Lectures on Chevalley groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 18 July 2013

§9. The orders of the finite Chevalley groups

Presently we will prove:

Theorem 24: Let W be a finite reflection group on a real space V of finite dimension ,S the algebra of polynomials on V, I(S) the subalgebra of invariants under W. Then:

(a) I(S) is generated by homogeneous algebraically independent elements I1,,I.
(b) The degrees of the Ij's, say d1,,d, are uniquely determined and satisfy Σj(dj-1)=N, the number of positive roots.
(c) For the irreducible Weyl groups the di's are as follows: W di's A B,C D E6 E7 E8 F4 G2 2,3,,+1 2,4,,2 2,4,,2-2, 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

Our main goal is:

Theorem 25:

(a) Let G be a universal Chevalley group over a field k of q elements and the di's as in Theorem 24. Then |G|= qNΠi (qdi-1) with N=Σ(di-1)= the number of positive roots.
(b) If G is simple instead, then we have to divide by c=|Hom(L1/L0,k*)|, given as follows: G A B,C D E6 E7 E8 F4 G2 c (+1,q-1) (2,q-1) (4,q-1) (3,q-1) (2,q-1) 111

Remark: We see that the groups of type B and C have the same order. If =2 the root systems are isomorphic so the groups are isomorphic. We will show later that if 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 di's be as in Theorem 24 and t an indeterminate. Then ΣwW tN(w) = Πi (1-tdi) (1-t).

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 |G|= qN(q-1) ΣwW qN(w).


Recall that, by Theorems 4 and 4', G=wWBwB (disjoint) and BwB=UHwUw with uniqueness of expression. Hence |G|= |U||H|· ΣwW|Uw|. Now by Corollary 1 to the proposition of §3, |U|=qN and |Uw|=qN(w). By Lemma 28, |H|=(q-1).

Corollary: U is a p-Sylow subgroup of G, if p denotes the characteristic of k.


p|qN(w) unless N(w)=0. Since N(w)=0 if and only if w=1, pΣqN(w).

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 Hom(L1/L0,k*) and the values of L1/L0 found in §3.

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: Here WS+1 permuting +1 linear functions ω1,,ω+1 such that σ1=Σωi=0. In this case the elementary symmetric polynomials σ2,,σ+1 are invariant and generate all other polynomials invariant under W.

Types B,C: Here W acts relative to a suitable basis ω1,,ω by all permutations and sign changes. Here the elementary symmetric polynomials in ω12,,ω2 are invariant and generate all other polynomials invariant under W.

Type D: Here only an even number of sign changes can occur. Thus we can replace the last of the invariants for B,ω12ω2 by ω1ω.

Theorem 26: Type A: Here WS+1 and N(w) is the number of inversions in the sequence (w(1),,w(+1)). If we write P(t)=wWS+1tN(w) then P+1(t)=P(t)(1+t+t2++t+1), as we see by considering separately the +2 values that w(+2) can take on. Hence the formula P(t)= Πj=2+1 (1-tj)/ (1-t) follows by induction.

Exercise: Prove the corresponding formulas for types B,C and D. 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 and I the algebra of polynomials on V invariant under G. Then:

(a) If G is generated by reflections, then I is generated by algebraically independent homogeneous elements (and 1).
(b) Conversely, if I is generated by algebraically independent homogeneous elements (and 1) then G is generated by reflections.

Example: Let =2 and V have coordinates x,y. If G={±id.}, then G is not a reflection group. I is generated by x2,xy, and y2 and no smaller number of elements suffices.

Notation: Throughout the proof we let S be the algebra of all polynomials on V, S0 the ideal in S generated by the homogeneous elements of I of positive degree, and Av stand for average over G (i.e. AvP= |G|-1 ΣgG gP).

Proof of (a). (Chevalley, Am. J. of Math. 1995.)

(1) Assume I1,I2, are elements of I such that I1 is not in the ideal in I generated by the others and that P1,P2, are homogeneous elements of S such that ΣPiIi=0. Then P1S0.


Suppose I1 ideal in S generated by I2,. Then I1=Σi2RiIi for some R2,S so that I1=AvI1=Σi2(AvRi)Ii belongs to the ideal in I generated by I2,, a contradiction. Hence I1 does not belong to the ideal in S generated by I2,.

