Invariants under simultaneous conjugation of SL 2 matrices

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Invariants under simultaneous conjugation of SL 2 matrices Master's colloquium, 4 November 2009

2 Outline 1 The problem 2 Classical Invariant Theory 3 Geometric Invariant Theory 4 Representation Theory 5 Conclusion

3 The Problem The problem Consider the space of m-tuples of matrices with determinant 1 (SL 2 matrices. Forget the basis: for any Q SL 2, (M 1,..., M m (QM 1 Q 1,..., QM m Q 1. We want to know the orbit space: {(M 1,..., M m SL m 2 }/. This space arises in monodromy of order 2 ordinary dierential equations with m + 1 singularities

4 Approaches Approach 1: (Classical Invariant Theory Well-known fact: trace, determinant of matrix are invariant under conjugation, so they tell orbits apart Invariant theory: classify these functions Notation: C[SL m 2 ] are polynomial functions on SL 2... SL 2 Notation: C[SL m 2 ]SL 2 are polynomial functions invariant under conjugation We calculated the structure of C[SL m 2 ]SL 2 using some classical results

5 Approaches Approach 2: Geometric Invariant Theory Assign geometric meaning to X = {(M 1,..., M m SL m 2 }/ in projective space Want: Y that separates the orbits as much as possible: every map ψ that is constant on orbits runs through Y : X φ Y ψ θ Z, (the universal mapping property This can be done for (semi-stable points

6 Approaches Approach 3: Representation Theory More abstractly, the action of SL 2 on the set of matrices can be seen as a representation of SL 2 Idea: forget the specic space SL 2 acts on Question: how does GIT work on representations? Question: does this correspond to our earlier results?

7 Approaches Summary of approaches Want to study: {(M 1,..., M m SL m 2 }/. Approach 1: CIT: Look at invariant function C[SL m 2 ]SL 2 Approach 2: GIT: Construct a quotient map with the universal mapping property Approach 3: RT: How does GIT work on representations?

8 Two matrices A theorem (I Let M 0 be the vector space of traceless matrices. Theorem C[M 0 M 0 ] SL 2 = C[TrM 2 1, TrM2 2, TrM 1M 2 ]. Sketch of proof Let f be an invariant function. Restrict it to the pairs {(( ( } t a 1, a, t 0, t c a so we have a function f C[a, c, t]. Almost any matrix pair can be conjugated to this form.

9 Two matrices A theorem (II Sketch of proof, cont'd Note that (( t t ( a 1, c a But then f C[c, a 2, t 2, at]. Now (( t t ( a 1, c a. t 2 = 1 2 TrM2 1, at = 1 2 TrM 1M 2, etcetera so we have found the invariant function that we restricted, which is a polynomial in TrM 1 2, TrM2 2, TrM 1M 2.

10 Two matrices Traceless, general matrices (I Note that for M SL 2 : ( a M i b i M 1 c i d i = M But then for functions: ( 12 a i 2 1 d i b i 1 c i 2 d i 1 2 a i M 1 + ( 12 a i d i a i d i C[SL 2 SL 2 ] SL 2 = (C[M1, M 2 ] SL 2 C[t 1, t 2 ]/(detm i t2 i = 1. So: we translate our results on traceless matrices

11 Two matrices Traceless, general matrices (II C[SL 2 SL 2 ] SL 2 = (C[M1, M 2 ] SL 2 C[t 1, t 2 ]/(detm i t2 i = 1 = C[TrM 2 1,TrM 2 2,TrM 1 M 2, t 1, t 2 ]/(... Correspondence: ( ( ( 12 a i b i a c i d i SL 2 i 1 2 d i b i Say (X 1, X 2 SL 2 2, then c i 1 2 d i 1 2 a i, 1 2 (a i + d i t i = 1 2 (a i + d i 1 2 TrX i TrM i M j = ( 1 2 a i 1 2 d i ( 1 2 a j 1 2 d j +... TrX i X j 1 2 TrX i TrX j

12 Two matrices Traceless, general matrices (III We are translating C[TrM 2 1,TrM2 2,TrM 1M 2, t 1, t 2 ]: t i = 1 2 (a i + d i TrX i TrM i M j = ( 1 2 a i 1 2 d i ( 1 2 a j 1 2 d j +... TrX i X j 1 2 TrX i TrX j. But for (X 1, X 2 SL 2 2 : TrX 2 i = a 2 i + d 2 i + 2b i c i = a 2 i + d 2 i + 2a i d i 2a i d i + 2 b i c i = (a i + d i 2 2 = (TrX i 2 2. Corollary C[SL 2 SL 2 ] SL 2 = C[TrX 1, TrX 2, TrX 1 X 2 ].

