Classical Groups, Character Degrees, Unitriangular Actions, and Quadratic Forms

Andrea Previtali Pisa, 8 April 2005

Sylow’s Theorems

Let G be a finite group, p a prime divisor of |G|. We say that P is a p-group if |P | is a p-power. A subgroup P ≤ G is a Sylow p-subgroup of G if P is a p-group and if P ≤ Q ≤ G, Q a p-group, then Q = P. We denote Syl_{p}(G) the collection of maximal p-subgroups of G. Theorem 1 (Sylow 1876). Let |G| = p^{a}m, p ∤ m, then if P ∈ Syl_{p}(G), then |P | = p^{a}; if P, Q ∈ Syl_{p}(G), then Q = P^{g}; |Syl_{p}(G)| = |G : N_{G}(P)| ≡_{p} 1.

General Linear Groups

Let F = 𝔽_{p}, V = F^{n}, and G = GL_{n}(p) = Aut(V).

Theorem 2 (Sharp transitivity). Let b, b′ be two ordered bases of V, then there exists a unique g ∈ GL_{n}(p) such that b^{g} = b′. In particular, |G| equals the number of ordered bases of V, so

|G| = n ; ∏ ; i=1 (p^{n} − p^{i}) = p^{\binom{n}{2}} n ; ∏ ; i=1 (p^{i} − 1).

Theorem 3. Let b = (v_{1}, …, v_{n}) be an ordered basis, V_{i} = ⟨v_{1}, …, v_{i}⟩ and P = {g ∈ G | v_{i}g − v_{i} ∈ V_{i−1}, i = 1, …, n}. Then P ∈ Syl_{p}(G).

Double Cosets

Let G ≥ H, K and set g ∼ x iff g = hxk, h ∈ H, k ∈ K. Equivalence classes are called (H, K)-double cosets.

Theorem 4. Let |H| = n, F = 𝔽_{p}, V = ⨁_{h∈H} F v_{h} and G = GL_{n}(p). Define R : H → G via v_{h}R(x) = v_{hx}. Then H ≃ R(H) ≤ G. Let P ∈ Syl_{p}(G), then there exists g ∈ G such that H ∩ P^{g} ∈ Syl_{p}(H).

Proof. Let G = ∐ HxP, then |HxP | = |H||P |/|H ∩ P^{x^{−1}}| and

|G : P | = _{x}^{∑} |H : H ∩ P^{x^{−1}}|.

Since |G : P | ≢_{p} 0 there exists x such that |H : H ∩ P^{x^{−1}}| ≢_{p} 0. Take g = x^{−1}.

Sesquilinear Forms

Let q = p^{a}, p prime, F = 𝔽_{q}, V = F^{n}. We call f a sesquilinear form on V if f : V × V → F is biadditive; f(av, w) = af(v, w) = f(v, a^{α}w), where α ∈ Aut(F).

Reflexive Forms

We say f is non-degenerate if f(v, V) = 0 implies v = 0 and reflexive if f(v, w) = 0 implies f(w, v) = 0. The only non-degenerate reflexive sesquilinear forms are: alternating: f(v, v) = 0; symmetric: f(v, w) = f(w, v); hermitian: f(v, w) = f(w, v)^{α}, α^{2} = id_{F} ≠ α ∈ Aut(F).

Finite Classical Groups

Given (V, f), f reflexive, the isometries of V constitute a classical group: linear: f ≡ 0; symplectic: f alternating; orthogonal : f symmetric; unitary: f hermitian.

Natural Sylow Subgroups

G classical, char (G) = char (F) = p, V = F^{2k+ϵ} , ϵ ∈ {0, 1}, then P ∈ Syl_{p}(G) has shape

P = A ⋊ U,

where U ∈ Syl_{p}(GL_{k}(q)) and A/Z(A) is abelian.

We call H = ⟨u, v⟩ hyperbolic plane if f(u, u) = f(v, v) = 0 and f(u, v) = 1.

Isotropic Case

Assumption: V is the orthogonal sum of k hyperbolic planes. Let R = (F)_{k}, τ ∈ End (R), (a_{ij})^{τ} = (a_{ji}^{α}) then: symmetric: A = ker (1 − τ), α = id_{F}; alternating: A = R^{(1−τ)} , α = id_{F}; hermitian: A = ker (1 − τ), α^{2} = id_{F} ≠ α.

Alternating≡Antisymmetric for p ≠ 2.

Symplectic Group

Let G = Sp_{2k}(q), q = p^{a}, F = 𝔽_{q}, R = (F)_{k}, τ transposition, A = ker (1 − τ) and U ∈ Syl_{p}(GL_{k}(q)) the group of lower unitriangular matrices. Then U acts on A via

a^{u} = u^{τ}au.

Proposition 5. P = A ⋊ U is a Sylow p-subgroup of Sp_{2k}(q).

