Weyl Group

3.1 Roots and Weights

3.1.1 Weyl Group

LetGbe a connected classical group and letHbe a maximal torus inG. Define the normalizerofHinGto be

Norm_G(H) ={g∈G:ghg⁻¹∈Hfor allh∈H},

and define the Weyl group W_G=Norm_G(H)/H. Since all maximal tori ofGare conjugate, the groupW_Gis uniquely defined (as an abstract group) byG, and it acts by conjugation as automorphisms ofH. We shall see in later chapters that many aspects of the representation theory and invariant theory for Gcan be reduced to questions about functions on H that are invariant underW_G. The success of this approach rests on the fact thatW_G is a finite group of known structure (either the symmetric group or a somewhat larger group). We proceed with the details.

SinceHis abelian, there is a natural homomorphismϕ:W_G //Aut(H)given byϕ(sH)h=shs⁻¹fors∈Norm_G(H). This homomorphism gives an action ofW_G on the character groupX(H), where forθ∈X(H)the characters·θis defined by

s·θ(h) =θ(s⁻¹hs), forh∈H.

(Note that the right side of this equation depends only on the coset sH.) Writing θ=e^λforλ∈P(G)as in Section 2.4.1, we can describe this action as

s·e^λ=e^s·λ , where hs·λ,xi=hλ,Ad(s)⁻¹xi forx∈h. This defines a representation ofW_Gonh^∗.

Theorem 3.1.1.W_Gis a finite group and the representation of W_Gonh^∗is faithful.

Proof. Lets∈Norm_G(H). Supposes·θ=θ for allθ∈X(H). Thens⁻¹hs=hfor allh∈H, and hences∈Hby Theorem 2.1.5. This proves that the representation of W_Gonh^∗is faithful.

To prove the finiteness ofW_G, we shall assume thatG⊂GL(n,C)is in the matrix form of Section 2.4.1, so thatHis the group of diagonal matrices inG. In the proof of Theorem 2.1.5 we noted thath∈Hacts on the standard basis forCⁿby

he_i=θi(h)eifori=1, . . . ,n,

whereθ_i∈X(H)andθ_i6=θ_jfori6=j. Lets∈Norm_G(H). Then

hsei=s(s⁻¹hs)ei=sθi(s⁻¹hs)ei= (s·θi)(h)sei. (3.1) Hencese_i is an eigenvector forhwith eigenvalue (s·θi)(h). Since the characters s·θ1, . . . ,s·θnare all distinct, this implies that there is a permutationσ∈S_nand there are scalarsλi∈C^×such that

se_i=λie_σ(i) fori=1, . . . ,n. (3.2)

3.1 Roots and Weights 129

Sinceshe_i=λ_iθ_i(h)e_σ(i) forh∈H, the permutationσ depends only on the coset sH. Ift∈Norm_G(H)andte_i=µ_ie_τ(i)withτ∈S_nandµ_i∈C^×, then

tsei=λite_σ(i)=µ_σ(i)λie_{τ σ(i)}.

Hence the maps7→σis a homomorphism from Norm_G(H)intoS_nthat is constant on the cosets ofH. If σ is the identity permutation, thenscommutes withH and therefores∈H by Theorem 2.1.5. Thus we have defined an injective homomor- phism fromW_GintoS_n, soW_Gis a finite group. ut We now describe W_G for each type of classical group. We will use the em- bedding ofWG intoS_n employed in the proof of Theorem 3.1.1. Forσ ∈S_n let sσ∈GL(n,C)be the matrix such thatsσei=e_σ(i)fori=1, . . . ,n. This is the usual representation ofS_nonCⁿaspermutation matrices.

Suppose G=GL(n,C). Then H is the group of all n×n diagonal matrices.

Clearlys_σ ∈Norm_G(H)for everyσ ∈S_n. From the proof of Theorem 3.1.1 we know that every coset inW_G is of the form s_σH for someσ ∈S_n. HenceW_G∼= S_n. The action ofσ∈S_non the diagonal coordinate functionsx₁, . . . ,x_nforH is σ·x_i=x_σ−1(i).

LetG=SL(n,C). NowH consists of all diagonal matrices of determinant 1.

