Show that every matrix M ∈Gld(C) of finite order is conjugate to a diagonal matrix

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Prof. Dr. T. Kr¨amer Due: 26 October 2017

Please bring your written solutions to the first problem session next Thursday, or hand them in earlier in my office (RUD 25, room 1.425).

Problem 1. (Representations of finite abelian groups)

(a) Let d∈N={1,2, . . .}. Show that every matrix M ∈Gld(C) of finite order is conjugate to a diagonal matrix.

(b) Deduce that forn∈N, any representationρ:Z/nZ→Gld(C) is a direct sum of 1-dimensional representations.

(c) Generalize this to representations of arbitrary finite abelian groups.

(d) Show that in general, part (c) fails if the wordfiniteorabelian is omitted.

Problem 2. LetGbe a finite group, fix an arbitrary fieldk, and letV =k[G] denote the vector space of all functions f :G→k. In the first lecture we have defined the right respectively left regular representations ρ:G →Gl(V) and λ: G→Gl(V) by the formula

(ρ(g)(f))(x) = f(xg) and (λ(g)(f))(x) = f(g⁻¹x) for x, g∈G, f∈k[G].

(a) Verify thatρandλ, as defined above, are indeed representations.

(b) Express these two representations by matrices in the basis ofV given by the characteristic functions

e_g(x) =

(1 ifx=g

0 otherwise forg∈G.

(c) Find an explicit isomorphism between the representationsρandλ.

Problem 3. Let G = Z/nZ for some n ∈ N. Using the basis from problem 2c, describe explicitly how the (right) regular representation V = C[G] decomposes into a direct sum of 1-dimensional representations.

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Prof. Dr. T. Kr¨amer Due: 2 November 2017

Please bring your written solutions to the problem session next Thursday, or hand them in earlier in my office (RUD 25, room 1.425).

Problem 4. LetGbe a group andV ∈Rep_k(G) a representation over a fieldk.

(a) We have defined the symmetric and exterior powers ofV as quotients of tensor powers. Show that forchar(k)-n! the quotient maps

V^⊗n Symⁿ(V) and V^⊗n Altⁿ(V)

can be written as the projection onto a direct summand ofV^⊗n∈Rep_k(G).

(b) Give an example wherechar(k)|n! and where the above property fails.

Problem 5. LetV ∈Rep_k(G). Show that for every normalsubgroupHEGwe get a subrepresentation

V^H = {v∈V | hv=vfor allh∈H } ∈ Rep_k(G).

In general, does this property hold for non-normal subgroups as well?

Problem 6. Let the group G = Sl2(C) act on the polynomial ring C[x, y] in two variables via

(g(f))(x, y) =f(ax+cy, bx+dy) for f ∈C[x, y], g= a b

c d

∈Sl2(C).

(a) Show that this is a group action and for eachn∈N0, the degreenmonomials span a finite-dimensional subrepresentation

Vn = nXⁿ

ν=0

aν·x^νy^d−ν∈C[x, y]

aν∈C

o ∈ Rep_C(G).

(b) Let T ⊂Gbe the subgroup of diagonal matrices. Show that for eachn∈N0

the restrictionVn|T ∈Rep_C(T) is semisimple by decomposing it explicitly into a direct sum of irreducible representations.

(c) Deduce that for eachn∈N0, the representationVn∈Rep_C(G) is irreducible.

Problem 7. Using the ideas from problem 6, deduce for all integersm≥n≥0 the decomposition

V_m⊗V_n ' V_m+n⊕V_m+n−2⊕ · · · ⊕V_m−n in Rep_C(Sl₂(C)).

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Please bring your written solutions to the problem session next Thursday, or hand them in earlier in my office (RUD 25, room 1.425).

Problem 8. Let Gbe a group andV ∈Rep_k(G) an irreducible representation over an algebraically closed fieldk. Show that the following holds:

(a) For n∈N, any subrepresentation ofV^⊕n has the formW 'V^⊕r withr≤n and the inclusion

V^⊕r ' W ,→ V^⊕n

is given by anr×nmatrix with linearly independent rows overk.

(b) If v1, . . . , vn∈V are linearly independent, then the map

k[G] → V^⊕n given by a 7→ (av₁, . . . , av_n) is surjective.

(c) The natural mapk[G]→End_k(V) is surjective (apply (b) to a basis).

