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Algebra III/Introduction to Algebra III: Representation Theory Due: Please upload solutions to NUCT by Tuesday, May 26, 2020.

Problem 1. Let π : G → GL(V ) be a real representation of finite dimension d.

Show that if π is irreducible and if d is odd, then the complexication π

C

: G → GL(V

C

)

is an irreducible complex representation.

Problem 2. Let f : k → k

0

be an extension of fields, let f

∗

: Vect

k

→ Vect

k0

be extension of scalars along f , and let f

∗

: Vect

k0

→ Vect

k

be restriction of scalars along f . Given k

0

-vector space V

10

and V

20

, we define

f

∗

(V

10

) ⊗

k

f

∗

(V

20

)

µ

// f

∗

(V

10

⊗

k0

V

20

) be the k-linear map given by µ(x

01

⊗

k

x

02

) = x

01

⊗

k0

x

02

.

(a) Give an example to show that, in general, the map µ is not and isomorphism.

Given k-vector spaces V

1

and V

2

, the composite k-linear map V

1

⊗

k

V

2

η⊗η

// f

∗

f

∗

(V

1

) ⊗

k

f

∗

f

∗

(V

2

)

µ

// f

∗

(f

∗

(V

1

) ⊗

k0

f

∗

(V

2

)) determines a unique k

0

-linear map

f

∗

(V

1

⊗

k

V

2

)

µ˜

// f

∗

(V

1

) ⊗

k0

f

∗

(V

2

).

(b) Calculate that for x

1

∈ V

1

, x

2

∈ V

2

, and b ∈ k

0

,

˜

µ((x

1

⊗

k

x

2

) ⊗

k

b) = (x

1

⊗

k

b) ⊗

k0

(x

2

⊗

k

1) = (x

1

⊗

k

1) ⊗

k0

(x

2

⊗

k

b).

(c) Show that the map ˜ µ is an isomorphism. [Hint: Write down a k

0

-linear map f

∗

(V

1

) ⊗

k0

f

∗

(V

2

) // f

∗

(V

1

⊗

k

V

2

)

and show that it is inverse to ˜ µ.]

1

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