On splitting of the normalizer of a maximal torus in groups of Lie - - PowerPoint PPT Presentation

On splitting of the normalizer of a maximal torus in groups of Lie type

Alexey Galt 07.08.2017

Example 1

Let G = SL2(Fp) be the special linear group of degree 2 over Fp. Then T = { λ λ−1

• , λ ∈ F

∗ p} is a maximal torus of G. The

normalizer NG(T) is the group of all monomial matrices of G and NG(T)/T ≃ Sym2. But G contains only one element of

• rder two:

−1 −1

• , and this element lies in T. Hence,

NG(T) does not split over T.

Example 2

Let G = GLn(Fp) be the general linear group of degree n over

• Fp. Then T = Dn(Fp) is a maximal torus of G. The normalizer

NG(T) is the group of all monomial matrices of G and NG(T)/T ≃ Symn. There is a canonical embedding of Symn into the group of all monomial matrices of G. If H is an image of Symn under this embedding, then H is a complement for T in NG(T). Since the center Z(G) of G is contained in T, then a maximal torus of PGLn(Fp) also has a complement in their normalizer. Moreover, PGLn(Fp) ≃ PSLn(Fp) and the same is true for PSLn(Fp).

Example 2

Let G = GLn(Fp) be the general linear group of degree n over

• Fp. Then T = Dn(Fp) is a maximal torus of G. The normalizer

NG(T) is the group of all monomial matrices of G and NG(T)/T ≃ Symn. There is a canonical embedding of Symn into the group of all monomial matrices of G. If H is an image of Symn under this embedding, then H is a complement for T in NG(T). Since the center Z(G) of G is contained in T, then a maximal torus of PGLn(Fp) also has a complement in their normalizer. Moreover, PGLn(Fp) ≃ PSLn(Fp) and the same is true for PSLn(Fp).

Problems

Let G be a simple connected linear algebraic group over the algebraic closure Fp of a finite field of positive characteristic p. Let σ be a Steinberg endomorphism and T a maximal σ-invariant torus of G. It’s well known that all the maximal tori are conjugated in G and the quotient NG(T)/T is isomorphic to the Weyl group W of G. The following problem arises.

Problem 1

Describe the groups G in which NG(T) splits over T.

Problems

Let G be a simple connected linear algebraic group over the algebraic closure Fp of a finite field of positive characteristic p. Let σ be a Steinberg endomorphism and T a maximal σ-invariant torus of G. It’s well known that all the maximal tori are conjugated in G and the quotient NG(T)/T is isomorphic to the Weyl group W of G. The following problem arises.

Problem 1

Describe the groups G in which NG(T) splits over T.

Problems

A similar problem arises in finite groups G of Lie type. Let T = T ∩ G be a maximal torus in a finite group of Lie type G, N(G, T) = NG(T) ∩ G an algebraic normalizer of G. Notice that N(G, T) NG(T), but the equality is not true in general.

Problem 2

Describe the groups G and their maximal tori T in which N(G, T) splits over T.

Problems

A similar problem arises in finite groups G of Lie type. Let T = T ∩ G be a maximal torus in a finite group of Lie type G, N(G, T) = NG(T) ∩ G an algebraic normalizer of G. Notice that N(G, T) NG(T), but the equality is not true in general.

Problem 2

Describe the groups G and their maximal tori T in which N(G, T) splits over T.

History

J.Tits ”Normalisateurs de tores I. Groupes de Coxeter ´ Etendus” // Journal of Algebra, 1966, V.4, 96–116. An answer to Problem 1 for simple Lie groups was given in

• M. Curtis, A. Wiederhold, B. Williams, ”Normalizers of

maximal tori” // Springer, Berlin, 1974, Lecture Notes in Math., V. 418, 31–47.

History

J.Tits ”Normalisateurs de tores I. Groupes de Coxeter ´ Etendus” // Journal of Algebra, 1966, V.4, 96–116. An answer to Problem 1 for simple Lie groups was given in

• M. Curtis, A. Wiederhold, B. Williams, ”Normalizers of

maximal tori” // Springer, Berlin, 1974, Lecture Notes in Math., V. 418, 31–47.

