REPRESENTATION THEORY WEEK 7


 Daniella Golden
 1 years ago
 Views:
Transcription
1 REPRESENTATION THEORY WEEK 7 1. Characters of L k and S n A character of an irreducible representation of L k is a polynomial function constant on every conjugacy class. Since the set of diagonalizable matrices is dense in L k, a character is defined by its values on the subgroup of diagonal matrices in L k. Thus, one can consider a character as a polynomial function of x 1,..., x k. Moreover, a character is a symmetric polynomial of x 1,...,x k as the matrices diag (x 1,...,x k ) and diag ( ) x s(1),...,x s(k) are conjugate for any s Sk. For example, the character of the standard representation in E is equal to x x k and the character of E n is equal to (x x k ) n. Let λ = (λ 1,...,λ k ) be such that λ 1 λ λ k. Let D λ denote the determinant of the k kmatrix whose i, j entry equals x λ j i. It is clear that D λ is a skewsymmetric polynomial of x 1,...,x k. If ρ = (k 1,...,1, ) then D ρ = i j (x i x j ) is the well known Vandermonde determinant. Let S λ = D λ+ρ D ρ. It is easy to see that S λ is a symmetric polynomial of x 1,...,x k. It is called a Schur polynomial. The leading monomial of S λ is the x λ 1...x λ k k (if one orders monomials lexicographically) and therefore it is not hard to show that S λ form a basis in the ring of symmetric polynomials of x 1,..., x k. Theorem 1.1. The character of W λ equals to S λ. I do not include a proof of this Theorem since it uses beautiful but hard combinatoric. The proof is much easier in general framework of Lie groups and is included in 61A course. Exercise. Check that dim W λ = ( λi λ ) j (ρ i<j i ρ j ) = i<j ( λi λ ) j (k 1)! (k )!...1!, if λ = λ + ρ. Now we use SchurWeyl duality to establish the relation between characters of S n and L k. Recall that the conjugacy classes in S n are given by partitions of n. Let C (µ) be the class associated with the partition µ in the natural way. Let ρ denote Date: March 15, 11. 1
2 REPRESENTATION THEORY WEEK 7 the representation of S n L k in E n. Let r be the number of rows in µ. Then one can see that (1.1) tr(ρ s g ) = (x µ x µ 1 k )...(xµr x µr k ), for any s C (µ) and a diagonal g L k. Denote by P µ the polynomial in the right hand side of the identity. Let χ λ be the character of V λ. Since tr (ρ s g ) = λ Γ n,k χ λ (s) S λ (g), one obtains the following remarkable relation (1.) P µ = λ Γ n,k χ λ (s) S λ.. Representations of compact groups Let be a group and a topological space. We say that is a topological group if the multiplication map and the inverse are continuous maps. Naturally, is compact if it is compact topological space. Examples. The circle S 1 = {z C z = 1}. A torus T n = S 1 S 1. Unitary group U n = { X L n X t X = 1 n }. Special unitary group Orthogonal group Special orthogonal group SU n = {X U n det X = 1}. O n = { x L n (R) X t X = 1 n }. SO n = {X O n det X = 1}. Theorem.1. Let be a compact group. There exists a unique measure on such that f (ts) dt = f (t) dt, for any integrable function f on and any s, and dt = 1. In the same way there exists a measure d t such that f (st) dt = f (t) d t, d t = 1. Moreover, for a connected compact group dt = d t.
3 REPRESENTATION THEORY WEEK 7 3 The measure dt (d t) is called rightinvariant (leftinvariant) measure, or Haar measure. We do not give the proof of this theorem in general. However, all examples we consider are smooth submanifolds in L k. Thus, to define the invariant measure we just need to define a volume in the tangent space at identity T 1 and then use right (left) multiplication to define it on the whole group. More precisely, let γ Λ top T 1. Then γ s = m s (γ), where m s : is the right (left) multiplication on s and m s is the induced map Λ top T 1 Λtop T s. After this normalize γ to satisfy γ = 1. Consider a vector space over C equipped with topology such that addition and multiplication by a scalar are continuous. We always assume that a topological vector space satisfies the following conditions (1) for any v V there exist a neighborhood of which does not contain v; () there is a base of convex neighborhoods of zero. Topological vector spaces satisfying above conditions are called locally convex. We do not go into the theory of such spaces. All we need is the fact that there is a nonzero continuous linear functional on a locally convex space. A representation ρ : L(V ) is continuous if the map V V given by (s, v) ρ s v is continuous. Regular representation. Let be a compact group and L () be the space of all complex valued functions on such that f (t) dt exists. Then L () is a Hilbert space with respect to Hermitian form f, g = f (t) g (t) dt. Moreover, a representation R of in L () given by R s f (t) = f (ts) is continuous and the Hermitian form is invariant. A representation ρ : L(V ) is called topologically irreducible if any invariant closed subspace of V is either V or. Lemma.. Every irreducible representation of is isomorphic to a subrepresentation in L (). Proof. Let ρ : L(V ) be irreducible. Pick a nonzero linear functional ϕ on V and define the map Φ : V L () which sends v to the matrix coefficient f v,ϕ (s) = ϕ, ρ s v. It is clear that a matrix coefficient is a continuous function on, therefore f v,ϕ L (). Furthermore Φ is a continuous intertwiner and KerΦ =.