Givenσ∈S_n, we may pickλ_i∈C^×such that the transformationsdefined by (3.2) has determinant 1 and hence is in Norm_G(H). To prove this, recall that every permu- tation is a product of cyclic permutations, and every cyclic permutation is a product of transpositions (for example, the cycle(1,2, . . . ,k)is equal to(1,k)···(1,3)(1,2)).

Consequently, it is enough to verify this whenσ is the transpositioni↔j. In this case we takeλ_j=−1 andλ_k=1 fork6=j. Since det(sσ) =−1, we obtain dets=1.

Thus the homomorphismWG //S_n constructed in the proof of Theorem 3.1.1 is surjective. HenceWG∼=S_n. Notice, however, that this isomorphism arises by choosing elements of Norm_G(H)whose adjoint action onhis given by permutation matrices; the group of all permutation matrices is not a subgroup ofG.

Next, consider the caseG=Sp(C^2l,Ω), withΩ as in (2.6). Lets_l∈GL(l,C) be the matrix for the permutation(1,l)(2,l−1)(3,l−2)···, as in equation (2.5).

Forσ ∈S_l let s_σ ∈GL(l,C)be the corresponding permutation matrix. Clearly s^t_σ=s⁻_σ¹,so if we define

π(σ) = sσ 0

0 s_lsσs_l

,

thenπ(σ)∈Gand henceπ(σ)∈Norm_G(H). Obviouslyπ(σ)∈Hif and only if σ=1, so we obtain an injective homomorphism ¯π:S_l //W_G.

To find other elements ofW_G, consider the transpositions(i,2l+1−i)inS_2l, where 1≤i≤l. Sete₋i=e2l+1−i, where{ei}is the standard basis forC^2l. Define τi∈GL(2l,C)by

τie_i=e_−i, τie_−i=−e_i, τie_k=e_kfork6=i,−i.

Since {τ_ie_j : j=±1, . . . ,±l} is an Ω-symplectic basis for C^2l, we have τ_i ∈ Sp(C^2l,Ω) by Lemma 1.1.5. Clearlyτ_i∈Norm_G(H)and τ_i²∈H. Furthermore, τiτj=τjτiif 1≤i,j≤l. GivenF⊂ {1, . . . ,l}, define

τF=

∏

i∈F

τi∈Norm_G(H).

Then theH-cosets of the elements{τ_F}form an abelian subgroupT_l∼= (Z/2Z)^lof W_G. The action ofτ_F on the coordinate functionsx₁, . . . ,x_l forH isx_i7→x⁻_i ¹for i∈Fandx_j7→x_jfor j∈/F. This makes it evident that

π(σ)τFπ(σ)⁻¹=τσF forF⊂ {1, . . . ,l}andσ∈S_l. (3.3) Clearly,T_l∩π¯(Sl)H={1}.

Lemma 3.1.2.For G=Sp(C^2l,Ω), the subgroup T_l⊂W_Gis normal, and W_Gis the semidirect product of T_landπ¯(S_l). The action of W_Gon the coordinate functions in O[H]is by x_i7→ x_σ(i)±1 (i=1, . . . ,l), for every permutationσand choice±1of exponents.

Proof. Recall that a groupKis asemidirect productof subgroupsLandMifMis a normal subgroup ofK,L∩M=1, andK=L·M.

By (3.3) we see that it suffices to prove that W_G =T_lπ(S_l). Suppose s∈ Norm_G(H). Then there existsσ∈S_2l such thatsis given by (3.2). Define

F={i:i≤landσ(i)≥l+1},

and letµ∈S_2lbe the product of the transpositions interchangingσ(i)withσ(i)−l for i∈F. Then µ σ stabilizes the set {1, . . . ,l}. Let ν ∈S_l be the corre- sponding permutation of this set. Thenπ(ν)⁻¹τFse_i=±λie_ifori=1, . . . ,l. Thus

s∈τFπ(ν)H. ut

Now consider the caseG=SO(C^2l+1,B), with the symmetric formBas in (2.9).

Forσ∈S_ldefine

ϕ(σ) =





s_σ 0 0

0 1 0

0 0s_ls_σs_l



.

