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Math 250: Higher Algebra Problem Set #5 (29 October 2004): Representations of finite groups
First, two exercises from Fulton-Harris on induced representations. Let G be a finite group, and H any subgroup.
- [Fulton-Harris, Exercise 3.16 on page 34] i) If U is a representation of G and W is a representation of H, show that
U ⊗ IndW = Ind(Res(U ) ⊗ W ).
In particular, Ind(Res(U )) = U ⊗ P , where P is the permutation representation of G on G/H. ii) If H ⊂ K ⊂ G with K also a subgroup, show that
IndGH W = IndGK (IndKH W )
(so induction, like restriction, is transitive).
- [Fulton-Harris, Exercise 3.19 on page 34] Let W be a finite-dimensional representation of H, and V = IndGH W. Quoting Fulton-Harris (p.34):
To compute [its] character, note that g ∈ G maps σW to gσW , so the trace is calculated from those cosets σ with gσ = σ, i.e., s−^1 gs ∈ H for [any, equivalently all] s ∈ σ. Therefore,
χInd W (g) =
gσ=σ
χW (s−^1 gs) (s ∈ σ arbitrary). (3.18)
i) If C is the conjugacy class of g in G, and C ∩ H decomposes into conjugacy classes D 1 ,... , Dr of H, then (3.18) can be rewritten as:
χInd W (g) = [G : H]
∑^ r
i=
|Di| |C|
χW (hi)
for hi ∈ Di. ii) If W is the trivial representation of H, then
χInd W (g) =
[G : H]
|C|
|C ∩ H|.
iii) Deduce the Frobenius reciprocity formula (Cor. 3.20 on p.35): for any finite-dimensional representations U, W of G, H respectively,
(χInd W , χU )G = (χW , χRes U )H.
We next describe “extraspecial 2-groups” and their representations. The 8-element dihedral and quaternion groups are the familiar first examples of extraspecial 2-groups.
- Let G be a finite group whose center contains a 2-element group {± 1 }, and let A be the quotient group G/{± 1 }. Let V be any irreducible representation of G, of dimension d. Show that the restriction of V to {± 1 } consists of either d copies of the trivial repre- sentation — in which case it comes from a representation of A — or d copies of the nontrivial 1-dimensional representation of {± 1 }. Equivalently, − 1 ∈ G acts on V by multiplication by a scalar, which is necessarily ei- ther +1 or −1. We’ll call these two kinds of representations of G “even” and “odd” respectively.
- Now suppose A is an “elementary abelian 2-group”, that is, a group isomorphic with (Z/ 2 Z)m^ for some m. Define a map (·, ·) : A × A → {± 1 } as follows: for any a, b ∈ A, let g ∈ G be either of the preimages of a, and let h ∈ G be either of the preimages of b; then (a, b) is the commutator ghg−^1 h−^1. Explain why (a, b) is in fact in {± 1 } and is well-defined (independent of the choice of g, h). Then show that (·, ·) is bilinear and alternating, i.e., that it satisfies the identities
(aa′, b) = (a, b)(a′, b), (a, bb′) = (a, b)(a, b′), (a, a) = 1, (a, b) = (b, a).
Prove that this pairing is nondegenerate if and only if {± 1 } is the center of G, and if and only if each g ∈ G − {± 1 } is conjugate to −g = (−1)g. In this case, G is said to be an “extraspecial 2-group”. Note that the 8-element dihedral and quaternion groups are indeed extraspecial 2-groups with m = 2.
- Now let G an extraspecial 2-group of order 2m+1. i) Show that G has 2m^ even representations, each of dimension 1. Deduce that G has a unique odd representation V and that V has dimension 2m/^2. [In particular it follows that m must be even — which we could also obtain from the nondegeneracy of the alternating form (·, ·).] Determine the character χV of this representation, and note that χV (g) ∈ R for all g ∈ G, whence V is necessarily either real or quaternionic. ii) For a ∈ A define Q(a) = g^2 where g is either of the preimages of a in G. Explain why this gives a well-defined map from A to {± 1 }, and show that it is a quadratic form whose associated bilinear form is (·, ·) (which, in our multiplicative notation for the G and its center, means that (a, b) = Q(ab)/(Q(a)Q(b)) for all a, b ∈ A). Use the formula of Exercise 3. 38 ∗^ (p.41) to conclude that V is real or quaternionic according as
a∈A Q(a) equals 2 m/ (^2) or − 2 m/ (^2).
Note that this confirms the known behavior of the 8-element extraspecial 2-groups. It turns out that any quadratic form on (Z/ 2 Z)m^ comes from some extraspecial 2-group, and thus in particular satisfies
a∈A Q(a) =^ ±^2
m/ (^2) ; and that two quadratic forms on (Z/ 2 Z)m (^) are equivalent under
Aut((Z/ 2 Z)m) = GL 2 m(Z/ 2 Z) if and only if their invariants 2−m/^2
a∈A Q(a) are equal. It follows that for each positive integer m there are two extraspecial 2-groups of order 2m+1, usually denoted 2 1+ + mand 21+ − m. For instance, the 8-element dihedral and quaternion groups are 21+2+ and 21+2 −.
Problem set is due in class Friday the 5th of November.
