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LANCASTER UNIVERSITY
2007 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 8 = { 1 , a, a^2 , a^3 , b, ba, ba^2 , ba^3 }, where a^4 = 1 = b^2 and ab = ba−^1 ; set
A =
, B =
Show that the map ρ : G → GL 2 (C) defined by
ρ(biaj^ ) = BiAj^ for 0 ≤ i ≤ 1 , 0 ≤ j ≤ 3
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 = 3v 1 − v 2 , av 2 = 7v 1 − 2 v 2.
Find all CG-submodules of V. [10]
please turn over
SECTION A continued
A3. Let G = C 4 = { 1 , a, a^2 , a^3 }, 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) State Maschke’s Theorem. (b) Define what it means for a CG-module to be completely reducible, and show that every non-zero CG-module is completely reducible. [10]
A5. Let G = C 4 = { 1 , a, a^2 , a^3 }, and let V = 〈v 1 , v 2 〉 and W = 〈w 1 , w 2 〉 be the CG-modules defined by
av 1 = −v 2 , av 2 = v 1 , aw 1 = − 3 w 1 − w 2 , aw 2 = 8w 1 + 3w 2.
Find if there are any non-zero CG-homomorphisms from V to W , and hence decide if V and W have a common composition factor. [10]
please turn over
SECTION B continued
B2. Let G be a finite group. (a) (i) State Schur’s Lemma as it applies to CG-modules. [4] (ii) Use Schur’s Lemma to deduce dim (HomCG(V, W )) when V and W are both irre- ducible. [4] (b) Let V be a CG-module, with an irreducible CG-submodule U. Suppose that V = U 1 ⊕ · · · ⊕ Ur with each Ui an irreducible CG-submodule of V. Given u ∈ U , we may then write u = u 1 + · · · + ur uniquely with ui ∈ Ui for all i; define maps πi : U → Ui for 1 ≤ i ≤ r by πi(u) = ui. (i) Show that each map πi is a CG-homomorphism. (You may assume that they are linear maps.) [3] (ii) Deduce that U ∼= Ui for some i. [2] (iii) Assume that the ordering is such that U ∼= Ui for i = 1,... , k and U 6 ∼= Ui for i = k + 1,... , r. Show that U ≤ U 1 ⊕ · · · ⊕ Uk. [4] (c) Let V and W be CG-modules. (i) Define what it means for V to have a given irreducible CG-module as a composition factor , and for V and W to have a common composition factor. [3] (ii) Let V = V 1 ⊕· · ·⊕Vr and W = W 1 ⊕· · ·⊕Ws, with each Vi and Wj irreducible. Using (a)(ii), show that dim (HomCG(V, W )) is the number of pairs (i, j) with Vi ∼= Wj. [Any result on additivity of dimensions of spaces of CG-homomorphisms may be used provided it is clearly stated.] [2] (d) Let G = C 2 = 〈a〉, and set V = 〈v 1 , v 2 , v 3 〉, W = 〈w 1 , w 2 , w 3 〉, where
av 1 = 7v 1 − 8 v 2 − 16 v 3 , aw 1 = w 1 + 2w 2 + 6w 3 , av 2 = − 2 v 1 + v 2 + 4v 3 , aw 2 = − 6 w 1 − 7 w 2 − 18 w 3 , av 3 = 4v 1 − 4 v 2 − 9 v 3 , aw 3 = 2w 1 + 2w 2 + 5w 3.
Write V 1 = 〈v 1 + 2v 2 − v 3 〉, V 2 = 〈 2 v 1 + 2v 2 − 3 v 3 〉, V 3 = 〈v 1 − v 2 − 2 v 3 〉, so that V = V 1 ⊕ V 2 ⊕ V 3 , and W 1 = 〈w 1 + w 2 + 3w 3 〉, W 2 = 〈 5 w 1 + 3w 2 + 4w 3 〉, W 3 = 〈 2 w 1 + w 2 + w 3 〉, so that W = W 1 ⊕ W 2 ⊕ W 3. Use (c)(ii) to determine dim (HomCG(V, W )). [8]
please turn over
SECTION B continued
B3. Let G = 〈a, b : a^5 = 1 = b^4 , ab = ba^2 〉; thus |G| = 20, and the elements of G are biaj^ for 0 ≤ i ≤ 3 and 0 ≤ j ≤ 4. (a) If u ∈ CG satisfies au = ζu for some ζ ∈ C, and v = bu, show that
av = ζ^2 v. [4]
(b) Let ζ ∈ C satisfy ζ^5 = 1. Find λ 1 , λ 2 , λ 3 , λ 4 ∈ C such that
w = 1 + λ 1 a + λ 2 a^2 + λ 3 a^3 + λ 4 a^4 ∈ CG
satisfies aw = ζw. [4] (c) Set ω = e^2 πi/^5 so ω^5 = 1; for j = 0, 1 ,... , 4 let wj be the element of CG given by (b) as satisfying awj = ωj^ wj. Check that^15
∑^4
j=
ωjkwj = ak^ and hence show that w 0 , w 1 ,... , w 4 is a basis of the subspace 〈 1 , a,... , a^4 〉 of CG. [6] (d) For j = 0, 1 ,... , 4 let xj = bwj , yj = bxj and zj = byj , and set Vj = 〈wj , xj , yj , zj 〉. Show that Vj is a CG-submodule of CG for j = 0, 1 ,... , 4. By noting that a similar argument to that in (c) shows that the xj , yj and zj form bases of certain subspaces, deduce that
CG = V 0 ⊕ V 1 ⊕ V 2 ⊕ V 3 ⊕ V 4. [8]
(e) By considering elements w 0 + λx 0 + λ^2 y 0 + λ^3 z 0 for appropriate values of λ, decompose V 0 as a direct sum of 1-dimensional CG-submodules. [8]
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