Invariants under simultaneous conjugation of SL 2 matrices

Similar documents
On the GIT moduli of semistable pairs consisting of a plane cubic curve and a line

AN INTRODUCTION TO MODULI SPACES OF CURVES CONTENTS

Mathematics 7800 Quantum Kitchen Sink Spring 2002

BEZOUT S THEOREM CHRISTIAN KLEVDAL

ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS

The Hilbert-Mumford Criterion

Complex Algebraic Geometry: Smooth Curves Aaron Bertram, First Steps Towards Classifying Curves. The Riemann-Roch Theorem is a powerful tool

GEOMETRIC STRUCTURES OF SEMISIMPLE LIE ALGEBRAS

OLIVIER SERMAN. Theorem 1.1. The moduli space of rank 3 vector bundles over a curve of genus 2 is a local complete intersection.

THE QUANTUM CONNECTION

Curtis Heberle MTH 189 Final Paper 12/14/2010. Algebraic Groups

INTRODUCTION TO GEOMETRIC INVARIANT THEORY

Algebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra

LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS

Sample algebra qualifying exam

D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties

We can choose generators of this k-algebra: s i H 0 (X, L r i. H 0 (X, L mr )

Semistable Representations of Quivers

Fundamental Lemma and Hitchin Fibration

Mini-Course on Moduli Spaces

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

2. Intersection Multiplicities

The tangent space to an enumerative problem

10. Smooth Varieties. 82 Andreas Gathmann

A matrix over a field F is a rectangular array of elements from F. The symbol

CSL361 Problem set 4: Basic linear algebra

Linear Algebra, 4th day, Thursday 7/1/04 REU Info:

Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013

Kähler configurations of points

Compactifications of Siegel Modular Varieties

COMMUTING PAIRS AND TRIPLES OF MATRICES AND RELATED VARIETIES

Dot Products. K. Behrend. April 3, Abstract A short review of some basic facts on the dot product. Projections. The spectral theorem.

K-stability and Kähler metrics, I

Lecture 1. Toric Varieties: Basics

Contents. 6 Systems of First-Order Linear Dierential Equations. 6.1 General Theory of (First-Order) Linear Systems

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2.

Algebra Workshops 10 and 11

where m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6

Contents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces

Combinatorics and geometry of E 7

Noncommutative compact manifolds constructed from quivers

Combinatorics for algebraic geometers

INTRODUCTION TO LIE ALGEBRAS. LECTURE 7.

ELEMENTARY LINEAR ALGEBRA

Mappings of elliptic curves

CS 147: Computer Systems Performance Analysis

Commuting nilpotent matrices and pairs of partitions

Definition 9.1. The scheme µ 1 (O)/G is called the Hamiltonian reduction of M with respect to G along O. We will denote by R(M, G, O).

Changing coordinates to adapt to a map of constant rank

Linear Algebra March 16, 2019

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

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.

(1) is an invertible sheaf on X, which is generated by the global sections

Maximum Rank Distance Codes are Generic Gabidulin Codes are Not

NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

Group Theory - QMII 2017

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS

Invariants and hyperelliptic curves: geometric, arithmetic and algorithmic aspects

REPRESENTATION THEORY OF S n

3 Galois Theory. 3.1 Definitions and Examples

Spring 2018 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 3

Some remarks on Nakajima s quiver varieties of type A

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

HARTSHORNE EXERCISES

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

z, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1

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

POLYNOMIAL IDENTITY RINGS AS RINGS OF FUNCTIONS

MAKSYM FEDORCHUK. n ) = z1 d 1 zn d 1.

Representations and Linear Actions

Varieties of Characters

Math 396. Bijectivity vs. isomorphism

K3 Surfaces and Lattice Theory

Computer Graphics II

Cover Page. The handle holds various files of this Leiden University dissertation.

Norm-induced partially ordered vector spaces

TRACE AND NORM KEITH CONRAD

LINKING INVARIANT FOR ALGEBRAIC PLANE CURVES

Margulis Superrigidity I & II

The Cayley-Hamilton Theorem and the Jordan Decomposition

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

August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1.

9. Birational Maps and Blowing Up

1. Affine Grassmannian for G a. Gr Ga = lim A n. Intuition. First some intuition. We always have to rst approximation

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

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort

THE MODULI SPACE OF RATIONAL FUNCTIONS OF DEGREE d

Gradings on Lie Algebras of Cartan and Melikian Type

Algebra I Fall 2007

The derived category of a GIT quotient

ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.

ELEMENTARY LINEAR ALGEBRA

(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

Logarithmic geometry and rational curves

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism then L = φ O P n

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

Transcription:

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

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

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

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

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

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?

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?

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.

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.

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 + 1 2 d i 0 0 1 2 a i + 1 2 d i C[SL 2 SL 2 ] SL 2 = (C[M1, M 2 ] SL 2 C[t 1, t 2 ]/(detm i + 1 4 t2 i = 1. So: we translate our results on traceless matrices

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 + 1 4 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

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 ].

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

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 2 16. Or: C[SL 3 2 ]SL 2 = C[a, b, c, d,e, f ] + C[a, b, c, d,e, f ] g

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.

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 :.

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.

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.

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 : 2 1 2 0 = SL 2 2. If we glue together these quotients, we get a new quotient.

A good quotient The ane subset SL 2 SL 2 SL 2 SL 2 is the ane subset where 1 2 0 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 2 1 1 2 ; TrM 1 M 2 TrM 1M 2 1 2. 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.

A good quotient Another subset Subset of Q Q where (a 1 + d 1 2 0 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

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.

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?

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.

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

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( 0 0 1 0 = 2; weight(0 0 1 0 = 2; weight(1 0 0 0,(0 0 0 1, = 0 So: non-semi-stable points are in span of ( 0 0 1 0 : nilpotent matrices All points are non-stable!

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.

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.

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 2 1 + x 2 2 + x 2 5 x 3 x 5 4x 2 4 0}.

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?

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

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 2 1 + 2ac x 1 x 2 + c 2 x 2 2 ; ( a b c d x1 x 2 = ab x 2 1 + (ad + bc x 1 x 2 + cd x 2 2 ; ( a b c d x 2 2 = b 2 x 2 1 + 2bd x 1 x 2 + d 2 x 2 2. Then [k] is the unique irreducible representation of dimension k + 1

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!

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 1 1 ( ( 0, 2 0 0 1 0 0 0 0 1,, 2 1 1 0 0 2 2 }, Theorem V = [2] + 2 [0] as representations.

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].

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

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?

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?

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?

Appendix Literature W. Drensky. Dening Relations for the Algebra of Invariants of 2 2 Matrices. Algebras and Representation Theory, 2003. A. Extra. The invariants of 2 2 matrices, their algebraic relations and the corresponding moduli problem. PhD Thesis, Katholieke Universiteit Nijmegen, 1976. See http://meilof.home.fmf.nl/scriptie/ for (the draft of my thesis.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback