Last update: 18 July 2013

Presently we will prove:

*Theorem 24:* Let $W$ be a finite reflection group on a real space $V$ of finite dimension
$\ell ,S$ the algebra of polynomials on $V,$
$I\left(S\right)$ the subalgebra of invariants under $W\text{.}$ Then:

(a) | $I\left(S\right)$ is generated by $\ell $ homogeneous algebraically independent elements ${I}_{1},\dots ,{I}_{\ell}\text{.}$ |

(b) | The degrees of the ${I}_{j}\text{'s,}$ say ${d}_{1},\dots ,{d}_{\ell},$ are uniquely determined and satisfy $\underset{j}{\Sigma}({d}_{j}-1)=N,$ the number of positive roots. |

(c) | For the irreducible Weyl groups the ${d}_{i}\text{'s}$ are as follows: $$\begin{array}{cc}W& {d}_{i}\text{'s}\\ \begin{array}{c}{A}_{\ell}\\ {B}_{\ell},{C}_{\ell}\\ {D}_{\ell}\\ {E}_{6}\\ {E}_{7}\\ {E}_{8}\\ {F}_{4}\\ {G}_{2}\end{array}& \begin{array}{c}2,3,\dots ,\ell +1\\ 2,4,\dots ,2\ell \\ 2,4,\dots ,2\ell -2,\ell \\ 2,5,6,8,9,12\\ 2,6,8,10,12,14,18\\ 2,8,12,14,18,20,24,30\\ 2,6,8,12\\ 2,6\end{array}\end{array}$$ |

Our main goal is:

*Theorem 25:*

(a) | Let $G$ be a universal Chevalley group over a field $k$ of $q$ elements and the ${d}_{i}\text{'s}$ as in Theorem 24. Then $\left|G\right|={q}^{N}\underset{i}{\Pi}({q}^{{d}_{i}}-1)$ with $N=\Sigma ({d}_{i}-1)=$ the number of positive roots. |

(b) | If $G$ is simple instead, then we have to divide by $c=\left|\text{Hom}({L}_{1}/{L}_{0},{k}^{*})\right|,$ given as follows: $$\begin{array}{ccccccccc}G& {A}_{\ell}& {B}_{\ell},{C}_{\ell}& {D}_{\ell}& {E}_{6}& {E}_{7}& {E}_{8}& {F}_{4}& {G}_{2}\\ c& (\ell +1,q-1)& (2,q-1)& (4,{q}^{\ell}-1)& (3,q-1)& (2,q-1)& 1& 1& 1\end{array}$$ |

*Remark:* We see that the groups of type ${B}_{\ell}$ and ${C}_{\ell}$
have the same order. If $\ell =2$ the root systems are isomorphic so the groups are isomorphic. We will show later that if
$\ell \ge 3$ the groups are isomorphic if and only if $q$ is even.

The proof of Theorem 25 depends on the following identity.

*Theorem 26:*
Let $W$ and the ${d}_{i}\text{'s}$ be as in Theorem 24 and $t$
an indeterminate. Then
$\underset{w\in W}{\Sigma}{t}^{N\left(w\right)}=\underset{i}{\Pi}(1-{t}^{{d}_{i}})(1-t)\text{.}$

We show first that Theorem 25 is a consequence of Theorems 24 and 26.

*Lemma 54:* If $G$ is as in Theorem 25(a) then
$\left|G\right|={q}^{N}{(q-1)}^{\ell}\underset{w\in W}{\Sigma}{q}^{N\left(w\right)}\text{.}$

Proof. | |

Recall that, by Theorems 4 and 4', $G=\bigcup _{w\in W}BwB$ (disjoint) and $BwB=UHw{U}_{w}$ with uniqueness of expression. Hence $\left|G\right|=\left|U\right|\left|H\right|\xb7\underset{w\in W}{\Sigma}\left|{U}_{w}\right|\text{.}$ Now by Corollary 1 to the proposition of §3, $\left|U\right|={q}^{N}$ and $\left|{U}_{w}\right|={q}^{N\left(w\right)}\text{.}$ By Lemma 28, $\left|H\right|={(q-1)}^{\ell}\text{.}$ $\square $ |

*Corollary:* $U$ is a $p\text{-Sylow}$ subgroup of $G,$
if $p$ denotes the characteristic of $k\text{.}$

Proof. | |

$p|{q}^{N\left(w\right)}$ unless $N\left(w\right)=0\text{.}$ Since $N\left(w\right)=0$ if and only if $w=1,$ $p\nmid \Sigma {q}^{N\left(w\right)}\text{.}$ $\square $ |

Proof of Theorem 25. | |

(a) follows from Lemma 54 and Theorem 26. (b) follows from the fact that the center of the universal group is isomorphic to $\text{Hom}({L}_{1}/{L}_{0},{k}^{*})$ and the values of ${L}_{1}/{L}_{0}$ found in §3. $\square $ |

Before giving general proofs of Theorems 24 and 26 we give independent (case by case) verifications of Theorems 24 and 26 for the classical groups.

*Theorem 24: Type ${A}_{\ell}\text{:}$* Here
$W\cong {S}_{\ell +1}$ permuting
$\ell +1$ linear functions ${\omega}_{1},\dots ,{\omega}_{\ell +1}$
such that ${\sigma}_{1}=\Sigma {\omega}_{i}=0\text{.}$
In this case the elementary symmetric polynomials ${\sigma}_{2},\dots ,{\sigma}_{\ell +1}$
are invariant and generate all other polynomials invariant under $W\text{.}$

*Types ${B}_{\ell},{C}_{\ell}\text{:}$*
Here $W$ acts relative to a suitable basis ${\omega}_{1},\dots ,{\omega}_{\ell}$
by all permutations and sign changes. Here the elementary symmetric polynomials in ${\omega}_{1}^{2},\dots ,{\omega}_{\ell}^{2}$
are invariant and generate all other polynomials invariant under $W\text{.}$

*Type ${D}_{\ell}\text{:}$* Here only an even number of sign changes
can occur. Thus we can replace the last of the invariants for
${B}_{\ell},{\omega}_{1}^{2}\dots {\omega}_{\ell}^{2}$
by ${\omega}_{1}\dots {\omega}_{\ell}\text{.}$

*Theorem 26: Type ${A}_{\ell}\text{:}$* Here
$W\cong {S}_{\ell +1}$ and $N\left(w\right)$
is the number of inversions in the sequence
$(w\left(1\right),\dots ,w(\ell +1))\text{.}$
If we write ${P}_{\ell}\left(t\right)=\sum _{w\in W\cong {S}_{\ell +1}}{t}^{N\left(w\right)}$
then ${P}_{\ell +1}\left(t\right)={P}_{\ell}\left(t\right)(1+t+{t}^{2}+\dots +{t}^{\ell +1}),$
as we see by considering separately the $\ell +2$ values that
$w(\ell +2)$ can take on. Hence the formula
${P}_{\ell}\left(t\right)=\underset{j=2}{\overset{\ell +1}{\Pi}}(1-{t}^{j})/(1-t)$
follows by induction.

*Exercise:* Prove the corresponding formulas for types
${B}_{\ell},{C}_{\ell}$ and
${D}_{\ell}\text{.}$ Here the proof is similar, the induction step being a bit more complicated.

Part (a) of Theorem 24 follows from:

*Theorem 27:* Let $G$ be a finite group of automorphisms of a real vector space $V$ of finite dimension
$\ell $ and $I$ the algebra of polynomials on $V$ invariant under
$G\text{.}$ Then:

(a) | If $G$ is generated by reflections, then $I$ is generated by $\ell $ algebraically independent homogeneous elements (and 1). |

(b) | Conversely, if $I$ is generated by $\ell $ algebraically independent homogeneous elements (and 1) then $G$ is generated by reflections. |

*Example:* Let $\ell =2$ and $V$ have coordinates
$x,y\text{.}$ If
$G=\{\pm \text{id.}\},$ then
$G$ is not a reflection group. $I$ is generated by
${x}^{2},xy,$ and
${y}^{2}$ and no smaller number of elements suffices.