4 4 REPRESENTATION THEORY WEEK 7 Recall that a Hilbert space is a space over C equipped with positive definite Hermitian form, complete in topology defined by the norm v = v, v 1/. We need the fact that a Hilbert space has an orthonormal topological basis. A continuous representation ρ : L (V ) is called unitary if V is a Hilbert space and v, v = ρ g v, ρ g v for any v V and g. The regular representation of in L () is unitary. In fact, Lemma. implies Corollary.3. Every topologically irreducible representation of a compact group is a subrepresentation in L (). Lemma.4. Every irreducible unitary representation of a compact group is finitedimensional. Proof. Let ρ : L(V ) be an irreducible unitary representation. Choose v V, v = 1. Define an operator T : V V by the formula Let Tx = v, x v. One can check easily that T is selfadjoint, i.e. x, Ty = Tx, y. Tx = ρ g T ( ρ 1 g x) dg. Then T : V V is an intertwiner and a selfadjoint operator. Furthermore, T is compact, i.e. if S = {x V x = 1}, then T (S) is a compact set in V. Every selfadjoint compact operator has an eigenvector. To construct an eigen vector find x S such that ( Tx, x ) is maximal. Then Tx = λx. Since Ker ( T λ Id ) is an invariant subspace in V, Ker ( T λ Id ) =. Hence T = λid. Note that for any orthonormal system of vectors e 1,..., e n V ei, Te i = ei, Te i 1, that implies λn 1. Hence dim V 1 λ. Corollary.5. Every irreducible continuous representation of a compact group is finitedimensional.
5 REPRESENTATION THEORY WEEK Orthogonality relations and PeterWeyl Theorem If ρ : L (V ) is a unitary representation. Define a matrix coefficient by the formula f v,w (g) = w, ρ g v. It is easy to check that (3.1) f v,w ( g 1 ) = f w,v (g) Theorem 3.1. For an irreducible unitary representation ρ : L(V ) f v,w, f v,w = f v,w (g)f v,w (g)dg = 1 dim ρ v, v w, w. The matrix coefficient of two nonisomorphic representation are orthogonal in L (). Proof. Define T End C (V ) and Tx = v, x v T = ρ g Tρ 1 g dg. As follows from Shur s lemma, T = λid. Since we obtain Hence On the other hand, w, Tw = tr T = tr T = v, v, T = v, v dim ρ. w, Tw = 1 dim ρ v, v w, w. w, v, ρ 1 g w ρ g v ( dg = f ) w,v g 1 f v,w (g)dg = = f v,w (g)f v,w (g)dg = 1 dim ρ f v,w, f v,w. In f v,w and f v,w are matrix coefficients of two nonisomorphic representation, the T =, and the calculation is even simpler. Corollary 3.. Let ρ and σ be two irreducible representations, then χ ρ, χ σ = 1 if ρ is isomorphic to σ and χ ρ, χ σ = otherwise. Theorem 3.3. (PeterWeyl) Matrix coefficient form a dense set in L () for a compact group.
