Table Of ContentMath 55a Lecture Notes
Evan Chen
Fall 2014
This is Math 55, which is usually referred to as “Math 55”. The instructor
this year is Dennis Gaitsgory.
Contents
1 September 2, 2014 5
1.1 Boring stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 September 4, 2014 7
2.1 Review of equivalence relations go here . . . . . . . . . . . . . . . . . . . . 7
2.2 Universal property of a quotient . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 September 9, 2014 9
3.1 Direct products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Commutative diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Sub-things . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.4 Let’s play Guess the BS! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.6 Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.7 Examples of normal groups . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 September 11, 2014 13
4.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Modules, and examples of modules . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Abelian groups are Z-modules . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.5 Homomorphisms of R-modules . . . . . . . . . . . . . . . . . . . . . . . . 15
4.6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.7 Sub-modules and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 September 16, 2015 17
5.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Direct Sums of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Direct Products of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Sub-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.5 Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.6 Return to the Finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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Math 55a Notes (Evan Chen) Contents
6 September 18, 2014 23
6.1 Linearly independent, basis, span . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 Dimensions and bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.3 Corollary Party . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.4 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7 September 23, 2014 28
7.1 Midterm Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.2 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.3 Given a map we can split into invertible and nilpotent parts . . . . . . . . 29
7.4 Eigen-blah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.5 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
8 September 25, 2014 33
8.1 Eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.2 Generalized eigenspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.3 Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.4 Lemmata in building our proof . . . . . . . . . . . . . . . . . . . . . . . . 35
8.5 Proof of spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8.6 Recap of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
9 September 30, 2014 38
9.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
9.2 Taking polynomials of an endomorphism . . . . . . . . . . . . . . . . . . . 38
9.3 Minimal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
9.4 Spectral Projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
9.5 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
10 October 2, 2014 42
10.1 Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10.2 A big proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10.3 Young Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
10.4 Proof of Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
11 October 7, 2014 46
11.1 Order of a Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
11.2 Groups of prime powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
11.3 Abelian groups and vector spaces are similar . . . . . . . . . . . . . . . . 47
11.4 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 48
11.5 Not algebraically closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
12 October 9, 2014 50
12.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
12.2 How do G-sets talk to each other? . . . . . . . . . . . . . . . . . . . . . . 50
12.3 Common group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
12.4 More group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
12.5 Transitive actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
12.6 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
12.7 Corollaries of Sylow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 54
12.8 Proof of (b) of Sylow’s Theorem assuming (a) . . . . . . . . . . . . . . . . 55
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Math 55a Notes (Evan Chen) Contents
13 October 14, 2014 56
13.1 Proof of the first part of Sylow’s Theorem . . . . . . . . . . . . . . . . . . 56
13.2 Abelian group structure on set of modules . . . . . . . . . . . . . . . . . . 56
13.3 Dual Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
13.4 Double dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
13.5 Real and Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 57
13.6 Obvious Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
13.7 Inner form induces a map . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14 October 16, 2014 60
14.1 Artificial Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.2 Orthogonal Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.3 Orthogonal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
14.4 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
14.5 Spectral theory returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
14.6 Things not mentioned in class that any sensible person should know . . . 64
14.7 Useful definitions from the homework . . . . . . . . . . . . . . . . . . . . 64
15 October 21, 2014 66
15.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
15.2 Basic Properties of Tensor Products . . . . . . . . . . . . . . . . . . . . . 66
15.3 Computing tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 67
15.4 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
16 October 23, 2014 69
16.1 Tensor products gain module structure . . . . . . . . . . . . . . . . . . . . 69
16.2 Universal Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
16.3 Tensor products of vector spaces . . . . . . . . . . . . . . . . . . . . . . . 70
16.4 More tensor stuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
16.5 Q & A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
17 October 28, 2014 73
17.1 Midterm Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
17.1.1 Problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
17.1.2 Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
17.1.3 Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
17.2 The space Λn (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
sub
17.3 The space Λn (V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
quot
17.4 The Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
17.5 Constructing the Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 78
17.6 Why do we care? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
18 October 30, 2014 81
18.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
18.2 Completing the proof that Λn (V) = Λn (V) . . . . . . . . . . . . . . . 82
sub quot
18.3 Wedging Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
19 November 4, 2014 85
19.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
19.2 Group Actions, and Sub-Representations . . . . . . . . . . . . . . . . . . . 85
19.3 Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
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Math 55a Notes (Evan Chen) Contents
19.4 Covariant subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
19.5 Quotient spaces and their representations . . . . . . . . . . . . . . . . . . 87
19.6 Tensor product of representations . . . . . . . . . . . . . . . . . . . . . . . 87
20 November 6, 2014 89
20.1 Representations become modules . . . . . . . . . . . . . . . . . . . . . . . 89
20.2 Subrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
20.3 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
20.4 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
20.5 Table of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
20.6 Induced and Restricted Representations . . . . . . . . . . . . . . . . . . . 93
21 November 11, 2014 94
21.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
21.2 Homework Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
21.3 A Theorem on Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
21.4 The Sum of the Characters . . . . . . . . . . . . . . . . . . . . . . . . . . 96
21.5 Re-Writing the Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
21.6 Some things we were asked to read about . . . . . . . . . . . . . . . . . . 98
22 November 13, 2014 100
22.1 Irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
22.2 Products of irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
22.3 Regular representation decomposes . . . . . . . . . . . . . . . . . . . . . . 101
22.4 Function invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
22.5 A Concrete Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
23 November 18, 2014 104
23.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
23.2 The symmetric group on five elements . . . . . . . . . . . . . . . . . . . . 104
23.3 Representations of S /(S ×S ) – finding the irreducible . . . . . . . . . . 106
5 3 2
23.4 Secret of the Young Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 107
23.5 The General Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
24 November 20, 2014 109
24.1 Reducing to some Theorem with Hom’s . . . . . . . . . . . . . . . . . . . 109
24.2 Reducing to a Combinatorial Theorem . . . . . . . . . . . . . . . . . . . . 110
24.3 Doing Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
25 December 2, 2014 113
26 December 4, 2014 114
26.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
26.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
26.3 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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Math 55a Notes (Evan Chen) 1 September 2, 2014
1 September 2, 2014
1.1 Boring stuff
Sets include R, Z, et cetera. A subset Y ⊆ X is exactly what you think it is.
