Representation Theory Graded Homework 3 Due: 1 December
Representation Theory Graded Homework 3
Due: 1 December
If you type (or scan) your solutions, please e-mail them to “iuliana.ciocanea
(at) gmail (dot) com” before or on 1 December. If you write your solutions,
please hand them in in class on the 1st.
Unless otherwise stated, G is a finite group of order n with s conjugacy classes
and that {ϕ(1) , . . . , ϕ(s) } is a complete set of representatives for the equivalence
classes of irreducible representations of G, and let d1 , . . . , ds be their degrees
and χ1 , . . . , χs their characters.
1. Let H and K be subgroups of the finite group G.
|H| · |K|
|H| · |K|
= −1
.
[2]
|H ∩ gKg −1 |
|g Hg ∩ K|
(b) For any H < G, prove that
exist elements z1 , . . . , zr where
S there S
[G : H] = r such that G = zi H = Hzi .
[3]
(a) Prove that |HgK| =
2. Prove that all the characters of Sn are real, in the sense that χ(g) = χ(g)
for all characters χ and all g ∈ G.
[2]
3. Let G be a non-abelian group and let ϕ be an irreducible representation
of the center Z(G) of G. Show that the induced representation IndG
Z(G) ϕ
is not irreducible.
[3]
4. Use Frobenius Reciprocity to give another proof that each irreducible representation ϕ(k) occurs in the regular representation L with multiplicity
dk .
[2]
5. Let H = {e, a, . . . , ak−1 } be a cyclic normal subgroup of G of order k such
that NG (a) = H, and let χ be the character on H defined by χ(am ) =
[4]
e2πim/k . Prove that IndG
H χ is an irreducible character of G.
6. Consider the discrete Heisenberg group
consisting of matrices of the form
1
M (a, b, c) = 0
0
H < GL3 (Fp ) modulo a prime p
a
1
0
b
c .
1
Our goal is to compute the character table of H.
(a) Show that H has p conjugacy classes of size 1 and p2 − 1 conjugacy
classes of size p.
[2]
(b) Show that there are p2 degree one representations of H. (You don’t
need to write them down explicitly.)
[2]
1
(c) Now suppose that p = 3. Let K be the abelian subgroup where
a = 0, and consider the p − 1 representations
ψr (M (0, b, c)) = e2πirb/p .
Construct the representations induced from these representations to
H and show that they are irreducible.
[3]
(d) Write down explicitly the character table when p = 3. (Don’t worry
about representatives for the conjugacy classes: you may just denote
them by their sizes.)
[2]
7. Let n ≥ 3 and let λ = (n − 1, 1) be a partition of n. Compute the
Sprecht representation associated to λ and show that it is isomorphic to
the standard representation ϕ. (Recall that the standard representation is
the representation ϕ such that ϕ ⊕ χ1 is the permutation representation
associated to the usual action of Sn on {1, 2, . . . , n}, where χ1 is the trivial
representation.)
[5]
Points available: [30]
Full marks: [30]
2
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