Proof. d(u) = diag (u, u^{−τ}), u ∈ U, and m(a) = ( \begin{matrix}I_{k} & 0 \\ a & I_{k} \end{matrix} ), a ∈ A, fix

f = ( \begin{matrix}0 & I_{k} \\ −I_{k} & 0 \end{matrix} ); m(a)^{d(u)} = m(u^{τ}au); |AU| = q^{\binom{k+1}{2}}q^{\binom{k}{2}} = q^{k^{2}} =

|Sp_{2k}(q)|_{p}.

Conjugacy Classes Sizes

G acts on G via h^{g} = g^{−1}hg, the conjugation action. The Orbits are the conjugacy classes g^{G} of size

|g^{G}| = |G : C_{G}(g)|.

Let ccs(G) = {|g^{G}| : g ∈ G} denote the (multi)-set of conjugacy classes sizes. If G is a group such that ccs(G) is a set of n-powers, we say G is an n-power size group.

A p-group P is a p-power size group.

Character Degrees

Let Irr(G) be the set of irreducible complex characters of G. Let cd(G) = {χ(1) : χ ∈ Irr(G)} denote the (multi)-set of character degrees. If G is a group such that cd(G) is a set of n-powers, we say G is an npower degree group. Theorem 6 (Itô 1951). If χ ∈ Irr(G), then χ(1)||G|. A p-group P is a p-power degree group.

Polynomial Equations over 𝔽_{q}

How can one count the number of solutions of polynomial equations over a finite field?

Theorem 7 (Warning). Let q = p^{a}, F = 𝔽_{q}, f ∈ F [x_{1}, …, x_{n}] with deg (f) < n, N = |{c ∈ F^{n} : f(c) = 0}|. Then N ≡_{p} 0.

Can we substitute the characteristic with the cardinality of F?

Theorem 8 (Ax). Let q = p^{a}, F = 𝔽_{q}, f ∈ F [x_{1}, …, x_{n}] with d = deg (f) < n, N = |\{c ∈ F^{n} : f(c) = 0\}|, b = ⌊\frac{n}{d}⌋. Then N ≡_{q^{b}} 0.

Conjugacy Class Sizes: II

Theorem 9 (P. 1999). Let G = G(q) be a finite classical group over 𝔽_{q^{m}}, m = 2 in the unitary case, m = 1 otherwise. Let q = p^{a} and P ∈ Syl_{p}(G). If p ≠ 2, then P is a q-power size group.

The same holds for G of exceptional type and p >> 0.

Conjecture: this holds for p a good prime

Character Degrees: II

Theorem 10 (P., Isaacs, Szegedy, Sangroniz). Let G = G(q) be a finite classical group over 𝔽_{q^{m}}, m = 2 in the unitary case, m = 1 otherwise. Let q = p^{a} and P ∈ Syl_{p}(G). Unless p = 2 and G symplectic or orthogonal, then P is a q-power degree group.

Power Series

Let F be a field, f a power series on F, A an associative F -algebra, y a nilpotent element of A. Then f(y) ∈ A. Define ρ_{y} : a ↦ ay and λ_{y} : a ↦ ya. They are commuting nilpotent F -linear maps on A. For x, y ∈ A nilpotent, i ∈ ℕ, let

ω_{i}(x, y) = i−1 ; ∑ ; t=0 λ_{x}^{t}ρ_{y}^{i−1−t},

then ω_{i}(x, y)(λ_{x} − ρ_{y}) = (λ_{x}^{i} − ρ_{y}^{i}). If x^{k} = y^{k} = 0, then ω_{i+1} (x, y)^{s} = 0 whenever si ≥ k.

Centralizers

Let f, g ∈ F [[x]], f_{1}g_{1} ≠ 0, x, y nilpotent element of an F -algebra A. Then [x, y] = 0 iff [f(x), g(y)] = 0. ⇒: obvious. ⇐: let t = f(x) and d = [y, t], then

0 = [g(y), t] = ∑ g_{i}ω_{i}(y, y).d = (g_{1}id + ν).d,

where ν is nilpotent.

Some examples

q = p^{a}, F = 𝔽_{q}, A ≤ (F)_{k} strictly lower triangular, f = exp , p ≥ k; same A, f(x) = 1 + x, any p;

B = \{( \begin{matrix}a & 0 \\ b & −a^{τ} \end{matrix} ) : a ∈ A, b = b^{τ} ∈ (F)_{k}\}, f(x) = \frac{1−x}{1+x} the

Cayley map, p ≠ 2. Notice that U = 1 + A = exp (A) is a Sylow p-subgroup of SL_{k}(q). Less obvious, but f(B) is a Sylow p-subgroup of Sp_{2k}(q).

Algebra Groups

Let J be a nilpotent associative 𝔽_{q}-algebra. We call G = 1 + J an algebra group and S ≤ G is a strong subgroup if |S ∩ (1 + L)| is a power of q for any subalgebra L of J.

Theorem 11 (Isaacs 1995). Let G be a 𝔽_{q}-algebra group. Then any strong subgroup S of G is a q-power degree group. In particular, G is a q-power degree group.

Corollary 12. The Sylow p-subgroup of SL_{k}(q) is a q-power degree group.