Q,{0},{1},∅,R ⊆ R. Yay.
X ∪X , X ∩X .
1 2 1 2
...Gaitsgory what are you doing
For a fixed universe X, we write Y, X \Y, X −Y for {x ∈ X | x ∈/ Y}.
Lemma 1.1
For Y ⊂ X,
(cid:0) (cid:1)
Y = Y.
Proof. Trivial.
darn this is being written out?
(cid:0) (cid:1)
x ∈ Y ⇐⇒ x ∈/ Y ⇐⇒ x ∈ Y.
(cid:0) (cid:1)
Hence Y = Y.
Lemma 1.2
(X ∩X ) = X ∪X .
1 2 1 2
Proof. Compute
x ∈ X ∩X ⇐⇒ x ∈/ X ∩X ⇐⇒ x ∈/ X ∨x ∈/ X ⇐⇒ x ∈ X ∨x ∈ X ⇐⇒ x ∈ X ∪X .
1 2 1 2 1 2 1 2 1 2
Lemma 1.3
X ∪X = X ∩X .
1 2 1 2
Proof. HW. But this is trivial and follows either from calculation or from applying the
previous two lemmas.
Given a set X we can consider its power set P(X). It has 2n elements.
1.2 Functions
f
Given two sets X and Y a map (or function) X −→ Y is an assignment ∀x ∈ X to an
element fx ∈ Y.
Examples: X = {55 students}, Y = Z. Then f(x) = $ in cents (which can be
negative).
Definition 1.4. A function f is injective (or a monomorphism) if x (cid:54)= y =⇒ fx (cid:54)= fy.
Definition 1.5. Afunctionf issurjective(oranepimorphism)if∀y ∈ Y∃x ∈ X : fx = y.
Composition;
f g
X −→ Y →− Z.
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Math 55a Notes (Evan Chen) 1 September 2, 2014
1.3 Equivalence relations
An equivalence relation ∼ must be symmetric, reflexive, and transitive. A relation ∼ will
partition its set X into cosets or equivalence classes. (The empty set is not a coset.)
Lemma 1.6
Let X be a set and ∼ an equivalence relation. Then for any x ∈ X there exists a
unique coset x with x ∈ x.
Proof. Very tedious manual work.
Now we can take quotients X/ ∼, and we have projections π : X → X/ ∼.
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Math 55a Notes (Evan Chen) 2 September 4, 2014
2 September 4, 2014
2.1 Review of equivalence relations go here
meow
2.2 Universal property of a quotient
f
(cid:45)
X Y
(cid:45)
˜
f
π
(cid:63)
X/ ∼
Proposition 2.1
Let X and Y be sets and ∼ an equivalence relation on X. Let f : X → Y be a
function which preserves ∼, and let π denote the projection X → X/ ∼. Prove that
there exists a unique function f˜such that f = f˜◦π.
The uniqueness follows from the following obvious lemma.
f
(cid:45)
X Y
(cid:45)(cid:45)
f1
g
f2
(cid:63)
X(cid:48)
Lemma 2.2
In the above commutative diagram, if g is surjective then f = f .
1 2
Proof. Just use g to get everything equal. Yay.
2.3 Groups
Definition 2.3. Asemi-groupisasetGendowedwithanassociative1 binaryoperation
∗ : G2 → G.
Lots of groups work.
Example 2.4 (Starfish)
Let G be an arbitrary set and fix a g ∈ G. Then let ab = g for any a,b ∈ G. This
0 0
is a semigroup.
A “baby starfish” has |G| = 1.
