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LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Third or Fourth Year)
MATHEMATICS & STATISTICS
Math 325 : Representation Theory 2 hours
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40.
SECTION A
A1. Let G = D 12 = {biaj^ : 0 ≤ i ≤ 1 , 0 ≤ j ≤ 5 }, where a^6 = 1 = b^2 and ab = ba−^1 ; let
A =
, B =
Show that the map ρ : G → GL 2 (C) defined by
ρ(biaj^ ) = BiAj^ for 0 ≤ i ≤ 1 , 0 ≤ j ≤ 5
is a representation of G. [10]
A2. Let G = C 6 = 〈a〉, and V = 〈v 1 , v 2 〉 be the CG-module defined by
av 1 = −v 1 − v 2 , av 2 = 3v 1 + 2v 2.
Find all CG-submodules of V. [10]
please turn over
SECTION A continued
A3. Let G = C 3 = { 1 , a, a^2 }, and consider the representation ρ : G → GL 3 (C) defined by
ρ(a) =
Let T be the matrix (^)
1 x y 0 1 2 0 0 1
and define the representation σ : G → GL 3 (C) by σ(g) = T −^1 ρ(g)T for all g ∈ G. Find values of x and y for which each matrix σ(g) is of the form
and give σ(a) and σ(a^2 ) explicitly. [10] A4. (a) Let G be a subgroup of the symmetric group Sn. Define the permutation module V for G. [2] (b) Let G = S 3 , and let V = 〈v 1 , v 2 , v 3 〉 be the permutation module. Let B 1 be the natural basis v 1 , v 2 , v 3 , and B 2 be the basis v 1 + v 2 + v 3 , v 1 − v 2 , v 2 − v 3. Calculate the matrices [g]B 1 and [g]B 2 as g runs through G. [8]
A5. Let G = C 6 = 〈a〉, and V = 〈v 1 , v 2 〉 and W = 〈w 1 , w 2 〉 be the CG-modules defined by
av 1 = −v 1 + 3v 2 , aw 1 = 3w 1 − w 2 , av 2 = −v 1 + 2v 2 , aw 2 = 7w 1 − 2 w 2.
Find a basis for the vector space HomCG(V, W ). [10]
please turn over
SECTION B continued
B2. Let G be a finite group. (a) Let θ : V → W be a CG-homomorphism. Show that ker θ and im θ are CG-submodules of V and W respectively. [4] (b) State and prove Schur’s Lemma as it applies to CG-modules. [12] (c) Show that if G is abelian then every irreducible CG-module is 1-dimensional. [4] (d) Let G = C 3 × C 3 = {aj^ bk^ : 0 ≤ j, k ≤ 2 }, with a^3 = b^3 = 1 and ba = ab. Obtain all the irreducible representations of G, showing that no two of them are equivalent; deduce that G has no faithful irreducible representation. [10]
please turn over
SECTION B continued
B3. Let G = 〈a, b : a^4 = 1 = b^3 , ba = ab^2 〉; thus |G| = 12, and the elements of G are aj^ bk^ for 0 ≤ j ≤ 3 and 0 ≤ k ≤ 2. (a) If v ∈ CG satisfies bv = γv for some γ ∈ C, and w = av, show that
bw = γ^2 w.
[4] (b) Set ζ = − 12 + i
√ 3 2 =^ e^2 πi/^3. Find in^ CG^ elements^ v^0 , v^1 , v^2 of the form 1 +^ λb^ +^ μb^2 which satisfy bv 0 = v 0 , bv 1 = ζv 1 , bv 2 = ζ^2 v 2 , and show that v 0 , v 1 , v 2 form a basis of the subspace 〈 1 , b, b^2 〉 of CG. [6] (c) For j = 0, 1 , 2 let wj = avj , xj = awj and yj = axj , and set Zj = 〈vj , wj , xj , yj 〉. Show that Z 0 , Z 1 , Z 2 are CG-submodules of CG, and that
CG = Z 0 ⊕ Z 1 ⊕ Z 2. (^) [5]
(d) Explain why at this stage it can be deduced that each Zj must be reducible. [1] (e) By considering elements v 0 + αw 0 + α^2 x 0 + α^3 y 0 for appropriate values of α, decompose Z 0 as a direct sum of 1-dimensional CG-submodules. [7] (f) By considering elements v 1 ± x 1 and w 1 ± y 1 , decompose Z 1 as a direct sum of two 2-dimensional CG-submodules. [7]
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