6 6 REPRESENTATION THEORY WEEK 7 Proof. We will prove the Theorem under assumption that L(E), in other words we assume that has a faithful finitedimensional representation. Let M = End C (E). The polynomial functions C [M] on M form a dense set in the space of continuous functions on (Weierstrass theorem), and continuous functions are dense in L (). On the other hand, C [M] is spanned by matrix coefficients of all representations in T (E) = n= E n. Hence matrix coefficients are dense in L (). Corollary 3.4. The characters of irreducible representations form an orthonormal basis in the subspace of class function in L (). Corollary 3.5. Let be a compact group and R denote the representation of in L () given by the formula Then R s,t f (x) = f ( s 1 xt ). L () = ρ b V ρ V ρ, where Ĝ denotes the set of isomorphism classes of irreducible unitary representations of and the direct sum is in the sense of Hilbert spaces. Remark 3.6. Note that it follows from the proof of Theorem 3.3, that if E is a faithful representation of a compact group, then all other irreducible representations appear in T (E) as subrepresentations. 4. Examples Example 1. Let S 1 = {z C z = 1}, z = e iθ. The invariant measure on S 1 is dθ The irreducible representations are one dimensional. They are given by the π characters χ n : S 1 C, where χ n (θ) = e inθ. Hence Ŝ1 = Z and L ( S 1) = n Z Ce inθ, this is wellknown fact that every periodic function can be extended in Fourier series. Example. Let = SU. Then consists of all matrices a b b ā, satisfying the relations a + b = 1. One also can realize SU as the subgroup of quaternions with norm 1. Thus, topologically SU is isomorphic to the threedimensional sphere S 3. To find all irreducible representation of SU consider the polynomial ring C [x, y] with the action of SU given by the formula ( ) a b ρ g (x) = ax + by, ρ g (y) = bx + āy, if g =. b a
7 REPRESENTATION THEORY WEEK 7 7 Let ρ n be the representation of in the space C n [x, y] of homogeneous polynomials of degree n. The monomials x n, x n 1 y,..., y n form a basis of C n [x, y]. Therefore dim ρ n = n + 1. We claim that all ρ n are irreducible and that every irreducible representation of SU is isomorphic to ρ n. Hence Ĝ = Z +. We will show this by checking that the characters χ n of ρ n form an orthonormal basis in the Hilbert space of class functions on. Note that every unitary matrix is diagonal in some orthonormal basis, therefore every conjugacy class of SU intersects the diagonal subgroup. Moreover, ( z z) and ( z z ) are conjugate. Hence the set of conjugacy classes can be identified with S1 quotient by the equivalence relation z z. Let z = e iθ, then (4.1) χ n (z) = z n + z n + + z n = zn+1 z n 1 sin (n + 1)θ =. z z 1 sin θ Now let us calculate the scalar product in the space of class function. It is clear that the invariant measure dg on is proportional to the standard volume form on the threedimensional sphere induced by the volume form on R 4. Let C (θ) denote the conjugacy class of all matrices with eigenvalues e iθ, e iθ. The characteristic polynomial of a matrix from C (θ) equals t cosθt + 1. Thus, we obtain a + ā = cosθ, or a = cos θ + yi for real y. Hence C (θ) satisfy the equation a + b = cos θ + y + b = 1, or y + b = sin θ. In other words, C (θ) is a twodimensional sphere of radius sin θ. Hence for a class function φ on φ (g)dg = 1 π φ (θ)sin θdθ. π All class function are even functions on S 1, i.e. they satisfy the condition φ ( θ) = φ (θ). One can see easily from (4.1) that χ n (θ) form an orthonormal basis in the space of even function on the circle with respect to the Hermitian product ϕ, η = 1 π π ϕ(θ) η (θ) sin θdθ. Example 3. Let = SO 3. Recall that SU can be realized as the set of quaternions with norm 1. Consider the representation γ of SU in H defined by the formula γ g (α) = gαg 1. One can see that the 3dimensional space H im of pure imaginary quaternions is invariant and (α, β) = Re ( α β ) is invariant positive definite scalar product on H im. Therefore ρ defines a homomorphism γ : SU SO 3. Check that Kerγ = {1, 1} and that γ is surjetive. Hence SO 3 = SU / {1, 1}. Thus, every representation of SO 3 can be lifted to the representations of SU, and a representation of SU factors to the representation of SO 3 iff it is trivial on 1. One can check easily that ρ n ( 1) = 1 iff n is even. Thus, an irreducible representations of SO 3 is
