3.1 Subgroups and Associated Groups
3.1.3 Weyl Group of GL(n, F )
S O T G L G
PROOF. It is immediate to see from (3.6) and (3.7), that UλT
Lλ = {In}. Normality of Uλ in Pλ follows from the fact that Uλ represents the kernel of the homomorphism ψ : Pλ → Lλ, where ψ acts onPλ by sending the main diagonal of an elementA ofPλ to the diagonal matrix having the same diagonal of A. Let A ∈ Pλ, be an arbitrary element. Then Aψ(A)−1 ∈ Uλ. It follows that A ∈ Uλψ(A) ⊆ UλLλ. Thus Pλ ⊆ UλLλ, and the equality of Pλ and UλLλ is established. Since UλEPλ,thenPλ=NPλ(Uλ).This completes the proof of the theorem.
Corollary 3.1.9. U T(n,F) =SU U T(n,F): O
n copies
F∗.
PROOF. The proof is a special case of combining Example 3.1.1 and Theorem 3.1.8.
Since the levi complement Lλ ∼=
k
O
i=1
GL(λi,F),then by Theorem 2.3.2, the irreducible characters of Lλ are
Irr(Lλ) = ( k
O
i=1
χi|χi∈Irr(GL(λi,F)) )
, (3.8)
where in the last equation, O
is to be understood the tensor product of characters.
Theorem 3.1.8 asserts that the exact sequence
Lλ−→Pλ−→Pλ/Uλ
is an isomorphism, where the first map is inclusion and the second projection. This means that an irreducible character of Lλ extends irreducibly to Pλ,by using the method of lifting of characters described in Section 2.4. By equation (3.8), we get
k
Y
i=1
|Irr(GL(λi, q))| irreducible characters of Pλ.The preceding irreducible characters ofPλ comes from characters of Lλ are used as a base for Frobenius method of induction of characters to build up characters of the group GL(n, q). The characters of the groupGL(n, q) appear into two series, namely Principal andDiscrete series. The Principal Seriescharacters are those which are obtained from characters of parabolic subgroups of GL(n, q). Any character which is not in the principal series characters is said to belong to the Discrete Series. The discussion of obtaining characters of GL(n, q) from those of Pλ, λ`n will be continued in Section 5.3. The discrete series characters will be discussed in Section 5.4.
Chapter 3 — Structure Of The General Linear Group
Theorem 3.1.10. The Weyl group W is isomorphic to the symmetric groupSn.
PROOF. Let B={e1, e2,· · ·, en} be the standard basis of V(n,F).The Weyl group W act on B on a natural way; that is if w ∈ W, then wei = ek, 1 ≤ i, k ≤ n. Let X = {1,2,· · ·, n}. For each w∈ W,the function ϕw :X −→ X given by ϕw(i) = k,for 1≤i, k ≤n is such that wei =ek,is well defined and a bijective. Hence ϕw ∈Sn.Now if we define ϕ:W −→Sn by ϕ(w) =ϕw,then it is not difficult to see that ϕis a bijective homomorphism and hence it is an isomorphism. The
result follows.
Remark 3.1.3. The above theorem asserts that the Weyl group of GL(n, q) is independent of the choice of the field F.It is characterized by the dimensionnonly.
In the next context, we introduce a special kind of matrices of GL(n,F) which are of great impor- tance in order to describe the elements of GL(n,F) and consequently SL(n,F).
Definition 3.1.9. A transvectionis a linear transformation T onV(n,F)with eigenvalues equal to 1 and satisfyingrank(T −In) = 1,where In is the identity transformation on V(n,F).
In matrix language, a transvection Aij(α) where i6= j and α ∈ F, is a matrix different from the identity matrix only that it has α in the (i, j)th position. It turns out that all transvections are elements of SL(n,F).
One can easily verify the following properties of transvections.
Lemma 3.1.11. Forα, β ∈F, i6=j, 1. Aij(0) =In.
2. det(Aij(α)) = 1.