*Notation:* Throughout the proof we let $S$ be the algebra of all polynomials on
$V,$ ${S}_{0}$ the ideal in $S$
generated by the homogeneous elements of $I$ of positive degree, and $Av$ stand for average over
$G$ (i.e.
$AvP={\left|G\right|}^{-1}\underset{g\in G}{\Sigma}gP\text{).}$

Proof of (a). (Chevalley, Am. J. of Math. 1995.) |
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(1) Assume ${I}_{1},{I}_{2},\dots $ are elements of $I$ such that ${I}_{1}$ is not in the ideal in $I$ generated by the others and that ${P}_{1},{P}_{2},\dots $ are homogeneous elements of $S$ such that $\Sigma {P}_{i}{I}_{i}=0\text{.}$ Then ${P}_{1}\in {S}_{0}\text{.}$
We choose a minimal finite basis ${I}_{1},\dots ,{I}_{n}$ for ${S}_{0}$ formed of homogeneous elements of $I\text{.}$ Such a basis exists by Hilbert's Theorem. (2) The ${I}_{i}\text{'s}$ are algebraically independent.
(3) The ${I}_{i}\text{'s}$ generate $I$ as an algebra.
(4) $n=\ell \text{.}$
By (2), (3) and (4) (a) holds. $\square $ |

Proof of (b). (Todd, Shephard Can. J. Math. 1954.) |
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Let ${I}_{1},\dots ,{I}_{\ell}$ be algebraically independent generators of $I$ of degrees ${d}_{1},\dots ,{d}_{\ell},$ respectively. (5) $\underset{i=1}{\overset{\ell}{\Pi}}{(1-{t}^{{d}_{i}})}^{-1}=\underset{g\in G}{Av}\hspace{0.17em}\text{det}{(1-gt)}^{-1},$ as a formal identity in $t\text{.}$
(6) $\Pi {d}_{i}=\left|G\right|$ and $\Sigma ({d}_{i}-1)=N=$ number of reflections in $G\text{.}$
(7) Let $G\prime $ be the subgroup of $G$ generated by its reflections. Then $G\prime =G$ and hence $G$ is a reflection group.
Thus Theorem 24(b) holds. $\square $ |

*Exercise:* For each reflection in $G$ choose a root $\alpha \text{.}$ Then
$\text{det}\hspace{0.17em}\frac{\partial ({I}_{1},{I}_{2},\dots )}{\partial ({x}_{1},{x}_{2},\dots )}=\Pi \alpha $
up to multiplication by a nonzero number.

*Remark:* The theorem remains true if $\mathbb{R}$ is replaced by any field of characteristic $0$ and "reflection" is
replaced by "automorphism of $V$ with fixed point set a hyperplane".

For the proof of Theorem 24(c) (determination of the ${d}_{i}\text{)}$ we use:

*Proposition:* Let $G$ and the ${d}_{i}$ be as in Theorem 27 and
$w={w}_{1}\dots {w}_{\ell},$
the product of the simple reflections (relative to an ordering of $V$ (see Appendix I.8)) in any fixed order. Let $h$ be the
order of $w\text{.}$ Then:

(a) | $N=\ell h/2\text{.}$ |

(b) | $w$ contains $\omega =\text{exp}\hspace{0.17em}2\pi i/h$ as an eigenvalue, but not $1\text{.}$ |

(c) | If the eigenvalues of $w$ are $\left\{{\omega}^{{m}_{i}}\hspace{0.17em}\right|\hspace{0.17em}1\le {m}_{i}\le h-1\}$ then $\{{m}_{i}+1\}=\left\{{d}_{i}\right\}\text{.}$ |