13 Three matrices & general case The general case In [Drensky2000], the complete structure of C[M k 0 ] is calculated. Proof uses invariant theory of SO 3 known in 1947 We wrote it down more clearly, and calculated C[SL m 2 ]SL 2 for any m in the same way as previously

14 Three matrices & general case The invariants for 3 SL 2 matrices Theorem Let a = Trx 1, b = Trx 2, c = Trx 3, d = Trx 1 x 2, e = Trx 1 x 3, f = Trx 2 x 3, g = Trx 1 x 2 x 3 Trx 1 x 3 x 2. Then C[SL 3 2 ]SL 2 = C[a, b, c, d,e, f, g]/(rel, where rel = g 2 + 4a 2 + 4b 2 + 4c 2 + 4d 2 + 4e 2 + 4f 2 +2a 2 bcf + 2abc 2 d + 2ab 2 ce 4ace 4abd 4bcf + 4def b 2 e 2 a 2 f 2 c 2 d 2 2bcde 2acdf 2abef a 2 b 2 c Or: C[SL 3 2 ]SL 2 = C[a, b, c, d,e, f ] + C[a, b, c, d,e, f ] g

15 Embedding into projective space Projective space: an introduction Consider the space R { }. Denote a point by (a : b, where (a : b = (λ a : λ b x R corresponds to (x : 1 corresponds to (1 : 0 (0 : 0 is excluded Geometry on projective spaces is much more elegant: in the projective plane every two lines intersect and other nice properties. Generalize to: P n = {(x 1 :... : x n+1 C n+1 }\(0 :... : 0 /(x1 :...:x n+1 =(λ x 1 :...:λ x n+1.

16 Embedding into projective space Embedding SL 2 in a projective space Obvious choice: add determinant as extra coordinate: let Q = {(a : b : c : d : P 4 ad bc = 2 }, and dene embedding ( a b c d φ (a : b : c : d : 1 = (( a b c d : 1. The conjugation can be extended to the whole of Q: conjugating an element by M SL 2 gives: ( M ( a b c d M 1 :.

17 Embedding into projective space Embedding SL m 2 in a projective space We can send tuples of matrices to tuples of Q's, e.g.: (( ( (( ( a 1 b 1 a c 1 d 1, 2 b 2 a c 2 d 2 1 b 1 a c 1 d 1 : 1, 2 b 2 c 2 d 2 : 2. Question: how is this a projective space? Answer: by considering this as an embedding into P 24 : Q Q P (( ( 24 a 1 b 1 a c 1 d 1 : 1, 2 b 2 c 2 d 2 : 2 (a 1 a 2 : a 1 b 2 : a 1 c 2 : a 1 d 2 : a 1 2 :... : 1 a 2 : 1 b 2 : 1 c 2 : 1 d 2 : 1 2.

18 A good quotient A good quotient Denitions A categorical quotient is (Y,φ that separates the orbits as much as possible: every map ψ that is constant on orbits runs through Y : Q Q φ Y ψ θ Z, Some additional properties: good quotient. If the quotient is good and all orbits go to dierent points, then it is called a geometric quotient.

19 A good quotient Constructing the quotient For ane spaces, the map induced by the algebra of invariants is a good quotient. For example: X = SL 2 SL 2, Y = C 3 : Idea: write φ(m 1, M 2 = (TrM 1,TrM 2,TrM 1 M 2. Q Q = U α : α A a cover by open, ane, dense, SL 2 -stable subsets. For example, {(( ( } a 1 b 1 a c 1 d 1 : 1, 2 b 2 c 2 d 2 : = SL 2 2. If we glue together these quotients, we get a new quotient.

20 A good quotient The ane subset SL 2 SL 2 SL 2 SL 2 is the ane subset where via: (( ( (( ( a 1 b 1 a c 1 d 1, 2 b 2 a c 2 d 2 1 b 1 a c 1 d 1 : 1, 2 b 2 c 2 d 2 : 1. Write invariants of SL 2 SL 2 as functions on Q Q: TrM 1 2TrM 1 1 2, TrM 2 TrM ; TrM 1 M 2 TrM 1M So a good quotient is Y = {(a : b : c : d P 3 d 0}, φ : ((M 1 : 1,(M 2 : 2 ( 2 M 1 : TrM 2 1 : TrM 1 M 2 : 1 2.