Problem 9. Show that the right or left adjoint of a functor, if it exists, is unique up to isomorphism. Deduce that there exist natural isomorphisms

Ind^G_H(U⊗Res^G_H(V)) ' Ind^G_H(U)⊗V and Ind^G_K(Ind^K_H(U)) ' Ind^G_H(U) for any subgroupsH ≤K≤Gof finite index and allU ∈Rep_k(H),V ∈Rep_k(G).

Problem 10. LetHEGbe a normal subgroup of finite index.

(a) Show that for any U ∈Rep_k(H) andV ∈Rep_k(G),

• ifV is semisimple, then Res^G_H(V) is semisimple.

• if Ind^G_H(U) is semisimple, thenU is semisimple.

(b) Show that ifchar(k)-n= [G:H], the converse implications also hold.

(c) Find counterexamples whenchar(k)|nor whenH ≤Gis not normal.

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Please hand in your written solutions before the lecture next Thursday. Note that there will be no problem session next week since we will exceptionally move the lecture to that time: 11:15 - 12:45 in room 1.013 (RUD25).

Problem 11. Suppose that the base fieldkis algebraically closed, and let NEGa normal subgroup for which the quotientG/Nis abelian. Show that for all irreducible representationsV1, V2∈Rep_k(G),

V1|N 'V2|N ⇐⇒ V1'V2⊗χ for some 1-dimensional χ∈Rep_k(G/N).

Problem 12. In this problem we discuss some character tables overk=C. (a) Find the character table ofS4and describe its irreducible representations.

(b) Determine which of these restricts to an irreducible representation of A₄ES₄ and find the character table of this alternating group.

(c) Find the character table of the groupsDandQpresented below. Are these two groups isomorphic?

D =

a, b|a⁴=b²=abab= 1

, Q =

a, b|a⁴= 1, bab=a, a²=b² .

Problem 13. Letm∈N. Theelementary symmetric polynomialsinx= (x1, . . . , xm) are defined by

en(x) = X

1≤i1<···<in≤m

xi₁· · ·xi_n for n ∈ N0.

(a) ExpandQm

i=1(t−xi) in terms of these elementary symmetric polynomials and deduce

nen(x) =

n

X

i=1

(−1)ⁱ⁺¹si(x)en−i(x) for si(x) = xⁱ₁+· · ·+xⁱ_m

ifn=m. Explain why the above formula then remains true also forn6=m.

(b) For finite groupsGandV ∈Rep_C(G), show that the characters of the exterior powers satisfy

χ_Altn(V)(g) = 1 n

n

X

i=1

(−1)ⁱ⁺¹χ_V(gⁱ)χ_Altn−i(V)(g) for g∈G, n∈N.

Write down a closed formula for these characters forn= 2,3.

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Please bring your written solutions to the problem session next Thursday as usual.

Problem 14. LetG=G₁×G₂ be the product of two groupsG₁, G₂, and let k be an algebraically closed field.

(a) Using the result from problem 8, show that for all irreducibleVi∈Rep_k(Gi) the tensor productV =V1V2∈Rep_k(G) is irreducible.

(b) Conversely, if V ∈ Rep_k(G) is irreducible and V1 ⊆V|G₁ is any irreducible constituent of its restriction toG1× {1} ⊆G, endow V2 = HomG₁(V1, V|G₁) with aG2-action and show that

V ' V1V2 in Rep_k(G).

(c) If|G|<∞andchar(k) = 0, find a shorter argument via character theory.

Problem 15. LetGbe a finite group.

(a) Show that thelower central series

G = G₁ D G₂ D · · · defined by G_i+1 = [G_i, G]

terminates with the trivial group iff theupper central series

{1} = Z₀EZ₁ E · · · defined by Z_i+1 = {z∈G | [z, g]∈Z_i∀g∈ G}

terminates with the whole group. We then sayGisnilpotent.

(b) Show that subgroups and quotients of nilpotent groups are nilpotent and that in a nilpotent groupGevery maximal abelian normal subgroupNEGis equal to its own centralizer: N=ZG(N).

Problem 16. Recall that a representation of a finite groupGis calledprimitiveif it is not induced from any proper subgroup.

(a) IfGhas a primitive faithful irreducible representation, show that any normal abelian subgroupNEGis contained in the centerZ(G).

(b) Using problem 16b, show that any irreducible representation of a nilpotent group ismonomial, i.e. induced from a 1-dimensional representation.

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Please bring your written solutions to the problem session next Thursday.

Problem 17. LetGbe a finite group of ordern. Show that forg∈Gthe following are equivalent:

(a) For everyV ∈Rep_C(G) we haveχV(g)∈Q.