We now prove (1) by induction on d=degP1. If d=0, P1=0S0. Assume d>0 and let gG be a reflection in a hyperplane L=0. Then for each i, L|(Pi-gPi). Hence Σ((Pi-gPi)/L)Ii=0, so by the induction assumption P1-gP1S0, i.e. P1gP1 (mod S0). Since G is generated by reflections this holds for all gG and hence P1AvP1 (mod S0). But AvP1S0 so P1S0.

We choose a minimal finite basis I1,,In for S0 formed of homogeneous elements of I. Such a basis exists by Hilbert's Theorem.

(2) The Ii's are algebraically independent.


If the Ii are not algebraically independent, let H(I1,,In)=0 be a nontrivial relation with all monomials in the Ii's of the same minimal degree in the underlying coordinates x1,,x. Let Hi=H(I1,,In)/Ii. By the choice of H not all Hi are 0. Choose the notation so that {H1,,Hm} (mn) but no subset of it generates the ideal in I generated by all the Hi. Let Hj=Σi=1mVj,iHi for j=m+1,,n where Vj,iI and all terms in the equation are homogeneous of the same degree. Then for k=1,2,, we have 0=H/xk= Σi=1nHiIi/xk= Σi=1mHi ( Ii/xk+ Σj=m+1n Vj,iIj/ xk ) . By (1) I1/xk+ Σj=m+1n Vj,1Ij/ xk S0. Multiplying by xk, summing over k, using Euler's formula, and writing dj=degIj we get d1I1+ Σj=m+1n Vj,1djIj =Σi=1nAiIi where Ai belongs to the ideal in S generated by the xk. By homogeneity A1=0. Thus I1 is in the ideal generated by I2,,In, a contradiction.

(3) The Ii's generate I as an algebra.


Assume PI is homogeneous of positive degree. Then P=ΣPiIi, PiS. By averaging we can assume that each PiI. Each Pi is of degree less than the degree of P, so by induction on its degree P is a polynomial in the Ii's.

(4) n=.


By (2) n. By Galois theory (I) is of finite index in (x1,x2,,xn), hence has transcendence degree over , whence n.

By (2), (3) and (4) (a) holds.

Proof of (b). (Todd, Shephard Can. J. Math. 1954.)

Let I1,,I be algebraically independent generators of I of degrees d1,,d, respectively.

(5) Πi=1 (1-tdi)-1 = AvgG det(1-gt)-1, as a formal identity in t.


Let ε1,,ε be the eigenvalues of g and x1,,x the corresponding eigenfunctions. Then det(1-gt)-1= Πi ( 1+εit+ εi2t2 + ) . The coefficient of tn is Σp1+p2+=n ε1p1ε2p2, i.e. the trace of g acting on the space of homogeneous polynomials in x1,,x of degree n, since the monomials x1p1x2p2 form a basis for this space. By averaging we get the dimension of the space of invariant homogeneous polynomials of degree n. This dimension is the number of monomials I1p1I2p2 of degree n, i.e., the number of solutions of p1d1+ p2d2+= n, i.e. the coefficient of tn in Πi=1 (1-tdi)-1 .

(6) Πdi=|G| and Σ(di-1)=N= number of reflections in G.


We have

det(1-gt)= { (1-t) ifg=1, (1-t)-1 (1+t) ifgis a reflection, a polynomial not divisible by(1-t)-1 otherwise.

Substituting this in (5) and multiplying by (1-t), we have Π ( 1+t++ tdi-1 ) -1 = |G|-1 ( 1+N(1-t)/ (1+t)+ (1-t)2P(t) ) where P(t) is regular at t=1. Setting t=1 we get Πdi-1=|G|-1. Differentiating and setting t=1 we get (Πdi-1)Σ (-(di-1)/2) = |G|-1 (-N/2), so Σ(di-1)=N.

(7) Let G be the subgroup of G generated by its reflections. Then G=G and hence G is a reflection group.


Let Ii,di, and N refer to G. The Ii can be expressed as polynomials in the Ii with the determinant of the corresponding Jacobian not 0. Hence after a rearrangement of the Ii, Ii/Ii0 for all i. Hence didi. But Σ(i-1)= N=N=Σ(di-1) by (6). Hence di=di for all i, so, again by (6), |G|=Πdi =Πdi= |G|, so G=G.

Corollary: The degrees d1,d2, above are uniquely determined and satisfy the equations (6).

Thus Theorem 24(b) holds.

Exercise: For each reflection in G choose a root α. Then det (I1,I2,) (x1,x2,) =Πα up to multiplication by a nonzero number.

Remark: The theorem remains true if 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 di) we use:

Proposition: Let G and the di be as in Theorem 27 and w=w1w, 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. Then:

(a) N=h/2.
(b) w contains ω=exp2πi/h as an eigenvalue, but not 1.
(c) If the eigenvalues of w are {ωmi|1mih-1} then {mi+1}={di}.


This was first proved by Coxeter (Duke Math. J. 1951), case by case, using the classification theory. For a proof not using the classification theory see Steinberg, T.A.M.S. 1959, for (a) and (b) and Coleman, Can. J. Math. 1958, for (c) using (a) and (b).

This can be used to determine the di for all the Chevalley groups. As an example we determine the di for E8. Here =8, N=120, so by (a) h=30. Since w acts rationally {ωn|(n,30)=1} are all eigenvalues. Since φ(30)=8= these are all the eigenvalues. Hence the di are 1, 7, 11, 13, 17, 19, 23, 29 all increased by 1, as listed previously. The proofs for G2 and F4 are exactly the same. E6 and E7 require further argument.

Exercise: Argue further.

Remark: The di's also enter into the following results, related to Theorem 24:

Let be the original Lie algebra, k a field of characteristic 0, G the corresponding adjoint Chevalley group. The algebra of polynomials on invariant under G is generated by algebraically independent elements of degree d1,,d, the di's as above.

This is proved by showing that under restriction from to the G-invariant polynomials on are mapped isomorphically onto the W-invariant polynomials on . The corresponding result for the universal enveloping algebra of then follows easily.

(b) If G acts on the exterior algebra on , the algebra of invariants is an exterior algebra generated by independent homogeneous elements of degrees {2di-1}.

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 in §8) is Π(1+t2di-1).

Proof of Theorem 26. (Solomon, Journal of Algebra, 1966.)

Let Π be the set of simple roots. If πΠ let Wπ be the subgroup generated by all wα, αΠ.

(1) If wWπ then w permutes the positive roots with support not in π.


If β is a positive root and supp βπ then β=ΣαΠeαα with some eα>0, απ. Now wβ is β plus a vector with support in π, hence its coefficient of α is positive, so wβ>0.

(2) Corollary: If wWπ then N(w) is unambiguous (i.e. it is the same whether we consider wW or wWπ).

(3) For πΠ define Wπ={wW|wπ>0}. Then:

(a) Every wW can we written uniquely w=ww with wWπ and wWπ.
(b) In (a) N(w)=N(w)+N(w).


(a) For any wW let wWπw be such that N(w) is minimal. Then wα>0 for all απ by Appendix II.19(a'). Hence wWπ so that wWπWπ. Suppose now w=ww=uu with w,uWπ and w,uWπ. Then wwu-1=u. Hence wwu-1π>0. Now w(-π)<0 so wu-1π has support π. Hence wu-1ππ so by Appendix 11.23 (applied to Wπ) wu-1=1. Hence w=u, w=u.

(b) follows from (a) and (1).

(4) Let W(t)= ΣwWtN(w), Wπ(t)= ΣwWπtN(w). Then ΣπΠ (-1)πW(t)/ Wπ(t) =tN, where N is the number of positive roots and (-1)π=(-1)|π|.


We have, by (3), W(t)/Wπ(t)= ΣwWπ tN(w). Therefore the contribution of the term for w to the sum in (4) is cwtN(w) where cw= Σ πΠ wπ>0 (-1)π. If w keeps positive exactly k elements of Π then

cw= { (1-1)k=0 ifk0 1 ifk=0.

Therefore the only contribution is made by w0, the element of w which makes all positive roots negative, so the sum in (4) is equal to tN as required.

Corollary: Σ(-1)π |W|/ |Wπ| =1.