Algebraic groups

The answer for Problem 1 is in the following table: Group Conditions of existence of a complement SLn(Fp) p = 2 or n is odd PSLn(Fp) No conditions Sp2n(Fp) p = 2 PSp2n(Fp) p = 2 or n 2 SO2n+1(Fp) No conditions SO2n(Fp) No conditions PSO2n(Fp) No conditions G2(Fp) No conditions F4(Fp) p = 2 Ek(Fp) p = 2

Algebraic groups

The answer for Problem 1 is in the following table: Group Conditions of existence of a complement SLn(Fp) p = 2 or n is odd PSLn(Fp) No conditions Sp2n(Fp) p = 2 PSp2n(Fp) p = 2 or n 2 SO2n+1(Fp) No conditions SO2n(Fp) No conditions PSO2n(Fp) No conditions G2(Fp) No conditions F4(Fp) p = 2 Ek(Fp) p = 2

Let Op′(Gσ) G Gσ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G. Let W be a Weyl group of G, π a natural homomorphism from N = NG(T) into W. Two elements w1, w2 are called σ-conjugate if w1 = (w−1)σw2w for some element w of W.

Proposition

There is a bijection between the G-classes of σ-stable maximal tori of G and the σ-conjugacy classes of W. Define CW,σ(w) = {x ∈ W|(x−1)σwx = w}.

Proposition

Let gσg−1 ∈ N and π(gσg−1) = w. Then (NG(T

g))σ/(T g)σ ≃ CW,σ(w).

Let Op′(Gσ) G Gσ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G. Let W be a Weyl group of G, π a natural homomorphism from N = NG(T) into W. Two elements w1, w2 are called σ-conjugate if w1 = (w−1)σw2w for some element w of W.

Proposition

There is a bijection between the G-classes of σ-stable maximal tori of G and the σ-conjugacy classes of W. Define CW,σ(w) = {x ∈ W|(x−1)σwx = w}.

Proposition

Let gσg−1 ∈ N and π(gσg−1) = w. Then (NG(T

g))σ/(T g)σ ≃ CW,σ(w).

Let Op′(Gσ) G Gσ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G. Let W be a Weyl group of G, π a natural homomorphism from N = NG(T) into W. Two elements w1, w2 are called σ-conjugate if w1 = (w−1)σw2w for some element w of W.

Proposition

There is a bijection between the G-classes of σ-stable maximal tori of G and the σ-conjugacy classes of W. Define CW,σ(w) = {x ∈ W|(x−1)σwx = w}.

Proposition

Let gσg−1 ∈ N and π(gσg−1) = w. Then (NG(T

g))σ/(T g)σ ≃ CW,σ(w).

Let Op′(Gσ) G Gσ be a finite group of Lie type. Two maximal tori in G are not necessary conjugate in G. Let W be a Weyl group of G, π a natural homomorphism from N = NG(T) into W. Two elements w1, w2 are called σ-conjugate if w1 = (w−1)σw2w for some element w of W.

Proposition

There is a bijection between the G-classes of σ-stable maximal tori of G and the σ-conjugacy classes of W. Define CW,σ(w) = {x ∈ W|(x−1)σwx = w}.

Proposition

Let gσg−1 ∈ N and π(gσg−1) = w. Then (NG(T

g))σ/(T g)σ ≃ CW,σ(w).

Linear groups

In case of linear group W ≃ Symn and the σ-conjugacy classes CW,σ(w) of W are coincide with ordinary conjugacy classes of symmetric group. Each such class corresponds to the cycle-type (n1)(n2) . . . (nm). Let {n1, . . . , nm} be a partition of n. We assume that n1 = . . . = nl1 < . . . < nl1+...+lr−1+1 = . . . = nl1+...+lr and a1 = nl1l1, a2 = nl1+l2l2, . . . , ar = nl1+...+lrlr.

Theorem

Let T be a maximal torus of G = SLn(q) with the cycle-type (n1)(n2) . . . (nm). Then T has a complement in N if and only if q is even or ai is odd for some 1 i r.