(b) The element gis conjugate tog^mfor allm∈Zwithgcd(m, n) = 1.

What does this say in the special case of symmetric and alternating groups?

Problem 18. LetGbe a finite group.

(a) Show that up to isomorphism, the character table of the group determines the centerZ ⊆C[G] of the group algebra.

(b) For any irreducible V ∈Rep_C(G) with central character ω :Z −→C, verify that

ω(eK) = χV(g)· |K|

dimV for g ∈ G, K = Cl(g), eK = X

h∈K

eh ∈ Z.

(c) Show that if

X

K∈Cl(G)

eK = c·Y

K

eK

for somec∈C, then the character of any non-trivial irreducibleV ∈Rep_C(G) vanishes somewhere, and deduce that thenG= [G, G].

Problem 19. Letλbe a partition ofd, and fix any Young tableau for it.

(a) Find the trace of right multiplication by the Young symmetrizer cλ =aλbλ

onA =C[Sd] and show

c²_λ = n_λ·c_λ for n_λ = d!

dim_CVλ

.

(b) ShowA ·aλbλ 'A ·bλaλ and deduce that for the transpose tableauλ^t one has

Vλ^t ' Vλ⊗sgn.

(c) Show that for the partitionλ= (d−1,1), the representationVλ∈Rep_C(Sd) is isomorphic to the standard representation of dimensiond−1.

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Prof. Dr. T. Kr¨amer Due: 14 December 2017

Problem 20. LetV =C² be the standard representation ofG=Gl2(C).

(a) Compute all the Schur functorsSλ(V)∈Rep_C(G).

(b) Determine all Schur polynomials in two variables, and verify directly that the product of any two of them is a sum of Schur polynomials.

(c) What does your result say about the representations ofSl2(C)?

Problem 21. Show by induction onnthat the ringS(x1, . . . , xn) =Z[x1, . . . , xn]^Sⁿ of symmetric functions is equal to a polynomial ring in the elementary symmetric polynomials:

Z[y₁, . . . , y_n] −→^∼ S(x₁, . . . , x_n),

y_ν 7→ e_ν(x₁, . . . , x_n) = X

1≤i1<···<iν≤n

x_i₁· · ·x_i_ν.

Hint: What is the kernel of the restriction map S(x₁, . . . , x_n)S(x₁, . . . , x_n−1)?

Problem 22. LetH, K ≤Gbe two subgroups of a finite group and S⊂Ga set of representatives for the double cosets: G = F

s∈SKsH.

(a) ForV ∈Rep_C(H) with induced representation Ind^G_H(V) =C[G]⊗_C[H]V show that

Res^G_KInd^G_H(V) = M

s∈S

W^s where W^s = C[K]·(s⊗V) ∈ Rep_C(K).

(b) PutV^s= Ad^∗_s(V) for the map Ads:sHs⁻¹∩K→H, x7→s⁻¹xs. Show that one gets a well-defined isomorphism

Ind^K_sHs−1∩K(V^s) −→^∼ W^s via k⊗v 7→ ks⊗v forv∈V,k∈K.

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Prof. Dr. T. Kr¨amer Due: 21 December 2017

For any partition λ, a Young tableau of shape λ is called astandard tableauif its rows and columns are increasing from left to right and top to bottom.

Problem 23. Letdλ∈Nbe the number of standard tableaux of shapeλ.

(a) Show that dλ =P

µ=λ\dµ where the sum runs over all Young diagrams µ that can be obtained by removing one box fromλ.

(b) Show that (deg(µ) + 1)·dµ=P

ν=µ∪dν where the sum runs over allν that can be obtained by adding one box to the Young diagramµ.

(c) Deduce inductively thatd! =P

λd²_λwhere the sum is over all λof degreed.

Problem 24. For any standard tableauT of degree d, letc_T ∈A =C[S_d] denote its Young symmetrizer.

(a) Let T₁ 6=T₂ be standard tableaux of the same shape. Suppose that reading row by row from left to right and top to bottom, the first entry where they differ is bigger inT2 than inT1. Show that thencT₂·cT₁ = 0.

(b) Deduce that for any set S of standard tableaux of degree d, the sum of the corresponding left submodulesP

T∈SAcT ⊆A is direct.

(c) Deduce via problem 23 that the decomposition of A into isotypic pieces is given by

A = M

deg(λ)=d

Aλ where Aλ = M

Tstandard of shapeλ

A ·cT ' V_λ^⊕d^λ

Problem 25. LetH≤Gbe finite groups andV ∈Rep_C(H).