Thenϕ(σ)∈Gand henceϕ(σ)∈Norm_G(H). Obviously,ϕ(σ)∈Hif and only if σ=1, so we get an injective homomorphism ¯ϕ:S_l //W_G.

We construct other elements ofW_G in this case by the same method as for the symplectic group. Set

e₋i=e2l+2−i fori=1, . . . ,l+1.

For each transposition (i,2l+2−i) in S_2l+1, where 1≤i≤l, we define γi ∈ GL(2l+1,C)by

3.1 Roots and Weights 131

γ_ie_i=e_−i, γ_ie_−i=e_i, γ_ie₀=−e₀, and γie_k=e_k fork6=i,0,−i.

Thenγi∈O(B,C)and by makingγiact onel+1by−1 we obtain detγi=1. Hence γi ∈Norm_G(H). Furthermore,γ_i²∈H andγiγj=γjγi if 1≤i,j≤l. Given F ⊂ {1, . . . ,l}, we define

γF=

∏

i∈F

γi∈Norm_G(H).

Then theH-cosets of the elements{γ_F}form an abelian subgroupT_l∼= (Z/2Z)^lof W_G. The action ofγ_Fon the coordinate functionsx₁, . . . ,x_lforO[H]is the same as that ofτ_Ffor the symplectic group.

Lemma 3.1.3.Let G=SO(C^2l+1,B). The subgroup Tl⊂W_Gis normal, and W_Gis the semidirect product of T_landϕ(S¯ _l). The action of W_Gon the coordinate functions inO[H]is by x_i7→ x_σ(i)±1

(i=1, . . . ,l), for every permutationσ and choice±1 of exponents.

Proof. Supposes∈Norm_G(H). Then there existsσ∈S_2l+1such thatsis given by (3.2), withn=2l+1. The action ofsas an automorphism ofHis

s·diag[a1, . . . ,a_n]s⁻¹=diag[a_σ−1(1), . . . ,a_σ−1(n)].

Sinceal+1=1 for elements ofH, we must haveσ(l+1) =l+1. Now use the same

argument as in the proof of Lemma 3.1.2. ut

Finally, we consider the caseG=SO(C^2l,B), withBas in (2.6). Forσ ∈S_l defineπ(σ)as in the symplectic case. Thenπ(σ)∈Norm_G(H). Obviously,π(σ)∈ Hif and only ifσ=1, so we have an injective homomorphism ¯π:S_l //W_G. The automorphism ofHinduced byσ∈S_lis the same as for the symplectic group.

We have slightly less freedom in constructing other elements ofW_Gthan in the case ofSO(C^2l+1,B). Set

e_−i=e_2l+1−i fori=1, . . . ,l.

For each transposition (i,2l+1−i) in S_2l, where 1≤i≤l, we define β_i ∈ GL(2l,C)by

β_ie_i=e_−i, β_ie_−i=e_i, βie_k=e_k fork6=i,−i.

Then βi∈O(C^2l,B)and clearly βiHβ_i⁻¹=H. However, detβ_i=−1, soβi∈/ SO(C^2l,B). Nonetheless, we still haveβ_i²∈Handβiβj=βjβiif 1≤i,j≤l. Given F⊂ {1, . . . ,l}, we define

βF=

∏

i∈F

βi.

If Card(F)iseven,then detβ_F=1 and henceβ_F∈Norm_G(H). Thus theHcosets of the elements{β_F : Card(F)even}form an abelian subgroupR_lofW_G.

Lemma 3.1.4.Let G=SO(C^2l,B). The subgroup R_l⊂W_Gis normal, and W_Gis the semidirect product of R_l andπ¯(S_l). The action of W_Gon the coordinate functions inO[H]is by x_i7→ x_σ(i)±1 (i=1, . . . ,l), for every permutationσand choice±1 of exponents with anevennumber of negative exponents.

Proof. By the same argument as in the proof of Lemma 3.1.2 we see that the normal- izer ofHinO(C^2l,B)is given by theHcosets of the elementsβ_Fπ(σ)asσ ranges overS_l andF ranges over all subsets of{1, . . . ,l}. Sinceπ(σ)∈Norm_G(H),we haveβ_Fπ(σ)∈Norm_G(H)if and only if Card(F)is even. ut