Exercise: Deduce from (4) that if α and β are complementary subsets of Π then Σπα (-1)π-α/ Wπ(t) = Σπβ (-1)π-β/ Wπ(t-1).

Set D={vV| (v,α)0 for all αΠ}, and for each πΠ set Dπ={vV| for all απ,(v,β)>0 for all βΠ-π}. Dπ is an open face of D.

(5) The following subgroups of W are equal:

(a) Wπ.
(b) The stabilizer of Dπ.
(c) The point stabilizer of Dπ.
(d) The stabilizer of any point of Dπ.


(a) (b) because π is orthogonal to Dπ. (b) (c) because D is a fundamental domain for W by Appendix III.33. Clearly (c) (d). (d) (a) by Appendix III.32.

(6) In the complex cut on real k-space by a finite number of hyperplanes let ni be the number of i-cells. Then Σ(-1)ini =(-1)k.


This follows from Euler's formula, but may be proved directly by induction. In fact, if an extra hyperplane H is added to the configuration, each original i-cell cut in two by H has corresponding to it in H an (i-1)-cell separating the two parts from each other, so that Σ(-1)ini remains unchanged.

(7) In the complex K cut from V by the reflecting hyperplanes let nπ(w) (πΠ,wW) denote the number of cells W-congruent to Dπ and w-fixed. Then ΣπΠ (-1)πnπ(w) =detw.


Each cell of K is W-congruent to exactly one Dπ. By (5) every cell fixed by w lies in Vw (Vw={vV|wv=v}). Applying (6) to Vw and using dimDπ=-|π| we get ΣπΠ (-1)πnπ(w) =(-1)-k, where k=dimVw. But w is orthogonal, so that its possible eigenvalues in V are +1, -1 and pairs of conjugate complex numbers. Hence (-1)-k= detw.

If χ is a character on W1, a subgroup of W, then χW denotes the induced character defined by (*) χW(w)= |W1|-1 Σ xW xwx-1εW1 χ(xwx-1). (See, e.g., W. Feit, Characters of finite groups.)

(8) Let χ be a character on W and χπ= (χ|Wπ)W (πΠ). then ΣπΠ (-1)πχπ(w) =χ(w) detw for all wW.


Assume first that χ1. Now xwx-1Wπ if and only if xwx-1 fixes Dπ (by (5)) which happens if and only if w fixes x-1Dπ. Therefore 1π(w)=nπ(w) by (*). By (7) this gives the result for χ1. If χ is any character then χπ=χ·1π so (8) holds.

(9) Let M be a finite dimensional real W-module, Iπ(M) be the subspace of Wπ-invariants, and Iˆ(M) be the space of W-skew-invariants (i.e. Iˆ(M)= {mM|wm= (detw)m for all wW}). Then ΣπΠ (-1)πdim Iπ(M) =dimIˆ(M).


In (8) take χ to be the character of M, average over wW, and use (*).

(10) If p=Πα, the product of the positive roots, then p is skew and p divides every skew polynomial on V.


We have wαp=-p=(detwα)p if α is a simple root by Appendix I.11. Since W is generated by simple reflections p is skew. If f is skew and α a root then wαf=(detwα)f=-f so α|f. By unique factorization p|f.

(11) Let P(t)= Π (1-tdi) /(1-t) and for πΠ let {dπi} and Pπ be defined for Wπ as {di} and P are for W. Then ΣπΠ (-1)πP(t)/ Pπ(t)=tN.


We must show (*) ΣπΠ (-1)πΠi (1-tdπi)-1 = tNΠi (1-tdi)-1 . Let S=Σk=0Sk be the algebra of polynomials on V, graded as usual. As in (5) of the proof of Theorem 27 the coefficient of tk on the left hand side of (*) is ΣπΠ (-1)πdimIπ (Sk). Similarly, using (10), the coefficient of of tk on the right hand side of (*) is dimIˆ(Sk). These are equal by (9).

(12) Proof of Theorem 26. We write (11) as (tN-(-1)Π) /P(t) = Σπ⫅̸Π (-1)π/Pπ(t) and (4) as (tN-(-1)Π) /W(t) = Σπ⫅̸Π (-1)π/Wπ(t) . Then, by induction on |Π|, W(t)=P(t).

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

Notes and References

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.

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