bar{z}}_r\end{array}\right)⟶\left(\begin{array}{cc}\begin{array}{c}{\bar{z}}_1\\ & z_2\\ & & \ddots \\ & & & z_r\end{array}& 0\\ 0& \begin{array}{c}{z}_1\\ & {\bar{z}}_2\\ & & \ddots \\ & & & {\bar{z}}_r\end{array}\end{array}\right)$$ which comes from conjugation by the matrix $$\left(\begin{array}{cc}\begin{array}{c}0\\ & 1\\ & & \ddots \\ & & & 1\end{array}& \begin{array}{c}1\\ & 0\\ & & \ddots \\ & & & 0\end{array}\\ \begin{array}{c}1\\ & 0\\ & & \ddots \\ & & & 0\end{array}& \begin{array}{c}0\\ & 1\\ & & \ddots \\ & & & 1\end{array}\end{array}\right)\text{.}$$ The Lie algebra $$\begin{array}{ccc}𝔲& =& \{x\in {𝔲}_n\hspace{0.17em}|\hspace{0.17em}{x}^{t}J+Jx=0\}\\ & =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{\bar{x}}^t+x=0,x^{t}J+Jx=0\}\\ & =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{\bar{x}}^t+x=0,-xJ+J\bar{x}=0\}\\ & =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}x=\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right),{\bar{a}}^t=-a,\hspace{0.17em}b=b^t\}\end{array}$$ has basis $$\left\{\begin{array}{c}{E}_{ij}+E_{i+n,J+n},\\ -E_{ji}-E_{j+n,i+n}\end{array}\hspace{0.17em}\right|\hspace{0.17em}1\le i<j\le n\}\bigcup \{E_{ij}-E_{ji}\hspace{0.17em}|\hspace{0.17em}\genfrac{}{}{0ex}{}{1\le i\le n,}{n+1\le j\le 2n}\}\text{.}$$ Then $$\begin{array}{ccc}𝔤& =& 𝔲\oplus i𝔲=ℂ{\otimes }_{ℝ}𝔲={𝔰𝔭}_{2r}\left(ℂ\right)\\ & =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{x}^{t}J+Jx=0\}\\ & =& \{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}x=\left(\begin{array}{cc}a& b\\ c& -a^t\end{array}\right),\hspace{0.17em}b=b^t,\hspace{0.17em}c=c^t\}\end{array}$$ and $𝔤$ has basis $$\begin{array}{ccccc}{X}_{{\epsilon }_i-{\epsilon }_j}=E_{ij}-E_{i+n,j+n},& & X_{{\epsilon }_j-{\epsilon }_i}=E_{ji}-E_{j+n,i+n},& & 1\le i<j\le n,\\ X_{{\epsilon }_i+{\epsilon }_j}=E_{i,j+n}+E_{j,i+n},& & X_{-{\epsilon }_i-{\epsilon }_j}=E_{i+n,j}+E_{j+n,i},& & 1\le i<j\le n,\\ X_{2{\epsilon }_k}=E_{k,k+m},& & X_{-2{\epsilon }_k}=E_{k+n,k},& & 1\le k\le n\text{.}\end{array}$$ Then $$\begin{array}{ccc}𝔱& =& \left\{\left(\begin{array}{cc}\begin{array}{c}i{h}_1\\ & \ddots \\ & & i{h}_r\end{array}& 0\\ 0& \begin{array}{c}-i{h}_1\\ & \ddots \\ & & -i{h}_r\end{array}\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{h}_i\in ℝ\}\\ 𝔥& =& 𝔱\oplus i𝔱=ℂ{\otimes }_{ℝ}𝔱=ℂ\text{-span}\{E_{ii}-E_{i+n,i+n}\hspace{0.17em}|\hspace{0.17em}1\le i\le r\}\text{.}\end{array}$$ Then $$𝔤=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)\text{where}{𝔤}_{\alpha }=ℂX_{\alpha }\text{and}R=\{\pm ({\epsilon }_i-{\epsilon }_j),\pm ({\epsilon }_i+{\epsilon }_j),\pm 2{\epsilon }_i\}\text{.}$$