21 A good quotient Another subset Subset of Q Q where (a 1 + d corresponds to M 1 SL 2 /(a 1 a 2 1 b 1 c 1 = z 2 1, a 2 d 2 b 2 c 2 = 1 via (( a 1 b 1 c 1 1 a 1,z 1, ( a 2 b 2 c 2 d 2 (( a 1 b 1 c 1 1 a 1 : z 1, ( a 2 b 2 c 2 d 2 : 1. In the classical way: C[M 1 SL 2 ] = C[z 1,TrM 2,TrM 1 M 2 ]. So a good quotient is Y = {(a : b : c : d P 3 d 0}, φ : ((M 1 : 1,(M 2 : 2 (TrM 1 TrM 2 : TrM 1 M 2 : 1 2 : TrM 1 2

22 A good quotient Glueing a quotient Setting 1 2 0, TrM 1 2 0, TrM 2 1 0, TrM 1 M 2 0, we get quotients mapping ((M 1 : 1,(M 2 : 2 to: (TrM 1 2 : TrM 2 1 : TrM 1 M 2 : 1 2 ; (TrM 1 TrM 2 : TrM 1 M 2 : 1 2 : TrM 1 2 ; (TrM 1 TrM 2 : TrM 1 M 2 : 1 2 : TrM 2 1 ; (TrM 2 1 : TrM 1 2 : TrM 1 M 2 : TrM 1 TrM 2. Combination Y = {(a : b : c : d : e ab = cd }\{(0 :... : 0 : 1}; φ maps ((M 1 : 1,(M 2 : 2 to (TrM 1 2 : TrM 2 1 : TrM 1 TrM 2 : 1 2 : TrM 1 M 2.

23 A good quotient Glueing a quotient (II LetY = {(a : b : c : d : e ab = cd }\{(0 :... : 0 : 1}; Is (Y,φ a quotient, where φ maps ((M 1 : 1,(M 2 : 2 to Theorem (TrM 1 2 : TrM 2 1 : TrM 1 TrM 2 : 1 2 : TrM 1 M 2? (Y,φ is a good quotient for the set (Q Q nn of non-nilpotent matrices. Proof. One checks: for all ane parts A, φ restricted to A is a good quotient. Can we do better than this?

24 Semi-stable and stable points Semi-stable and stable points Mumford's Geometric Invariant Theory: for what points does a good quotient exist? Denitions Let V be projective. Then x V is semi-stable if there exists a homogeneous SL 2 -invariant polynomial with strictly positive degree which does not vanish at x; x is stable if the orbit of x in the underlying ane space is closed with maximal dimension. Then: Theorem There exists a good quotient φ : X ss Y and φ restricted to the stable points is a geometric quotient.

25 Semi-stable and stable points Mumford's criterion Theorem (Mumford's criterion for SL 2 (1 Find a basis for V so that {( λ λ 1 } λ C acts on basis elements of V as λ v i = c i λ w i vi (2 A basis vector v i is said to have weight w i (3 Up to change of basis, points that are not semi-stable are spanned by positive weight vectors (4 Up to change of basis, points that are not stable are spanned by non-negative weight vectors

26 Semi-stable and stable points Mumford's criterion for one matrix Calculate action: Weights: (( a b c d ( λ 0 0 λ 1 acts as follows: : (( a λ 2 b λ 2 c d : weight( = 2; weight( = 2; weight( ,( , = 0 So: non-semi-stable points are in span of ( : nilpotent matrices All points are non-stable!

27 Semi-stable and stable points Mumford's criterion for two matrices (I The action of diag(λ,λ 1 gives as result: (( ( : 1, a 1 λ 2 b 1 a 2 λ 2 b 2 : 2. Recall: basis vectors (in P 24 were a 1 a 2, a 1 b 2,..., 1 2. Theorem A pair ((M 1 : 1,(M 2, 2 is not semi-stable i 1 matrix is nilpotent and the pair is reducible, i.e., can be conjugated to: (( 0 (, 0.

28 Semi-stable and stable points Mumford's criterion for two matrices (II The action of diag(λ,λ 1 gives as result: (( ( : 1, a 1 λ 2 b 1 a 2 λ 2 b 2 : 2. Recall: basis vectors (in P 24 were a 1 a 2, a 1 b 2,..., 1 2. Theorem A pair ((M 1 : 1,(M 2, 2 is not stable i 1 matrix is nilpotent or the pair is reducible, i.e., can be conjugated to: (( 0 (, 0.

