课程名称︰代数二
课程性质︰数学系大二必修
课程教师︰于靖
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2016/06/20
考试时限（分钟）：210
试题 :
In answering the following problems, please give complete arguments as much
as possible. You may ask for any definition. You may use freely any Theorem
already proved (or Lemmas, Propositions) from the Course Lectures, or previous
courses on Linear Algebra. Previous Exercises assignments are NOT allowed to use
in doing these problems. You MUST write down the complete statements of the
theorems on which your arguments are based.
Notations:
Let K/F be a finite extension of fields, with α ∈ K. Then Tr (α) :=
K/F
Σ σ(α), where σ runs through all the embeddings of K into an algebraic
σ
closure of F which fix F.
We let Ｚ denote the cyclic group of order p.
p
For any group G, Z(G) denotes the center of G.
Let H ≦ G be a subgroup of a finite group. The induced representation of a
representation ρ of the subgroup H on the vector space V is the |CG-module
ρ
G
given by Ind (ρ) := |CG \otimes V . To work on induced representations, the
H |CH ρ
Frobenius reciprocity is a key.
Let M be a complex finite-dimensional representation of a finite group G. By
Maschke's theorem, M is equivalent to a direct sum of copies of irreducible
representations of G. Given any fixed irreducible representation N of G, its
multiplicity in M is the maximal number m such that there is a subrepresentation
of M which is equivalent to a direct sum of m copies of N.
Let ρ ,ρ be complex representations of a finite group G on vector
1 2
spaces, V , i = 1, 2. Their tensor product ρ \otimes ρ is given by:
ρ_i 1 2
(ρ \otimes ρ )(g) := ρ (g) \otimes ρ (g) ∈ GL(V \otimes V ).
1 2 1 2 ρ_1 ρ_2
We let V* denote the dual space Hom (V, |C). If h: V → W is a linear
|C
transformation, its transpose h* goes from W* to V*. Given ρ: G → GL(V) a
finite dimensional representation of a finite group G, there is a contragredient
representation ρ*: G → GL(V*) defined by:
-1
ρ*(g) := (ρ(g)*) .
Problems:
3 3 3 2 2 2 3
1. Let I := (x + y + z , x + y + z , (x + y + z) ) ⊂ |C[x, y, z]. Show that
3 3
x, y, z are in the radical of this ideal I. (Hint: consider the ideal (x + y +
3 2 2 2 3
z , x + y + z , (x + y + z) , 1 - wx) ⊂ |C[x, y, z, w], and use Gröbner
basis)
2. (1) Let K/F be a finite Galois extension. Show that Tr : K → F is a
K/F
surjective F-linear map.
(2) Suppose the Galois group of K/F is cyclic of order n, generated by σ.
Take θ ∈ K with Tr (θ) ≠ 0 and α ∈ K with Tr (α) = 0. Let β ∈ K be
K/F K/F
given as
2
β := (ασ(θ) + (α + σ(α))σ (θ) + ...
n-2 n-1
+ (α + σ(α) + ... + σ (α)) σ (θ)) / Tr (θ).
K/F
Show that α = β - σβ.
(3) Suppose F is a field of characteristic p ≠ 0, and K/F is a cyclic Galois
extension of degree p. Prove that K = F(α), where α is a root of the
p
polynomial x - x - a ∈ F[x], with a ∈ F.
3
3. Let p be an odd prime number. Consider a non-abelian group G of order p . In
last semester, we have classified these groups: there are two isomorphism
classes:
2
(Ｚ ) \rtimes Ｚ , Ｚ \rtimes Ｚ .
p p p^2 p
2
(1) Show that such group G always has p characters of degree 1.
(2) Show that such group has (p-1) irreducible characters of degree p.
Together with the characters of degree 1 these give the whole table of
irreducible characters for G.
(3) Let g be a fixed generator of Z(G). Verify that the irreducible
characters of degree p for such groups are given by (1 ≦ l ≦ p-1): χ (g) =
l
2πil/p
pe , and χ (h) = 0 if h \notin Z(G).
l
4. Let G be a finite group which has r conjugacy classes. Let ρ : G → W , i =
i i
1, ..., r be distinct irreducible complex representations of G up to
isomorphisms. Extend ρ by linearity as homomorphism from the group algebra |CG
i
~
to End (W ), denoted by ρ . Then, by Wedderburn's theorem, we arrive at the
|C i i
~
algebra isomorphism, for u := Σ a g ∈ |CG, u \mapsto (ρ (u)):
g∈G g i
~ r r
ρ: |CG → Π End (W ) \cong Π M (|C).
i |C i i=1 n_i
where n := dim W .
i |C i
r
On the other hand, given (u ) ∈ Π End (W ), verify the following
i i |C i
~-1
(analogue of the Fourier Inversion) formula: the element ρ (u ) = Σ a g
i g∈G g
∈ |CG is
1 r -1
a =── Σ n tr (ρ (g )u ).
g |G| i=1 i W_i i i
(Hint: use the orthogonality of the distinct irreducible characters.)
5. Let n be a positive integer. Consider the natural permutation representation
S → V, where V is the complex vector space with basis {e , ..., e } and
n 1 n
permutation σ ∈ S acting via e \mapsto e .
n i σ(i)
(1) Show that V is the direct sum of two subrepresentations of S , one is the
n
trivial representation, another is an irreducible representation ρ of degree
n
(n-1).
(2) Let S ⊂ S be the subgroup permuting the set {1, ..., n}. Prove that
n n+1
the representation ρ of S is restricted to a representation of S on V.
n+1 n+1 n
(3) Show that the irreducible representation ρ has multiplicity one in
n+1
S_{n+1}
the induced representation Ind (ρ ).
S_n n
6. Let ρ , ρ be complex representations of finite group G on vector spaces,
1 2
V , i = 1, 2. Let V := Hom (V , V ). For g ∈ G and v ∈ V, define the
ρ_i |C ρ_1 ρ_2
action:
-1
ρ(g)(v) := ρ (g) 。 v 。 ρ (g).
2 1
Verify that ρ is a representation of G which is isomorphic to the
representation
ρ* \otimes ρ ,
1 2
where ρ* is the contragredient representation of G on (V )*.
1 ρ_1
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