Proof. | |

This was first proved by Coxeter ( This can be used to determine the ${d}_{i}$ for all the Chevalley groups. As an example we determine the ${d}_{i}$ for ${E}_{8}\text{.}$ Here $\ell =8,$ $N=120,$ so by (a) $h=30\text{.}$ Since $w$ acts rationally $\left\{{\omega}^{n}\hspace{0.17em}\right|\hspace{0.17em}(n,30)=1\}$ are all eigenvalues. Since $\phi \left(30\right)=8=\ell $ these are all the eigenvalues. Hence the ${d}_{i}$ are 1, 7, 11, 13, 17, 19, 23, 29 all increased by 1, as listed previously. The proofs for ${G}_{2}$ and ${F}_{4}$ are exactly the same. ${E}_{6}$ and ${E}_{7}$ require further argument. $\square $ |

*Exercise:* Argue further.

*Remark:* The ${d}_{i}\text{'s}$ also enter into the following results, related to Theorem 24:

Let $\mathcal{L}$ be the original Lie algebra, $k$ a field of characteristic 0, $G$ the corresponding adjoint Chevalley group. The algebra of polynomials on $\mathcal{L}$ invariant under $G$ is generated by $\ell $ algebraically independent elements of degree ${d}_{1},\dots ,{d}_{\ell},$ the ${d}_{i}\text{'s}$ as above.

This is proved by showing that under restriction from $\mathcal{L}$ to $\mathscr{H}$ the $G\text{-invariant}$ polynomials on $\mathcal{L}$ are mapped isomorphically onto the $W\text{-invariant}$ polynomials on $\mathscr{H}\text{.}$ The corresponding result for the universal enveloping algebra of $\mathcal{L}$ then follows easily.

(b) If $G$ acts on the exterior algebra on $\mathcal{L},$ the algebra of invariants is an exterior algebra generated by $\ell $ independent homogeneous elements of degrees $\{2{d}_{i}-1\}\text{.}$

This is more difficult. It implies that the Poincaré polynomial (whose coefficients are the Betti numbers) of the corresponding compact semisimple Lie group (the group $K$ constructed from $\u2102$ in §8) is $\Pi (1+{t}^{2{d}_{i}-1})\text{.}$