29 Semi-stable and stable points Our quotient for Q Q Recall: We have a good quotient for (Q Q nn : (Y,φ where Y = {(a : b : c : d : e ab = cd}\{(0 : 0 : 0 : 0 : 1}, φ maps ((M 1 : 1,(M 2 : 2 to (TrM 1 2 : 1 TrM 2 : TrM 1 TrM 2 : 1 2 : TrM 1 M 2. Note that (Q Q ss (Q Q nn (Q Q s φ is a well-dened map (Q Q ss {(a : b : c : d : e ab = cd}. Is it a good quotient? Theorem φ is a geometric quotient from (Q Q s to {(x 1 : x 2 : x 3 : x 4 : x 5 x 1 x 2 = x 3 x 4, x x x 2 5 x 3 x 5 4x 2 4 0}.

30 Semi-stable and stable points More than 2 matrices Ane cover of (Q... Q nn is always possible Stable, semi-stable points can always be classied Image of (Q... Q nn : harder to determine Proving it is a geometic quotient for (Q... Q s?

31 The representation theory of SL2 Representation theory Let a group G act linearly on a vector space V : g v V. Determine possible actions up to isomorphism: forget the space V For good G, one can uniquely write V i irreducible V = V 1 V 2... V n, Classify irreducible representations Apply Mumford's criterion

32 The representation theory of SL2 Representation theory of SL 2 Let [k] = {f C[x 1, x 2 ] f is homogeneous of degree k} of dimension k + 1. For example, [2] = span{x 2 1, x 1x 2, x 2 2 }. SL 2 acts on this as follows: ( a b c d f (x1, x 2 = f (a x 1 + c x 2, b x 1 + d x 2, so ( a b c d x 2 1 = a 2 x ac x 1 x 2 + c 2 x 2 2 ; ( a b c d x1 x 2 = ab x (ad + bc x 1 x 2 + cd x 2 2 ; ( a b c d x 2 2 = b 2 x bd x 1 x 2 + d 2 x 2 2. Then [k] is the unique irreducible representation of dimension k + 1

33 The representation theory of SL2 Mumford's criterion for [k] ( λ 0 Look at weights of action of 0 λ 1 : ( λ 0 0 λ 1 f (x 1, x 2 = f (λ x 1,λ 1 x 2. So the weight of a basis vector x1 k x 2 l is k l. So: if we know the structure of Q Q as a representation, the problem is really simple!

34 Representation theory of Q... Q Representation theory of Q Fact M 0 = [2] as representations. Q was (( embedded in a 5-dimensional vector space: a b V =, c d Choose basis {( ( ( ( 0, ,, }, Theorem V = [2] + 2 [0] as representations.

35 Representation theory of Q... Q Representation theory of Q Q Theorem Let V = ([2] + 2 [0] ([2] + 2 [0], then V = [4] + 5 [2] + 5 [0]. Fact [2] [2] = [4] + [2] + [0] as SL 2 -representations. Proof. ([2] + 2 [0] 2 = [2] [2] + 2 [0] [2] + 2 [2] [0] + 4 [0] [0] = ([4] + [2] + [0] + 2 [2] + 2 [2] + 4 [0] = [4] + 5 [2] + 5 [0].

36 Representation theory of Q... Q Mumford's criterion revisited Unstable points: V = [4] + 5 [2] + 5 [0]. x 4 1 ( x 3 1 x 2 + y 2 1 ( x1 3 x 2 y1 2 ( y2 2, y3 2 ( ( ( y 2 4, y 2 5 ( ( ( 1 0 (, ( 12 0, ( Corresponds to our earlier results!

37 Representation theory of Q... Q Generalization to m > 2 Finding stable/semi-stable points is very easy Harder to keep track of the representation isomorphism Computer algebra packages might help?

38 Representation theory the executive summary Recall: irreducible representations of SL 2 are [k] of dimension k + 1 Weight of basis vector x k 1 x l 2 = k l Let Q Q be embedded in V = A 5 A 5, we constructed an explicit isomorphism V = [4] + 5 [2] + 5 [0]. Verify that Mumford's criterion gives the same result. To generalize, use computer algebra packages?

39 Conclusion & Evaluation In the ane case, C[SL m 2 ]SL 2 was determined This gives rise to a good quotient on the projective (Q... Q ss (Q... Q nn (Q... Q s Semi-simple and stable points can also be interpreted with representation theory. A good quotient for (Q... Q ss? A geometric quotient for (Q... Q s? Embedding with one coordinate added?

40 Appendix Literature W. Drensky. Dening Relations for the Algebra of Invariants of 2 2 Matrices. Algebras and Representation Theory, A. Extra. The invariants of 2 2 matrices, their algebraic relations and the corresponding moduli problem. PhD Thesis, Katholieke Universiteit Nijmegen, See for (the draft of my thesis.