Examples of Lie groups

${GL}_n\left(ℝ\right)$
${SL}_n\left(ℝ\right)$
$O_n\left(ℝ\right)$
$S{O}_n\left(ℝ\right)$
${SP}_{2n}$
$U_n=\{\left(\begin{array}{c}1\\ & \ddots & *\\ & & \ddots \\ & & & 1\end{array}\right)\in {GL}_n\left(ℝ\right)\}\text{.}$
$B_n=\{\left(\begin{array}{c}*\\ & \ddots & *\\ & & \ddots \\ & & & *\end{array}\right)\in {GL}_n\left(ℝ\right)\}\text{.}$
$T_n=\{\left(\begin{array}{c}*\\ & \ddots \\ & & *\end{array}\right)\in {GL}_n\left(ℝ\right)\}\text{.}$
Any closed subgroup of ${GL}_n\left(ℝ\right)\text{.}$
Any finite group.
$S^1=\left\{e^{2\pi ix}\hspace{0.17em}\right|\hspace{0.17em}x\in ℝ\}=ℝ/\mathbb{Z}\text{.}$
$S^1\times \cdots \times S^1=$ torus.
$U_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1\}\text{.}$

Complex groups

${GL}_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)\ne 0\}\text{.}$
${SL}_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)=1\}\text{.}$
${SO}_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1,\hspace{0.17em}\text{det}\left(g\right)=1\}\text{.}$
${SP}_{2n}\left(ℂ\right)=\{g\in M_{2n}\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}gJ{g}^t=J\}\text{.}$
$O_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1\}\text{.}$

Compact subgroups

${SO}_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1,\hspace{0.17em}\text{det}\left(g\right)=1\}\text{.}$
$U_n=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1\}\text{.}$
${SU}_n=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1,\hspace{0.17em}\text{det}\left(g\right)=1\}\text{.}$
${SP}_{2n}=\{g\in M_{2n}\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1,\hspace{0.17em}gJ{g}^t=J\}\text{.}$

Real noncompact groups

${GL}_n\left(ℝ\right)$
${SL}_n\left(ℝ\right)$
${SL}_n\left(ℍ\right)$
${SO}_n\left(ℝ\right)$
$SO{(m,n)}^{\circ }$
$SU(m,n)$
$Sp(m,n)$
${SO}^{*}\left(2n\right)$

Nilpotent groups

Heisenberg groups
Circle groups
Tori

Covers

Spin groups
Pin groups
Metaplectic groups

Real reductive groups

${GL}_n\left(ℝ\right)$
${SL}_n\left(ℝ\right)$
${GL}_n\left(ℂ\right)$ (as a subgroup of $GL(2n,ℝ)\text{)}$
${SL}_n\left(ℂ\right)$
$O(p,q)$
$SO(p,q)$
$U(p,q)$
$SU(p,q)$
$Sp(2n,ℝ)$

Linear algebraic groups

${GL}_n\left(K\right)$
${SL}_n\left(K\right)$
$O_n\left(K\right)$
${SO}_n\left(K\right)$
${Sp}_{2n}\left(K\right)$
${𝔾}_a$
${𝔾}_m$
Any finite subgroup of ${GL}_n\left(K\right)$
$T_n=\{\left(\begin{array}{c}{x}_1\\ & \ddots \\ & & x_n\end{array}\right)\in {GL}_n\left(K\right)\}\text{.}$
$B_n$ the group of upper triangular matrices in ${GL}_n\left(K\right)$
$U_n$ the group of upper unitriangular matrices in ${GL}_n\left(K\right)$

$K$ could be an algebraically closed field of characteristic $p,$
a $p\text{-adic}$ field,
a locally compact topological field,
a local field,
a global field.

Finite Chevalley groups

${GL}_n\left({𝔽}_q\right)$
${SL}_n\left({𝔽}_q\right)$
$O_n\left({𝔽}_q\right)$
${SO}_n\left({𝔽}_q\right)$
${Sp}_{2n}\left({𝔽}_q\right)$
where ${𝔽}_q$ is a field with $q=p^k$ elements.

Notes and references

This is a typed exert of Representation theory Lecture notes: Chapter 5 by Arun Ram. Research supported in part by National Science Foundation grant DMS-9622985.

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