3. If α6= 0,thenAij(α)∈U T(n,F)⇐⇒i < j.
4. Aij(α)Aij(β) =Aij(α+β).
5. (Aij(α))−1 =Aij(−α).
6. Fori6=j6=k6=i, the commutator [Aij(α), Ajk(β)] =Aik(αβ).
PROOF. Direct results from the definition.
As a quick result of this lemma, we have
Corollary 3.1.12. For fixediandj, the setAij ={Aij(α)| α∈F}forms a subgroup ofSL(n,F).
S O T G L G
PROOF. It follows directly by parts (2), (3) and (4) of Lemma 3.1.11.
The subgroups defined this way are refer as theroot subgroups of GL(n,F).
Now, we come to a known theorem concerning the structure of the groupGn=GL(n,F).
Theorem 3.1.13 (Bruhat Decomposition Theorem). GL(n,F) =U T(n,F)·W ·U T(n,F).
PROOF. In Singh [70], it is shown that any matrixA∈GL(n,F) splits into a productA=L1wdL2, where L1, L2 ∈ SU U T(n,F), d ∈ O
n copies
F∗ and w ∈ W. It follows that any element of GL(n,F) is a product of an upper triangular matrix, a permutation matrix, and another upper triangular
matrix. One can refer also to Alperin [3] for the details.
Remark 3.1.4. Bruhat Decomposition Theorem asserts thatGL(n,F) is a union (disjoint) of the double cosetsU T(n,F)wU T(n,F) as wranges over all elements ofW.ThusGL(n,F) is a union of n! disjoint double cosetsU T(n,F)w U T(n,F).
The next theorem gives a smaller generating set for GL(n,F) than that given by Bruhat Decom- position Theorem, but we first mention a lemma without proof, which will be helpful in the proof of the theorem.
Lemma 3.1.14. For each b∈U T(n,F),there exists a product T of transvections such that T b is a diagonal matrix having the same main diagonal entries as b.
PROOF. See Alperin [3].
Theorem 3.1.15. The group GL(n,F) is generated by the set of all invertible diagonal matrices and all transvections.
PROOF. By Bruhat Decomposition Theorem, we haveGL(n,F) =U T(n,F)·W·U T(n,F).Thus if we could write all the elements ofU T(n,F) andW in terms of diagonal matrices and transvection, then we done. Using Lemma 3.1.14, we can see that U T(n,F) has this property. By Theorem 3.1.10, every permutation matrices can be written in terms of permutations ofSn, which are generated by the set of transpositions. The action of a transposition on the standard basis B={e1, e2,· · ·, en} is that it sendsei 7−→ej 7−→ei for somei6=jand fixes the rest ofB. Now the action of the matrix Aji(1)Aij(−1)Aji(1) on Bis that it sends ei 7−→ej 7−→ −ei for i6=j and fixes the other elements of B. Multiplying this latter matrix by the diagonal matrix diag(1,· · · ,1,−1,1,· · ·,1), where−1 is in the (i, i) position, the resulting matrix sends ei 7−→ ej 7−→ ei for i6= j and fixes the other elements ofB, which shows thatW can be written in terms of diagonal matrices and transvections.
The result follows.
Theorem 3.1.16. The groupSL(n,F) is generated by the root subgroups Aij.
Chapter 3 — Structure Of The General Linear Group
PROOF. We give the idea of the proof, which rests on the following three main points. Full details of the proof can be found in Alperin [3].
• Every element of the groupSL(n,F) can be transformed into an element of the groupU T(n,F) by multiplying by some suitably transvections.
• Every element of the group U T(n,F) can be transformed into an element of the group SU U T(n,F) by multiplying by some suitably transvections.
• Every element of the groupSU U T(n,F) can be transformed into the identity elementIn by multiplying by some suitably transvections.
Thus any element ofSL(n,F) is a product of transvections, which completes the proof.
Theorem 3.1.17. All transvections are conjugate in GL(n, q) and ifn≥3,then all transvections are conjugate in SL(n, q).
PROOF. See Alperin [3] or Rotman [65].