(a) Show that χ_IndG

H(V)(g) = _|H|¹ P

x∈G,xgx⁻¹∈H χV(xgx⁻¹) for allg∈G.

(b) Using problem 22 (=Mackey) or otherwise, show that Ind^G_H(V) is irreducible if and only if

• V ∈Rep_C(H) is irreducible, and

• Hom_sHs⁻¹_∩H(V^s, V|_sHs⁻¹_∩H) = 0 for all non-trivial [s]∈H\G/H.

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Prof. Dr. T. Kr¨amer Due: 11 January 2018

Problem 27. Ford, e∈Nconsider the standard embeddingSd ,→Sd+e. (a) Show that for any partitionµof degreed+eone has

V_µ|_S_d ' M

λ

m_µλ·V_λ

where λ runs over partitions ofd whose Young diagram is contained in the diagram of µ, and mµλ is the number of ways to fill the labels 1,2, . . . , e into the boxes of the complement µ\λsuch that the rows and columns are increasing:

• • 1 2

• 3 4

• • 1 3

• 2 4

• • 1 4

• 2 3

• • 2 3

• 1 4

• • 2 4

• 1 3

m_(4,3)(2,1)= 5

(b) What does this say in the special cases whered= 1 ore= 1?

Problem 28. LetW ∈Rep(Sd) be the standard representation of dimensiond−1.

(a) Using the previous result, show by induction on d∈N that for everyν < d one has

V(d−ν,1,...,1) ' Alt^ν(W).

(b) Show similarly that

Sym²(W) ' 1⊕W ⊕V_(d−2,2) for d ≥ 4.

Problem 29. Show by induction on d ∈N that the only irreducible V ∈Rep(S_d) with 1<dim(V)< dare the standard representation and its tensor product with the sign character, plus the following exceptions:

• d= 4 andV 'V_(2,2),

• d= 6 andV 'V_(3,3)orV 'V_(2,2,2).

What are the dimensions dim(V) in these exceptional cases?

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Problem 30. LetM=M_n×n(R) be the vector space of realn×nmatrices andS⊂M the subspace of all symmetric matrices.

(a) Identifying these spaces with their respective tangent spaces, show that the derivative off : M →S, A7→ A^tA at any point A ∈ O(n) = f⁻¹(1) is the linear map

(Df)(A) : TA(M) −→ T1(S), X 7→ A^tX+X^tA.

(b) Show that this linear map is surjective, and deduce that the subsetO(n)⊂M is a compact Lie group. What is its dimension?

(c) Prove the corresponding statements forU(n) andSp(n).

Problem 31. Viewing the Hamiltonian quaternions H = C⊕Cj as a C-algebra, find an embedding M_n×n(H) ,→ M_2n×2n(C) of matrix rings that gives rise to an isomorphism

Sp(n) ' U(2n)∩Sp2n(C).

Problem 32. LetGbe a Lie group.

(a) Show that there is a unique homomorphismc:G→R^×such that for any left invariant volume formωonGthe pull-back under right translations is given by

ρ^∗_h(ω) =c(h)·ω for h∈G.

(b) Compute cfor the Lie group

G = n a b 0 1

b∈R, a∈R^×

o ⊂ Gl2(R).

(c) IfGis compact, show that c(h)∈ {±1} for allh∈G.

(d) Looking at orientations, show thatcis non-trivial for G=O(2).

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Problem 33. LetV =Rⁿ for somen∈N.

(a) Show that every continuous map f :V →Rwith f(x+y) =f(x) +f(y) for allx, y∈Rmust beR-linear.

(b) Deduce from this that any continuous homomorphism ρ:V →U(1) has the formρ(x) =e^if(x)for somef ∈Hom_R(V,R).

(c) Find all irreducible continuous representations of toriU(1)× · · · ×U(1).

Problem 34. Let G be a compact Lie group. If V₁, V₂ ∈ Rep_C(G) are isomorphic representations, show that for anyG-invariant Hermitian inner productsh·,·i_ionV_i there is an isomorphism

f ∈ HomG(V1, V2) with

f(u), f(v)

2 = hu, vi1 for allu, v ∈V1,g∈G.

Problem 35. For a compact Lie groupGand V ∈Irr_C(G), a bilinear form on V is called G-invariantif it is preserved under the action of the group. Show that there is a nondegenerate such bilinear form iff

HomG(V, V) 6= {0}.