Theorem 13 (P., Szegedy, Sangroniz 2001). Let G = G_{n}(q) be a classical group, P ∈ Syl_{p}(G). Then P is a strong subgroup of U ∈ Syl_{p}(SL_{n}(q)), unless p = 2 and G is symplectic or orthogonal. In particular, P is a q-power degree group.

Even Characteristic

Let q = 2^{a} and P ∈ Syl_{2}(Sp_{4}(q)), then \frac{q}{2} ∈ cd(P) and ccs(P). How far are cd(P) and ccs(P) from being q-powers, when P is a Sylow 2-subgroup of Sp_{n}(q) or SO _{n}^{±}(q), q = 2^{a}? Assumption: Isotropic case, the natural module V =⊥_{i=1}^{k} H_{i}, H_{i} hyperbolic.

Orbit Sizes

Let F = 𝔽_{q^{m}}, m = 2 in the unitary case, m = 1 otherwise, R = (F)_{k}, P = A ⋊ U, where A ≲ R equals ker (1 − τ) or R^{(1−τ)} , (a_{ij})^{τ} = (a_{ji}^{α}).

U acts via a^{u} = u^{τ}au.

Theorem 14 (P., Gow, Marjoram 2001). The orbits of U on A have q-power sizes and contain a unique element in canonical form (essentially a monomial matrix).

Use Bruhat decomposition for the classical group G = ⨆ Bw_{i}B, B a Borel subgroup and w_{i} elements of the Weyl group.

Dual Actions

ψ : R × R → F_{0}, ψ(a, b) = Tr (ab^{τ}), F_{0} = C_{F} (α), defines a perfect pairing. So we may identify Â = hom (A, F_{0}) with R/A^{⊥}. If p ≠ 2 then R = ker (1 − τ) ⊥ R^{(1−τ)} , so Â ≃ R/A^{⊥} ≃ A. U acts on Â via λ^{u}(a^{u}) = λ(a) and any λ has shape λ_{b}(a) = Tr (ab^{τ}), b ∈ A. Let σ denote the reflection with respect to the second main diagonal. Theorem 15 (P. 1995). A and Â are NOT equivalent as U-sets, but

|λ_{b}^{U}| = |(b^{σ})^{U}|,

namely the same orbit sizes occur. (Weak equivalence.)

Dual Actions: Even case

Let p = 2 and G symplectic, then A = ker (1 − τ). Given r ∈ R define the quadratic form Q_{r}(x) = xrx^{τ} on V = 𝔽_{q}^{k}. Define a U-action on the set 𝒬(V) of quadratic forms on V via Q_{r}^{u} = Q_{uru^{τ}}. Theorem 16 (Gow 2001). Â ≃ R/A^{⊥} ≃ 𝒬(V) as U-sets. Theorem 17 (P. 2005). The orbit sizes set of U on 𝒬(V) is

\{\frac{q^{s}}{2^{j}} | 0 ≤ j ≤ k, \binom{2j}{2} ≤ s ≤ \binom{2k}{2}\} .

Linear Recurrence Relations

We obtain detailed information on the orbit sizes of U acting on sesquilinear and quadratic forms. We define enumerating polynomials and recover linear recurrence relations for them.

We denote ℓ_{n}(t) the polynomial ∑_{i≥0} ℓ_{i}t^{i}, where ℓ_{i} counts the number of orbits of order q^{i}.

Theorem 18 (P. 2005). Let a_{n} be the orbit size polynomial for alternating forms over 𝔽_{q} in dimension n. Set w = q − 1, then a_{n} ∈ ℕ[w, t] and, for n ∈ ℕ,

a_{n+2} = a_{n+1} + wt^{n}\frac{t^{n+1} − 1}{t − 1}a_{n},

a_{0} = a_{1} = 1, deg _{w} a_{n} = ⌊\frac{n}{2}⌋, deg _{t} a_{n} = \binom{n}{2} − ⌊\frac{n}{2}⌋;

Clifford Theory

Let ε ∈ ℂ^{∗}, |ε| = p, then μ_{b}(a) = ε^{ς(λ_{b}(a))} defines all elements of Irr(A), where ς : F → 𝔽_{p} denotes the field trace. Let χ ∈ Irr(P) and χ_{A} ≥ μ, μ ∈ Irr(A). Let I ≤ P be the Inertia subgroup of μ, then μ^{I} = ∑_{β∈Irr(I/A)} ϕβ, where ϕ_{A} = μ and χ = (ϕβ)^{P} ∈ Irr(G). So χ(1) = β(1)|P : I| and cd(P) = {β(1)|P : I| : β ∈ Irr(I/A), I = I(μ_{b}), μ_{b} ∈ Irr(A)}. If P ∈ Syl_{2}(Sp_{2k}(q)), then \{\frac{q^{s}}{2^{j}} | 0 ≤ j ≤ k, \binom{2j}{2} ≤ s ≤ \binom{2k}{2}\} ⊆ cd(P).

Problem: is there equality?