Mathematics 7800 Quantum Kitchen Sink Spring 2002

Mathematics 7800 Quantum Kitchen Sink Spring 2002 4. Quotients via GIT. Most interesting moduli spaces arise as quotients of schemes by group actions. We will first analyze such quotients with geometric

AN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS

AN INTRODUCTION TO MODULI SPACES OF CURVES MAARTEN HOEVE ABSTRACT. Notes for a talk in the seminar on modular forms and moduli spaces in Leiden on October 24, 2007. CONTENTS 1. Introduction 1 1.1. References

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 16. Symplectic resolutions of µ 1 (0)//G and their deformations 16.1. GIT quotients. We will need to produce a resolution of singularities for C 2n

Mini-Course on Moduli Spaces

Mini-Course on Moduli Spaces Emily Clader June 2011 1 What is a Moduli Space? 1.1 What should a moduli space do? Suppose that we want to classify some kind of object, for example: Curves of genus g, One-dimensional

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES ROBERT M. GURALNICK AND B.A. SETHURAMAN Abstract. In this note, we show that the set of all commuting d-tuples of commuting n n matrices that

Noncommutative compact manifolds constructed from quivers

Noncommutative compact manifolds constructed from quivers Lieven Le Bruyn Universitaire Instelling Antwerpen B-2610 Antwerp (Belgium) lebruyn@wins.uia.ac.be Abstract The moduli spaces of θ-semistable representations

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

The derived category of a GIT quotient

September 28, 2012 Table of contents 1 Geometric invariant theory 2 3 What is geometric invariant theory (GIT)? Let a reductive group G act on a smooth quasiprojective (preferably projective-over-affine)

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2)

GALOIS GROUPS OF CUBICS AND QUARTICS (NOT IN CHARACTERISTIC 2) KEITH CONRAD We will describe a procedure for figuring out the Galois groups of separable irreducible polynomials in degrees 3 and 4 over

Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing

Construction of Mixed-Level Orthogonal Arrays for Testing in Digital Marketing Vladimir Brayman Webtrends October 19, 2012 Advantages of Conducting Designed Experiments in Digital Marketing Availability

Changing coordinates to adapt to a map of constant rank

Introduction to Submanifolds Most manifolds of interest appear as submanifolds of others e.g. of R n. For instance S 2 is a submanifold of R 3. It can be obtained in two ways: 1 as the image of a map into

NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques

MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS

MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MIHNEA POPA 1. Lecture II: Moduli spaces and generalized theta divisors 1.1. The moduli space. Back to the boundedness problem: we want

Algebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?

Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials

Group Theory - QMII 2017

Group Theory - QMII 017 Reminder Last time we said that a group element of a matrix lie group can be written as an exponent: U = e iαaxa, a = 1,..., N. We called X a the generators, we have N of them,

TRACE AND NORM KEITH CONRAD 1. Introduction Let L/K be a finite extension of fields, with n = [L : K]. We will associate to this extension two important functions L K, called the trace and the norm. They

Representations and Linear Actions

Representations and Linear Actions Definition 0.1. Let G be an S-group. A representation of G is a morphism of S-groups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category

Norm-induced partially ordered vector spaces

Stefan van Lent Norm-induced partially ordered vector spaces Bachelor thesis Thesis Supervisor: dr. O.W. van Gaans Date Bachelor Exam: 30th of June 016 Mathematisch Instituut, Universiteit Leiden Contents

Mappings of elliptic curves

Mappings of elliptic curves Benjamin Smith INRIA Saclay Île-de-France & Laboratoire d Informatique de l École polytechnique (LIX) Eindhoven, September 2008 Smith (INRIA & LIX) Isogenies of Elliptic Curves

LINKING INVARIANT FOR ALGEBRAIC PLANE CURVES

LINKING INVARIANT FOR ALGEBRAIC LANE CURVES BENOÎT GUERVILLE-BALLÉ Introduction The subject of our interest is the topology of algebraic plane curves (see [8, 12] for basic results). They are geometric

CHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998

CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic

THE MODULI SPACE OF RATIONAL FUNCTIONS OF DEGREE d

THE MODULI SPACE OF RATIONAL FUNCTIONS OF DEGREE d MICHELLE MANES. Short Introduction to Varieties For simplicity, the whole talk is going to be over C. This sweeps some details under the rug, but it s

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea

Torsion of dierentials on toric varieties Klaus Altmann Institut fur reine Mathematik, Humboldt-Universitat zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de Abstract