Deduce that in this case there exists

J ∈ Hom_G(V, V) with J◦J =±id_V and that the bilinear form is

(a) symmetric iffV 'W ⊗_RCfor some real representationW ∈Rep_R(G), (b) alternating iffV is a vector space over the quaternionsHandV ∈Rep_H(G).

Looking at the character ofSym²(V) andAlt²(V), show

Z

G

χ_V(g²)dg =







+1 in case (a),

−1 in case (b), 0 otherwise.

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Prof. Dr. T. Kr¨amer Due: 01 February 2018

Problem 36. Letn∈N. The action ofG=O(n) on Rⁿ induces for eachd∈Nan action on

Vd =

homogenous polynomialsP :Rⁿ−→Cof degreed ⊂ C[x1, . . . , xn], given by

(g·P)(x₁, . . . , x_n) := P(y₁, . . . , y_n) for y_j :=

n

X

i=1

g_ij·x_i

where gij are the matrix entries ofg∈G. Show that h·,·i: Vd×Vd −→ C, hP, Qi := P _∂x^∂

1, . . . ,_∂x^∂

n

Q(x1, . . . , xn) defines a G-invariant Hermitian inner product, and that for this inner product one has an orthogonal splitting

V_d = ker(∆)⊕ x²₁+· · ·+x²_n

V_d−2 where ∆ :=

n

X

i=1

∂²

∂x²_i ∈ Hom_G(V_d, V_d−2).

Problem 37. Compute the Weyl groups ofSO(2n),SO(2n+ 1) andSp(n).

Problem 38. Consider theleftaction ofSp(n) onHⁿ (as aright H-module).

(a) Show that everyA∈Sp(n) has an eigenvector with a complex eigenvalue in the sense that

A·v = v·λ for some v ∈ Hⁿ\ {0} and λ ∈ C ⊂ H. (b) Deduce that the rightH-moduleHⁿ has an orthonormal basis of eigenvectors

for A with complex eigenvalues, and that there exists a matrix B ∈ Sp(n) such that

B⁻¹AB =







λ1 0

. ..

0 λn





 with λ1, . . . , λn ∈ U(1).

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Problem 39. Show thatR³ forms a Lie algebra with respect to the bracket defined by the cross-product

×: R³×R³ −→ R³, ~v×w~ :=





v2w3−v3w2

v3w1−v1w3

v1w2−v2w1





and find an isomorphism of Lie algebras

f : R³ −→^∼ so(3) := Lie(SO(3))

such that for the standard action ofso(3) onR³ one has~x×~v= (f(~x))(~v).

Problem 40. ForG=Sp(n), SO(2n) or SO(2n+ 1), determine the weights of the adjoint representation

Lie(G)⊗_RC, Ad

|_T ∈ Rep_C(T)

with respect to the maximal torusT ⊂Gthat we considered in the lecture.

Problem 41. Show thatSp(1)'SU(2) and this group is simply connected. Look at its adjoint representation to find a homomorphism SU(2) →SO(3) with finite kernel, soSO(3) is not simply connected. What is its fundamental group?

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LetGbe an arbitrary Lie group with Lie algebrag=Lie(G).

Problem 42. LetX1, X2,· · · ∈g\ {0}be a sequence of nonzero elements converging to zero. Passing to a subsequence we may assume that for suitableλ1, λ2,· · · ∈R>0

andYn:=λn·Xn the limit

Y := lim

n→∞Yn

exists and is nonzero. IfH ⊂Gis a closed topological subgroup withexp(Xn)∈H for alln∈N, show that

exp(sY) ∈ H for all s ∈ R.

Problem 43. Verify that for any closed topological subgroup H ⊂G the following subset is a real vector subspace:

h := n

dα dt|t=0

α:R→G smooth with α(R)⊆H and α(0) =eo

⊆ g.

Locally writing α=exp◦β, observe that ^dα_dt|_t=0 = lim_n→∞Y_n for Y_n =n·β(_n¹) and deduce

h = n

Y ∈g

exp(sY)∈H for all s∈R o

.

Problem 44. In the above setting, show that for any subspaceh⁰ ⊆gwithg=h⊕h⁰ there are open neighborhoods

0 ∈ U ⊆ h and 0 ∈ U⁰ ⊆ h⁰ such that the map

ϕ: U×U⁰ −→ G, (u, u⁰) 7→ exp(u)·exp(u⁰)

is an open embedding withϕ⁻¹(H) =U× {0}. Deduce that H is a Lie subgroup.