Proof of Theorem 26. (Solomon, Journal of Algebra, 1966.) |
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Let $\Pi $ be the set of simple roots. If $\pi \subseteq \Pi $ let ${W}_{\pi}$ be the subgroup generated by all ${w}_{\alpha},$ $\alpha \in \Pi \text{.}$ (1) If $w\in {W}_{\pi}$ then $w$ permutes the positive roots with support not in $\pi \text{.}$
(2) (3) For $\pi \subseteq \Pi $ define ${W}_{\pi}^{\prime}=\{w\in W\hspace{0.17em}|\hspace{0.17em}w\pi >0\}\text{.}$ Then:
(4) Let $W\left(t\right)=\underset{w\in W}{\Sigma}{t}^{N\left(w\right)},$ ${W}_{\pi}\left(t\right)=\underset{w\in {W}_{\pi}}{\Sigma}{t}^{N\left(w\right)}\text{.}$ Then $\underset{\pi \subseteq \Pi}{\Sigma}{(-1)}^{\pi}W\left(t\right)/{W}_{\pi}\left(t\right)={t}^{N},$ where $N$ is the number of positive roots and ${(-1)}^{\pi}={(-1)}^{\left|\pi \right|}\text{.}$
Set $D=\{v\in V\hspace{0.17em}|$ $(v,\alpha )\ge 0$ for all $\alpha \in \Pi \},$ and for each $\pi \subseteq \Pi $ set ${D}_{\pi}=\{v\in V\hspace{0.17em}|$ for all $\alpha \in \pi ,(v,\beta )>0$ for all $\beta \in \Pi -\pi \}\text{.}$ ${D}_{\pi}$ is an open face of $D\text{.}$ (5) The following subgroups of $W$ are equal:
(6) In the complex cut on real $k\text{-space}$ by a finite number of hyperplanes let ${n}_{i}$ be the number of $i\text{-cells.}$ Then $\Sigma {(-1)}^{i}{n}_{i}={(-1)}^{k}\text{.}$
(7) In the complex $K$ cut from $V$ by the reflecting hyperplanes let ${n}_{\pi}\left(w\right)$ $(\pi \subseteq \Pi ,w\in W)$ denote the number of cells $W\text{-congruent}$ to ${D}_{\pi}$ and $w\text{-fixed.}$ Then $\underset{\pi \subseteq \Pi}{\Sigma}{(-1)}^{\pi}{n}_{\pi}\left(w\right)=\text{det}\hspace{0.17em}w\text{.}$
If $\chi $ is a character on ${W}_{1},$ a subgroup of $W,$
then ${\chi}^{W}$ denotes the induced character defined by $(*)$
${\chi}^{W}\left(w\right)={\left|{W}_{1}\right|}^{-1}\underset{\underset{xw{x}^{-1}\epsilon {W}_{1}}{x\in W}}{\Sigma}\chi \left(xw{x}^{-1}\right)\text{.}$
(See, e.g., W. Feit, (8) Let $\chi $ be a character on $W$ and ${\chi}_{\pi}={\left(\chi \right|{W}_{\pi})}^{W}$ $(\pi \subseteq \Pi )\text{.}$ then $\underset{\pi \subseteq \Pi}{\Sigma}{(-1)}^{\pi}{\chi}_{\pi}\left(w\right)=\chi \left(w\right)$ $\text{det}\hspace{0.17em}w$ for all $w\in W\text{.}$
(9) Let $M$ be a finite dimensional real $W\text{-module,}$ ${I}_{\pi}\left(M\right)$ be the subspace of ${W}_{\pi}\text{-invariants,}$ and $\stackrel{\u02c6}{I}\left(M\right)$ be the space of $W\text{-skew-invariants}$ (i.e. $\stackrel{\u02c6}{I}\left(M\right)=\{m\in M\hspace{0.17em}|\hspace{0.17em}wm=\left(\text{det}\hspace{0.17em}w\right)m$ for all $w\in W\}\text{).}$ Then $\underset{\pi \subseteq \Pi}{\Sigma}{(-1)}^{\pi}\text{dim}\hspace{0.17em}{I}_{\pi}\left(M\right)=\text{dim}\hspace{0.17em}\stackrel{\u02c6}{I}\left(M\right)\text{.}$
(10) If $p=\Pi \alpha ,$ the product of the positive roots, then $p$ is skew and $p$ divides every skew polynomial on $V\text{.}$
(11) Let $P\left(t\right)=\Pi (1-{t}^{{d}_{i}})/(1-t)$ and for $\pi \subseteq \Pi $ let $\left\{{d}_{\pi i}\right\}$ and ${P}_{\pi}$ be defined for ${W}_{\pi}$ as $\left\{{d}_{i}\right\}$ and $P$ are for $W\text{.}$ Then $\underset{\pi \subseteq \Pi}{\Sigma}{(-1)}^{\pi}P\left(t\right)/{P}_{\pi}\left(t\right)={t}^{N}\text{.}$
(12) Proof of Theorem 26. We write (11) as $({t}^{N}-{(-1)}^{\Pi})/P\left(t\right)=\underset{\pi \u2ac5\u0338\Pi}{\Sigma}{(-1)}^{\pi}/{P}_{\pi}\left(t\right)$ and (4) as $({t}^{N}-{(-1)}^{\Pi})/W\left(t\right)=\underset{\pi \u2ac5\u0338\Pi}{\Sigma}{(-1)}^{\pi}/{W}_{\pi}\left(t\right)\text{.}$ Then, by induction on $\left|\Pi \right|,$ $W\left(t\right)=P\left(t\right)\text{.}$ $\square $ |

*Remark.* Step (7), the geometric step, represents the only simplification of Solomon's original proof.

This is a typed excerpt of *Lectures on Chevalley groups* by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson.
This work was partially supported by Contract ARO-D-336-8230-31-43033.