Harvard-55a

Math 55a Lecture Notes
Evan Chen
Fall 2014
This is Harvard College’s famous Math 55a, instructed by Dennis Gaitsgory.
The formal name for this class is “Honors Abstract and Linear Algebra” but
it generally goes by simply “Math 55a”.
The permanent URL is http://web.evanchen.cc/~evanchen/coursework.
html, along with all my other course notes.
Contents
1 September 2, 2014 5
1.1 Boringstuff................................... 5
1.2 Functions .................................... 5
1.3 Equivalencerelations.............................. 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 Directproducts................................. 9
3.2 Commutativediagrams ............................ 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 Ringhomomorphisms ............................. 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
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Evan Chen (Fall 2014) Contents
5.4 Sub-Modules .................................. 19
5.5 FreeModules.................................. 20
5.6 ReturntotheFinite.............................. 21
6 September 18, 2014 23
6.1 Linearly independent, basis, span . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 Dimensionsandbases ............................. 24
6.3 CorollaryParty................................. 24
6.4 ProofofTheorem ............................... 26
7 September 23, 2014 28
7.1 MidtermSolutions............................... 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 SpectralTheorem ............................... 34
8.4 Lemmata in building our proof . . . . . . . . . . . . . . . . . . . . . . . . 35
8.5 Proof of spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
8.6 RecapofProof ................................. 37
9 September 30, 2014 38
9.1 Review ..................................... 38
9.2 Taking polynomials of an endomorphism . . . . . . . . . . . . . . . . . . . 38
9.3 MinimalPolynomials.............................. 39
9.4 SpectralProjector ............................... 40
9.5 Polynomials................................... 40
10 October 2, 2014 42
10.1JordanCanonicalForm ............................ 42
10.2Abigproposition................................ 42
10.3YoungDiagrams ................................ 43
10.4ProofofExistence ............................... 44
11 October 7, 2014 46
11.1OrderofaGroup................................ 46
11.2Groupsofprimepowers ............................ 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
12.1GroupActions ................................. 50
12.2 How do G-sets talk to each other? . . . . . . . . . . . . . . . . . . . . . . 50
12.3Commongroupactions ............................ 50
12.4Moregroupactions............................... 52
12.5Transitiveactions ............................... 53
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Evan Chen (Fall 2014) Contents
12.6Orbits...................................... 54
12.7 Corollaries of Sylow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 54
12.8 Proof of (b) of Sylow’s Theorem assuming (a) . . . . . . . . . . . . . . . . 55
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
13.3DualModule .................................. 56
13.4Doubledual................................... 57
13.5 Real and Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 57
13.6ObviousTheorems............................... 58
13.7 Inner form induces a map . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
14 October 16, 2014 60
14.1 Artificial Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
14.2OrthogonalSubspace.............................. 60
14.3OrthogonalSystems .............................. 61
14.4Adjointoperators................................ 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
15.1Generators ................................... 66
15.2 Basic Properties of Tensor Products . . . . . . . . . . . . . . . . . . . . . 66
15.3 Computing tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 67
15.4Complexification ................................ 67
16 October 23, 2014 69
16.1 Tensor products gain module structure . . . . . . . . . . . . . . . . . . . . 69
16.2UniversalProperty............................... 69
16.3 Tensor products of vector spaces . . . . . . . . . . . . . . . . . . . . . . . 70
16.4Moretensorstuff................................ 71
16.5Q&A...................................... 72
17 October 28, 2014 73
17.1MidtermSolutions............................... 73
17.1.1Problem1................................ 73
17.1.2Problem2................................ 74
17.1.3Problem3................................ 75
17.2 The space Λn
sub(V)............................... 76
17.3 The space Λn
quot(V) .............................. 77
17.4TheWedgeProduct .............................. 78
17.5 Constructing the Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . 78
17.6Whydowecare?................................ 80
18 October 30, 2014 81
18.1Review ..................................... 81
18.2 Completing the proof that Λn
sub(V)=Λn
quot(V) ............... 82
18.3WedgingWedges................................ 83
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19 November 4, 2014 85
19.1Representations................................. 85
19.2 Group Actions, and Sub-Representations . . . . . . . . . . . . . . . . . . 85
19.3InvariantSubspaces .............................. 86
19.4Covariantsubspace............................... 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
20.2Subrepresentations............................... 89
20.3Schur’sLemma................................. 90
20.4Splittings .................................... 91
20.5 Table of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
20.6 Induced and Restricted Representations . . . . . . . . . . . . . . . . . . . 93
21 November 11, 2014 94
21.1Review ..................................... 94
21.2HomeworkSolutions.............................. 94
21.3 A Theorem on Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
21.4 The Sum of the Characters . . . . . . . . . . . . . . . . . . . . . . . . . . 96
21.5Re-WritingtheSum .............................. 97
21.6 Some things we were asked to read about . . . . . . . . . . . . . . . . . . 98
22 November 13, 2014 100
22.1Irreducibles...................................100
22.2 Products of irreducibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
22.3 Regular representation decomposes . . . . . . . . . . . . . . . . . . . . . . 101
22.4Functioninvariants...............................102
22.5AConcreteExample..............................103
23 November 18, 2014 104
23.1Review .....................................104
23.2 The symmetric group on five elements . . . . . . . . . . . . . . . . . . . . 104
23.3 Representations of S5/(S3×S2) – finding the irreducible . . . . . . . . . 106
23.4 Secret of the Young Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 107
23.5TheGeneralTheorem .............................108
24 November 20, 2014 109
24.1 Reducing to some Theorem with Hom’s . . . . . . . . . . . . . . . . . . . 109
24.2 Reducing to a Combinatorial Theorem . . . . . . . . . . . . . . . . . . . . 110
24.3DoingCombinatorics..............................111
25 December 2, 2014 113
26 December 4, 2014 114
26.1Categories....................................114
26.2Functors.....................................114
26.3 Natural Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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Evan Chen (Fall 2014) 1 September 2, 2014
§1September 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.
X1∪X2,X1∩X2.
. . . Gaitsgory what are you doing
For a fixed universe X, we write Y,X\Y,X−Yfor {x∈X|x /∈Y}.
Lemma 1.1
For Y⊂X,Y=Y.
Proof. Trivial.
darn this is being written out?
x∈Y⇐⇒ x /∈Y⇐⇒ x∈Y.
Hence Y=Y.
Lemma 1.2
(X1∩X2) = X1∪X2.
Proof. Compute
x∈X1∩X2⇐⇒ x /∈X1∩X2⇐⇒ x /∈X1∨x /∈X2⇐⇒ x∈X1∨x∈X2⇐⇒ x∈X1∪X2.
Lemma 1.3
X1∪X2=X1∩X2.
Proof. HW. But this is trivial and follows either from calculation or from applying the
previous two lemmas.
Given a set Xwe can consider its power set P(X). It has 2nelements.
§1.2 Functions
Given two sets X and Y a map (or function) Xf −→ 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 fis injective (or a monomorphism) if x6=y=⇒fx 6=fy.
Definition 1.5. A function f is surjective (or an epimorphism) if ∀y∈Y∃x∈X : fx = y .
Composition;
Xf −→ Yg −→ Z.
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