1(ab)c=a(bc)
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Math 55a Notes (Evan Chen) 2 September 4, 2014
Definition 2.5. A semi-group G is a monoid if there exists an identity 1 ∈ G such that
∀g ∈ G, g·1 = 1·g = g.
Proposition 2.6
The identity of any semi-group G is unique.
Proof. Let 1, 1(cid:48) be identities. Then
1 = 1·1(cid:48) = 1(cid:48).
Definition 2.7. A group is a monoid G with inverses: for any g ∈ G there exists g−1
with
gg−1 = g−1g = 1.
Proposition 2.8
Inverses are unique.
Proof. Suppose x , x are both inverses of g. Then
1 2
x = x gx = x .
1 1 2 2
Definition 2.9. A group is abelian if it is commutative.
2.4 Homomorphisms
Definition 2.10. Let G and H be groups. A group homomorphism is a map f :
G → H that preserves multiplication.
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Math 55a Notes (Evan Chen) 3 September 9, 2014
3 September 9, 2014
3.1 Direct products
Given two sets X and Y we can define the direct product
X ×Y = {(x,y) | x ∈ X,y ∈ Y}.
For example, R2 is the Euclidean plane.
Hence the operation of a semigroup should be thought of as
mult
G×G −−−→ G.
Given f : Y → Y we define id ×f : X ×Y → X ×Y by
1 2 f 1 2
(x,y ) (cid:55)→ (x,fy ).
1 1
3.2 Commutative diagrams
We can then rephrase associativity using the following commutative diagram.
id ×∗
G (cid:45)
G×G×G G×G
∗×id ∗
G
(cid:63) (cid:63)
∗
(cid:45)
G×G G
We can also rephrase homomorphisms as follows: given ϕ : G → H we require the
following diagram to commute.
ϕ×ϕ
(cid:45)
G×G H ×H
∗ ∗
G H
(cid:63) (cid:63)
ϕ
(cid:45)
G H
3.3 Sub-things
Definition 3.1. Let G be a semigroup / monoid / group. We say H ≤ G is a sub-
semigroup if h ,h ∈ H =⇒ h h ∈ H. Moreover, if G is a monoid and 1 ∈ H then H
1 2 1 2
is a sub-monoid. Finally, if H is closed under inverses as well then H is a subgroup.
Example 3.2
Let G be the additive group of integers. Then N is a sub-semigroup, Z is a
≥0
sub-monoid, and 2Z is a subgroup.
Lemma 3.3
If H ≤ G and H ≤ G are subgroups, then H ∩H is a subgroup.
1 2 1 2
Proof. Obvious.
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Math 55a Notes (Evan Chen) 3 September 9, 2014
3.4 Let’s play Guess the BS!
“In what follows I’ll state some false statements. You will be asked to prove them at your
own risk.”
Lemma 3.4
Given ϕ : G → H a homomorphism, ϕ(G) is a subgroup of H.
Proof. Given any ϕ(a), ϕ(b) in ϕ(G) we have
ϕ(a)ϕ(b) = ϕ(ab) ∈ ϕ(G).
Then use ϕ(1) = 1 to get the rest of the conditions.
Lemma 3.5
If ϕ : G → H and H(cid:48) ≤ H, then ϕ−1(H(cid:48)) is a subgroup of G.
Proof. This one turns out to be true.
Fact 3.6. Given H , H subgroups of G, H ∪H need not be a subgroup of G.
1 2 1 2
Proof. Take G = Z, H = 100Z, H = 101Z.
1 2
3.5 Kernels
Definition 3.7. Given a homomorphism ϕ : G → H, the kernel kerϕ is defined by
kerϕ = ϕ−1({1}).
Proposition 3.8
Let ϕ : G → H be a homomorphism. Then ϕ is injective as a map of sets if and
only if kerϕ = {1}.
Proof. In all cases 1 ∈ kerϕ. If |kerϕ| =(cid:54) 1 then clearly ϕ is not injective. On the other
hand, suppose kerϕ = {1}. If ϕa = ϕb we get ϕ(ab−1) = 1, so if we must have ab−1 = 1
or a = b.
Definition 3.9. Let G be a group and let H ≤ G be a subgroup. We define the right
equivalence relation on G with respect to H ∼r as follows: g ∼r g if ∃h ∈ H such that
1 2
g = g h.
2 1
Define ∼(cid:96) similarly.
To check this is actually an equivalence relation, note that
1 ∈ H =⇒ g ∼r g
and
g ∼ g =⇒ g = g h =⇒ g h−1 = g =⇒ g ∼r g .
1 2 1 2 1 2 2 1
Finally, if g = g h(cid:48) and g = g h(cid:48)(cid:48) then g = g (h(cid:48)h(cid:48)(cid:48)), so transitivity works as well.
1 2 2 3 1 3
Note that g ∼r g ⇐⇒ g−1g ∈ H.
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Definition 3.10. Let G/H be the set of equivalence classes of G with respect to ∼r.
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