Linear groups

In case of linear group W ≃ Symn and the σ-conjugacy classes CW,σ(w) of W are coincide with ordinary conjugacy classes of symmetric group. Each such class corresponds to the cycle-type (n1)(n2) . . . (nm). Let {n1, . . . , nm} be a partition of n. We assume that n1 = . . . = nl1 < . . . < nl1+...+lr−1+1 = . . . = nl1+...+lr and a1 = nl1l1, a2 = nl1+l2l2, . . . , ar = nl1+...+lrlr.

Theorem

Let T be a maximal torus of G = SLn(q) with the cycle-type (n1)(n2) . . . (nm). Then T has a complement in N if and only if q is even or ai is odd for some 1 i r.

Symplectic and orthogonal groups

Let n = n′ + n′′, {n1, . . . , nk} and {nk+1, . . . , nm} be partitions

• f n′ and n′′, respectively. A set {−n1, . . . , −nk, nk+1, . . . , nm}

will be called a cycle-type and denoted by (n1) . . . (nk)(nk+1) . . . (nm). As above we assume that n1 = . . . = nl1 < . . . < nl1+...+lr−1+1 = . . . = nl1+...+lr Let a1 = nl1l1, a2 = nl1+l2l2, . . . , ar = nl1+...+lrlr.

Theorem

Let q be a power of a prime p. Let T a maximal σ-invariant torus of G, T a corresponding maximal torus of G with the cycle-type (n1) . . . (nk)(nk+1) . . . (nm) and m > 4. Then Group Conditions of existence of a complement PSp2n(q) q is even q ≡ 1 (mod 4) Ω2n+1(q) ai is odd for some 1 i r q ≡ 3 (mod 4) and ni is even for all 1 i m q ≡ 1 (mod 4) PΩ+

2n(q) ai is odd for some 1 i r

q ≡ 3 (mod 4) and ni is even for all 1 i m q is even PSLn(q) ai is odd for some 1 i r (n)2 < (q − 1)2

The answer for Problem 2 for groups E6(q) is in the following table:

No Representative ω |ω| |CW (ω)| Structure of CW (w) Torus T ⋊ 1 1 1 51840 O5(3) : Z2 (q − 1)6 — 2 ω1 2 1440 S2 × S6 (q − 1)4 × (q2 − 1) — 3 ω1ω2 2 192 D8 × S4 (q − 1)2 × (q2 − 1)2 — 4 ω3ω1 3 216 Z3 × (S2

3 : Z2)

(q − 1)3 × (q3 − 1) + 5 ω2ω3ω5 2 96 Z2 × Z2 × S4 (q2 − 1)3 — 6 ω1ω3ω5 6 36 Z6 × S3 (q − 1) × (q2 − 1) × (q3 − 1) + 7 ω1ω3ω4 4 32 Z4 × D8 (q − 1)2 × (q4 − 1) — 8 ω1ω4ω6ω36 2 1152 (q + 1)2 × (q2 − 1)2 — 9 ω1ω2ω3ω5 6 24 Z3 × D8 (q2 − 1) × (q + 1)(q3 − 1) + 10 ω1ω5ω3ω6 3 108 Z3 × S3 × S3 (q − 1) × (q2 + q + 1) × (q3 − 1) + 11 ω1ω4ω6ω3 4 16 Z4 × Z2 × Z2 (q2 − 1) × (q4 − 1) — 12 ω1ω4ω3ω2 5 10 Z2 × Z5 (q − 1) × (q5 − 1) + 13 ω3ω2ω5ω4 6 36 Z6 × S3 (q2 − 1) × (q − 1)(q3 + 1) + 14 ω3ω2ω4ω14 4 96 SL2(3) : Z4 (q − 1)(q2 + 1)2 — 15 ω1ω5ω3ω6ω2 6 36 Z6 × S3 (q2 + q + 1) × (q + 1)(q3 − 1) + 16 ω1ω4ω6ω3ω36 4 96 Z4 × S4 (q + 1)2 × (q4 − 1) — 17 ω1ω4ω5ω3ω36 10 10 Z10 (q + 1)(q5 − 1) + 18 ω1ω4ω6ω3ω5 6 12 Z6 × Z2 (q2 + q + 1) × (q − 1)(q3 + 1) + 19 ω2ω5ω3ω4ω6 8 8 Z8 (q2 − 1)(q4 + 1) + 20 ω20ω5ω4ω3ω2 12 12 Z12 (q − 1)(q2 + 1)(q3 + 1) + 21 ω1ω5ω2ω3ω6ω36 3 648 (((Z2