Algebra I Fall 2007

MIT OpenCourseWare http://ocw.mit.edu 18.701 Algebra I Fall 007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.701 007 Geometry of the Special Unitary

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

1.1. Introduction SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

When 2 and 3 are invertible in A, L A is the scheme

8 RICHARD HAIN AND MAKOTO MATSUMOTO 4. Moduli Spaces of Elliptic Curves Suppose that r and n are non-negative integers satisfying r + n > 0. Denote the moduli stack over Spec Z of smooth elliptic curves

1 Linear transformations; the basics

Linear Algebra Fall 2013 Linear Transformations 1 Linear transformations; the basics Definition 1 Let V, W be vector spaces over the same field F. A linear transformation (also known as linear map, or

CHAPTER 3: THE INTEGERS Z

CHAPTER 3: THE INTEGERS Z MATH 378, CSUSM. SPRING 2009. AITKEN 1. Introduction The natural numbers are designed for measuring the size of finite sets, but what if you want to compare the sizes of two sets?

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

1. Find the solution of the following uncontrolled linear system. 2 α 1 1

Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +

The Rationality of Certain Moduli Spaces of Curves of Genus 3

The Rationality of Certain Moduli Spaces of Curves of Genus 3 Ingrid Bauer and Fabrizio Catanese Mathematisches Institut Universität Bayreuth, NW II D-95440 Bayreuth, Germany Ingrid.Bauer@uni-bayreuth.de,

Math 396. Bijectivity vs. isomorphism

Math 396. Bijectivity vs. isomorphism 1. Motivation Let f : X Y be a C p map between two C p -premanifolds with corners, with 1 p. Assuming f is bijective, we would like a criterion to tell us that f 1

Math 396. Quotient spaces

Math 396. Quotient spaces. Definition Let F be a field, V a vector space over F and W V a subspace of V. For v, v V, we say that v v mod W if and only if v v W. One can readily verify that with this definition

ELEMENTARY LINEAR ALGEBRA

ELEMENTARY LINEAR ALGEBRA K. R. MATTHEWS DEPARTMENT OF MATHEMATICS UNIVERSITY OF QUEENSLAND Corrected Version, 7th April 013 Comments to the author at keithmatt@gmail.com Chapter 1 LINEAR EQUATIONS 1.1

Math 396. An application of Gram-Schmidt to prove connectedness

Math 396. An application of Gram-Schmidt to prove connectedness 1. Motivation and background Let V be an n-dimensional vector space over R, and define GL(V ) to be the set of invertible linear maps V V

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

REPRESENTATION THEORY. WEEK 4

REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a B-module. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators

This is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:

Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS

THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS KEITH CONRAD. Introduction The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise

A new parametrization for binary hidden Markov modes

A new parametrization for binary hidden Markov models Andrew Critch, UC Berkeley at Pennsylvania State University June 11, 2012 See Binary hidden Markov models and varieties [, 2012], arxiv:1206.0500,

THE MODULI SPACE OF CURVES

THE MODULI SPACE OF CURVES In this section, we will give a sketch of the construction of the moduli space M g of curves of genus g and the closely related moduli space M g,n of n-pointed curves of genus

10. Classifying Möbius transformations: conjugacy, trace, and applications to parabolic transformations

10. Classifying Möbius transformations: conjugacy, trace, and applications to parabolic transformations 10.1 Conjugacy of Möbius transformations Before we start discussing the geometry and classification

DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of D n, including the normal subgroups. We will also introduce an infinite

Math 2030 Assignment 5 Solutions

Math 030 Assignment 5 Solutions Question 1: Which of the following sets of vectors are linearly independent? If the set is linear dependent, find a linear dependence relation for the vectors (a) {(1, 0,

1 Fields and vector spaces

1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary

Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 As usual, k is a perfect field and k is a fixed algebraic closure of k. Recall that an affine (resp. projective) variety is an

Lecture 4 and 5 Controllability and Observability: Kalman decompositions

1 Lecture 4 and 5 Controllability and Observability: Kalman decompositions Spring 2013 - EE 194, Advanced Control (Prof. Khan) January 30 (Wed.) and Feb. 04 (Mon.), 2013 I. OBSERVABILITY OF DT LTI SYSTEMS

COMPLEX MULTIPLICATION: LECTURE 14

COMPLEX MULTIPLICATION: LECTURE 14 Proposition 0.1. Let K be any field. i) Two elliptic curves over K are isomorphic if and only if they have the same j-invariant. ii) For any j 0 K, there exists an elliptic

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices

ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex

T -equivariant tensor rank varieties and their K-theory classes

T -equivariant tensor rank varieties and their K-theory classes 2014 July 18 Advisor: Professor Anders Buch, Department of Mathematics, Rutgers University Overview 1 Equivariant K-theory Overview 2 Determinantal

Chapter 7. Linear Algebra: Matrices, Vectors,

Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.

