REPRESENTATION THEORY

K will denote a field and all algebras will be finite dimensional K-algebras.

1) Suppose that A is a central simple K-algebra and that B is a simple Kalgebra. Suppose that j_{1} and j_{2} are embeddings B ↪ A of K-algebras. Show that j_{1} and j_{2} are conjugate by an element of A^{×}. This is the Noether-Skolem theorem. [Hint: Consider A as a B ⊗_{K} A^{op} in two ways and show that these give isomorphic B ⊗_{K} A^{op}-modules.]

2) If K is an algebraicly closed field show that the Brauer group of K is trivial.

3) Show that the Brauer group of ℝ is ℤ/2ℤ. [Hint: If D is a central division algebra over ℝ choose an maximal field L and an element j ∈ D−L which normalises L and determine the relations they need to satisfy.]

4) Show that the Brauer group of a finite field K is trivial. [Hint: If D is a central division algebra, choose a maximal subfield L and show that D^{×} = ⋃_{g∈D^{×}/L^{×}} gL^{×}g^{−1}.]

5) Let A be a central simple K-algebra of dimension n^{2} and assume that K is perfect (every algebraic extension is separabale). Show that there is an isomorphism ϕ : A ⊗_{K} \overline{K} →^{∼} M_{n}(\overline{K}). If σ ∈ Gal (\overline{K}/K) show that σ ∘ ϕ|_{A} is GL_{n}(\overline{K})-conjugate to ϕ|_{A}. Deduce that for a ∈ A the characteristic polynomial of ϕ(a) lies in K[X]. Also show that it does not depend on the choice of ϕ. This polynomial is called the characteristic polynomial of a. Minus the coefficient of X^{n−1} is called the reduced trace tr_{A/K}a of a and (−1)^{n} times the constant coefficient is called the reduced norm N_{A/K}a of a.

Show that if a, a′ ∈ A and c ∈ K then tr_{A/K}(ca + a′) = ctr_{A/K}a + tr_{A/K}a′. Show also that if a, a′ ∈ A then N_{A/K}(aa′) = (N_{A/K}a)(N_{A/K}a′).

Also show that if a ∈ A and K(a) is a field then tr_{A/K}a = (n/[K(a) : K])tr_{K(a)/K}a and N_{A/K}a = (N_{K(a)/K}a)^{n/[K(a):K]}.

6) Let K be a local field. Show that the Brauer group of K^{nr} is trivial. [Hint: Suppose that D is a central division algebra. Show that v_{K} extends to a homomorphism v_{K} : D^{×} → ℚ satsifying v_{K}(1 + d) ≥ min{0, v_{K}(d)} for all d ∈ D. Let O_{D} denote the subset of d ∈ D with v_{K}(d) ≥ 0 and let Π be an element of O_{D} with minimal strictly positive valuation. Show that O_{D} is a (non-commutative) ring, that D = O_{D}[1/π_{K}] and that O_{D} = O_{K^{nr}} + ΠO_{D} and hence that O_{K^{nr}}[Π] is dense in O_{D}. Deduce that D is a field.]

7) Suppose that L is a finite field, G is a group and that ρ : G → GL_{n}(L) is a semisimple representation. Let K be the sub-field of L generated by tr ∧^{i} ρ(g) for all g ∈ G and i ∈ ℤ_{>0}. Show that ρ is conjugate to a representation into GL_{n}(K).

8) Let K be a perfect field and let g ∈ GL_{n}(K). Show that there are unique elements s and u in GL_{n}(K) such that g = su = us, such that s is semi-simple (i.e. its minimal polynomial has distinct roots) and u is unipotent (i.e. its characteristic polynomial is (X − 1)^{n}). [Hint: First treat the case that K is algebraicly closed and consider the Jordan normal form for g.]

9) Suppose that H is a subgroup of a finite group G and that ρ is a representation of H over a field K. Show that

tr Ind _{H}^{G}(ρ)(g) = ∑_{k∈H\G} tr ρ(kgk^{−1})

where the sum is over those k with kgk^{−1} ∈ H.

10) Let G be a finite group and (ρ, V) a representation of G over an algebraicly closed field of characteristic 0.

(a) Suppose that G is nilpotent but not abelian and that ρ is faithful and irreducible. Show that G has a normal abelian subgroup A which properly contains the centre of G. Show further that ρ(A) does not consist of scalars. Let χ be a character of A such that the χ-eigenspace V^{χ} of A in V is non-trivial and let H denote the subgroup of h ∈ G with ρ(h)V^{χ} = V^{χ}. Show that V ≅ Ind _{H}^{G}V^{χ}.

(b) Show that if G is nilpotent and ρ is irreducible then there is a subgroup H ⊂ G and a homomorphism χ : H → K^{×} such that ρ ≅ Ind _{H}^{G}χ. [Hint: Use induction on #G.]

(c) If G is nilpotent and ρ is irreducible, show that

ρ − (dim ρ)1 = (Ind _{H}^{G}σ) − (dim σ)(Ind _{H}^{G}1) + (dim σ) ∑_{χ}(χ − 1)

where H is a maximal proper subgroup of G, where σ is an irreducible representation of H and where χ runs over characters (one dimensional representations) of G/H.

(d) If G is nilpotent show that

ρ − (dim ρ)1 = ∑_{i}((Ind _{H_{i}}^{G}χ_{i}) − (Ind _{H_{i}}^{G}1))

where the H_{i} are subgroups of G and where χ_{i} : H_{i} → K^{×}.

(e) Now for any G, use Brauer’s theorem to show that we can find nilpotent subgroups H_{i} ⊂ G and representations σ_{i} of H_{i} and integers n_{i} such that

1 = ∑_{i} n_{i}Ind _{H_{i}}^{G}σ_{i}.

Deduce that

ρ − (dim ρ)1 = ∑_{i} n_{i}(Ind _{H_{i}}^{G}(ρ|_{H_{i}} ⊗ σ_{i}) − (dim ρ)Ind _{H_{i}}^{G}σ_{i}).

(f) Show that we can find nilpotent subgroups H_{i} ⊂ G and representations χ_{i} : H → K^{×} of H_{i} and integers n_{i} such that

ρ − (dim ρ)1 = ∑_{i} n_{i}(Ind _{H_{i}}^{G}χ_{i} − Ind _{H_{i}}^{G}1).