3 ) : Z3) : Q8) : Z3 (q2 + q + 1)3

+ 22 ω1ω4ω6ω3ω5ω36 6 36 Z6 × S3 (q + 1) × (q5 + q4 + q3 + q2 + q + 1) + 23 ω1ω4ω6ω3ω2ω5 12 12 Z12 (q2 + q + 1)(q4 − q2 + 1) + 24 ω1ω4ω14ω3ω2ω6 9 9 Z9 (q6 + q3 + 1) + 25 ω1ω4ω14ω3ω2ω31 6 72 Z3 × SL2(3) (q2 − q + 1) × (q4 + q2 + 1) +

The answer for Problem 2 for groups E6(q) is in the following table:

No Representative ω |ω| |CW (ω)| Structure of CW (w) Torus T ⋊ 1 1 1 51840 O5(3) : Z2 (q − 1)6 — 2 ω1 2 1440 S2 × S6 (q − 1)4 × (q2 − 1) — 3 ω1ω2 2 192 D8 × S4 (q − 1)2 × (q2 − 1)2 — 4 ω3ω1 3 216 Z3 × (S2

3 : Z2)

(q − 1)3 × (q3 − 1) + 5 ω2ω3ω5 2 96 Z2 × Z2 × S4 (q2 − 1)3 — 6 ω1ω3ω5 6 36 Z6 × S3 (q − 1) × (q2 − 1) × (q3 − 1) + 7 ω1ω3ω4 4 32 Z4 × D8 (q − 1)2 × (q4 − 1) — 8 ω1ω4ω6ω36 2 1152 (q + 1)2 × (q2 − 1)2 — 9 ω1ω2ω3ω5 6 24 Z3 × D8 (q2 − 1) × (q + 1)(q3 − 1) + 10 ω1ω5ω3ω6 3 108 Z3 × S3 × S3 (q − 1) × (q2 + q + 1) × (q3 − 1) + 11 ω1ω4ω6ω3 4 16 Z4 × Z2 × Z2 (q2 − 1) × (q4 − 1) — 12 ω1ω4ω3ω2 5 10 Z2 × Z5 (q − 1) × (q5 − 1) + 13 ω3ω2ω5ω4 6 36 Z6 × S3 (q2 − 1) × (q − 1)(q3 + 1) + 14 ω3ω2ω4ω14 4 96 SL2(3) : Z4 (q − 1)(q2 + 1)2 — 15 ω1ω5ω3ω6ω2 6 36 Z6 × S3 (q2 + q + 1) × (q + 1)(q3 − 1) + 16 ω1ω4ω6ω3ω36 4 96 Z4 × S4 (q + 1)2 × (q4 − 1) — 17 ω1ω4ω5ω3ω36 10 10 Z10 (q + 1)(q5 − 1) + 18 ω1ω4ω6ω3ω5 6 12 Z6 × Z2 (q2 + q + 1) × (q − 1)(q3 + 1) + 19 ω2ω5ω3ω4ω6 8 8 Z8 (q2 − 1)(q4 + 1) + 20 ω20ω5ω4ω3ω2 12 12 Z12 (q − 1)(q2 + 1)(q3 + 1) + 21 ω1ω5ω2ω3ω6ω36 3 648 (((Z2

3 ) : Z3) : Q8) : Z3 (q2 + q + 1)3

+ 22 ω1ω4ω6ω3ω5ω36 6 36 Z6 × S3 (q + 1) × (q5 + q4 + q3 + q2 + q + 1) + 23 ω1ω4ω6ω3ω2ω5 12 12 Z12 (q2 + q + 1)(q4 − q2 + 1) + 24 ω1ω4ω14ω3ω2ω6 9 9 Z9 (q6 + q3 + 1) + 25 ω1ω4ω14ω3ω2ω31 6 72 Z3 × SL2(3) (q2 − q + 1) × (q4 + q2 + 1) +