BRILL-NOETHER THEORY. This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve.

BRILL-NOETHER THEORY TONY FENG This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. INTRODUCTION Brill-Noether theory is concerned

1 Holomorphic functions

Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

The Jordan Canonical Form

The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finite-dimensional vector space over an algebraically closed field. Here we develop

Vector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle

Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 7-9 December 2010 1 The notion of vector bundle In affine geometry,

A Crash Course in Topological Groups

A Crash Course in Topological Groups Iian B. Smythe Department of Mathematics Cornell University Olivetti Club November 8, 2011 Iian B. Smythe (Cornell) Topological Groups Nov. 8, 2011 1 / 28 Outline 1

disc f R 3 (X) in K[X] G f in K irreducible S 4 = in K irreducible A 4 in K reducible D 4 or Z/4Z = in K reducible V Table 1

GALOIS GROUPS OF CUBICS AND QUARTICS IN ALL CHARACTERISTICS KEITH CONRAD 1. Introduction Treatments of Galois groups of cubic and quartic polynomials usually avoid fields of characteristic 2. Here we will

Toric Varieties. Madeline Brandt. April 26, 2017

Toric Varieties Madeline Brandt April 26, 2017 Last week we saw that we can define normal toric varieties from the data of a fan in a lattice. Today I will review this idea, and also explain how they can

Topics in linear algebra

Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over

Moduli of Pointed Curves. G. Casnati, C. Fontanari

Moduli of Pointed Curves G. Casnati, C. Fontanari 1 1. Notation C is the field of complex numbers, GL k the general linear group of k k matrices with entries in C, P GL k the projective linear group, i.e.

ISOMETRIES OF THE HYPERBOLIC PLANE

ISOMETRIES OF THE HYPERBOLIC PLANE ALBERT CHANG Abstract. In this paper, I will explore basic properties of the group P SL(, R). These include the relationship between isometries of H, Möbius transformations,

Linear Algebra- Final Exam Review

Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.

2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b

. FINITE-DIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,

Group Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.

Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite

Abstract Vector Spaces and Concrete Examples

LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.

Homogeneous Constant Matrix Systems, Part II

4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues

Algebra SEP Solutions

Algebra SEP Solutions 17 July 2017 1. (January 2017 problem 1) For example: (a) G = Z/4Z, N = Z/2Z. More generally, G = Z/p n Z, N = Z/pZ, p any prime number, n 2. Also G = Z, N = nz for any n 2, since

Highest-weight Theory: Verma Modules

Highest-weight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or,

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra

SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman

Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 17, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Integral Domains and Fraction Fields 0.1.1 Theorems Now what we are going

EXTERIOR POWERS KEITH CONRAD 1. Introduction Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate R to. A bilinear

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY

LECTURE VI: SELF-ADJOINT AND UNITARY OPERATORS MAT 204 - FALL 2006 PRINCETON UNIVERSITY ALFONSO SORRENTINO 1 Adjoint of a linear operator Note: In these notes, V will denote a n-dimensional euclidean vector

In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation.

1 2 Linear Systems In these chapter 2A notes write vectors in boldface to reduce the ambiguity of the notation 21 Matrix ODEs Let and is a scalar A linear function satisfies Linear superposition ) Linear

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

Algebraic group actions and quotients

23rd Autumn School in Algebraic Geometry Algebraic group actions and quotients Wykno (Poland), September 3-10, 2000 LUNA S SLICE THEOREM AND APPLICATIONS JEAN MARC DRÉZET Contents 1. Introduction 1 2.

z = f (x; y) f (x ; y ) f (x; y) f (x; y )

BEEM0 Optimization Techiniques for Economists Lecture Week 4 Dieter Balkenborg Departments of Economics University of Exeter Since the fabric of the universe is most perfect, and is the work of a most

Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

. Consider the linear system dx= =! = " a b # x y! : (a) For what values of a and b do solutions oscillate (i.e., do both x(t) and y(t) pass through z

Preliminary Exam { 1999 Morning Part Instructions: No calculators or crib sheets are allowed. Do as many problems as you can. Justify your answers as much as you can but very briey. 1. For positive real

CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE

CLASSIFICATIONS OF THE FLOWS OF LINEAR ODE PETER ROBICHEAUX Abstract. The goal of this paper is to examine characterizations of linear differential equations. We define the flow of an equation and examine

where Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset

Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring

The geography of irregular surfaces

Università di Pisa Classical Algebraic Geometry today M.S.R.I., 1/26 1/30 2008 Summary Surfaces of general type 1 Surfaces of general type 2 3 and irrational pencils Surface = smooth projective complex

Gordon solutions. n=1 1. p n

Gordon solutions 1. A positive integer is called a palindrome if its base-10 expansion is unchanged when it is reversed. For example, 121 and 7447 are palindromes. Show that if we denote by p n the nth

COMPLEX MULTIPLICATION: LECTURE 13

COMPLEX MULTIPLICATION: LECTURE 13 Example 0.1. If we let C = P 1, then k(c) = k(t) = k(c (q) ) and the φ (t) = t q, thus the extension k(c)/φ (k(c)) is of the form k(t 1/q )/k(t) which as you may recall

ALGORITHMS FOR COMPUTING QUARTIC GALOIS GROUPS OVER FIELDS OF CHARACTERISTIC 0

ALGORITHMS FOR COMPUTING QUARTIC GALOIS GROUPS OVER FIELDS OF CHARACTERISTIC 0 CHAD AWTREY, JAMES BEUERLE, AND MICHAEL KEENAN Abstract. Let f(x) beanirreducibledegreefourpolynomialdefinedover afieldf and

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

Mathematics I. Exercises with solutions. 1 Linear Algebra. Vectors and Matrices Let , C = , B = A = Determine the following matrices:

Mathematics I Exercises with solutions Linear Algebra Vectors and Matrices.. Let A = 5, B = Determine the following matrices: 4 5, C = a) A + B; b) A B; c) AB; d) BA; e) (AB)C; f) A(BC) Solution: 4 5 a)

Spring, 2012 CIS 515. Fundamentals of Linear Algebra and Optimization Jean Gallier

Spring 0 CIS 55 Fundamentals of Linear Algebra and Optimization Jean Gallier Homework 5 & 6 + Project 3 & 4 Note: Problems B and B6 are for extra credit April 7 0; Due May 7 0 Problem B (0 pts) Let A be

RUSSELL S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW

Hedén, I. Osaka J. Math. 53 (2016), 637 644 RUSSELL S HYPERSURFACE FROM A GEOMETRIC POINT OF VIEW ISAC HEDÉN (Received November 4, 2014, revised May 11, 2015) Abstract The famous Russell hypersurface is

arxiv: v1 [math.rt] 15 Oct 2008

CLASSIFICATION OF FINITE-GROWTH GENERAL KAC-MOODY SUPERALGEBRAS arxiv:0810.2637v1 [math.rt] 15 Oct 2008 CRYSTAL HOYT AND VERA SERGANOVA Abstract. A contragredient Lie superalgebra is a superalgebra defined

On algebras arising from semiregular group actions

On algebras arising from semiregular group actions István Kovács Department of Mathematics and Computer Science University of Primorska, Slovenia kovacs@pef.upr.si . Schur rings For a permutation group

A TRANSCENDENTAL METHOD IN ALGEBRAIC GEOMETRY

Actes, Congrès intern, math., 1970. Tome 1, p. 113 à 119. A TRANSCENDENTAL METHOD IN ALGEBRAIC GEOMETRY by PHILLIP A. GRIFFITHS 1. Introduction and an example from curves. It is well known that the basic

Group Law for elliptic Curves

Group Law for elliptic Curves http://www.math.ku.dk/ verrill/grouplaw/ (Based on Cassels Lectures on Elliptic curves, Chapter 7) 1 Outline: Introductory remarks Construction of the group law Case of finding

ON SOME STRUCTURES OF LEIBNIZ ALGEBRAS

Contemporary Mathematics ON SOME STRUCTURES OF LEIBNIZ ALGEBRAS Ismail Demir, Kailash C. Misra and Ernie Stitzinger Abstract. Leibniz algebras are certain generalization of Lie algebras. In this paper

Section 13.4 The Cross Product

Section 13.4 The Cross Product Multiplying Vectors 2 In this section we consider the more technical multiplication which can be defined on vectors in 3-space (but not vectors in 2-space). 1. Basic Definitions

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom Justin Campbell August 3, 2017 1 Representations of SU 2 and SO 3 (R) 1.1 The following observation is long overdue. Proposition

Mapping Class Groups MSRI, Fall 2007 Day 8, October 25

Mapping Class Groups MSRI, Fall 2007 Day 8, October 25 November 26, 2007 Reducible mapping classes Review terminology: An essential curve γ on S is a simple closed curve γ such that: no component of S