8 8 REPRESENTATION THEORY WEEK 7 isomorphic to ρ m and dim ρ m = m + 1. Below we give an independent realization of irreducible representation of SO 3. Harmonic analysis on a sphere. Consider the sphere S in R 3 defined by the equation x + y + z = 1. It is clear that SO 3 acts in the space of complexvalued functions on S. Introduce differential operators in R 3 : e = 1 ( x + y + z ), h = x x + y y + z z + 3, f = 1 ( ) x + y + z, note that e, f, and h commute with the action of SO 3 and satisfy the relations [e, f] = h, [h, e] = e, [h, f] = f. Let P n be the space of homogeneous polynomial of degree n and H n = Ker f P n. The polynomials of H n are harmonic polynomials since they are annihilated by Laplace operator. For any ϕ P n ( h (ϕ) = n + 3 ) ϕ. If ϕ H n, then ( fe (ϕ) = ef (ϕ) h (ϕ) = n + 3 ) ϕ, and by induction fe k (ϕ) = efe k 1 (ϕ) he k 1 (ϕ) = In particular, this implies that (4.) fe k (H n ) = e k 1 (H n ). We prove that (4.3) P n = H n e (H n ) e (H n 4 ) +... by induction on n. Indeed, by induction assumption P n = H n e (H n 4 ) +..., ( nk + k (k 1) + 3k ) e k 1 ϕ. then (4.) implies fe (P n ) = P n. Hence H n ep n =. On the other hand, f : P n P n is surjective, and therefore dim H n + dim P n = dim P n. Therefore (4.4) P n = H n P n, which implies (4.3). Note that after restriction on S, the operator e acts as the multiplication on 1. Hence (4.3) implies that C [ S ] = n H n. To calculate the dimension of H n use (4.4) dim H n = dim P n dim P n = (n + 1)(n + ) n (n 1) = n + 1.
9 REPRESENTATION THEORY WEEK 7 9 Finally, we claim that the representation of SO 3 in H n is irreducible and isomorphic to ρ n. Check that ϕ = (x + iy) n H n and the rotation on the angle θ about z axis maps ϕ to e inθ ϕ. Since this rotation is the image of e iθ/ e iθ/, under the homomorphism γ : SU SO 3, the statement follows from (4.1). Recall now the following theorem (Lecture Notes 1). A convex centrally symmetric solid in R 3 is uniquely determined by the areas of the plane crosssections through the origin. A convex solid B can be defined by an even continuous function on S. Indeed, for each unit vector v let ϕ (v) = sup { t / tv B }. Define a linear operator T in the space of all even continuous functions on S by the formula Tϕ (v) = π ϕ (w)dθ, where w runs the set of unit vectors orthogonal to v, and θ is the angular parameter on the circle S v. Check that Tϕ (v) is the area of the cross section by the plane v. We have to prove that T is invertible. Obviously T commutes with the SO 3 action. Therefore T can be diagonalized. Moreover, T acts on H n as the scalar operator λ n Id. We have to check that λ n for all n. Let ϕ = (x + iy) n H n. Then ϕ (1,, ) = 1 and Tϕ (1,, ) = π π (iy) n dθ = ( 1) n sin n θdθ, here we take the integral over the circle y + z = 1, and assume y = sin θ, z = cosθ. Since Tϕ = λ n ϕ, we obtain π λ n = ( 1) n sin n θdθ.
Math 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationAlgebra 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
More informationCHAPTER 6. Representations of compact groups
CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
More informationRepresentation 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
More informationREPRESENTATION THEORY WEEK 5. B : V V k
REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationALGEBRA 8: Linear algebra: characteristic polynomial
ALGEBRA 8: Linear algebra: characteristic polynomial Characteristic polynomial Definition 8.1. Consider a linear operator A End V over a vector space V. Consider a vector v V such that A(v) = λv. This
More informationTopics 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
More informationExercises Lie groups
Exercises Lie groups E.P. van den Ban Spring 2009 Exercise 1. Let G be a group, equipped with the structure of a C manifold. Let µ : G G G, (x, y) xy be the multiplication map. We assume that µ is smooth,
More informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NWSE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 18811940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
More information11. Representations of compact Lie groups
11. Representations of compact Lie groups 11.1. Integration on compact groups. In the simplest examples like R n and the torus T n we have the classical Lebesgue measure which defines a translation invariant
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationHighestweight Theory: Verma Modules
Highestweight Theory: Verma Modules Math G4344, Spring 2012 We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semisimple Lie algebra (or,
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationSome notes on Coxeter groups
Some notes on Coxeter groups Brooks Roberts November 28, 2017 CONTENTS 1 Contents 1 Sources 2 2 Reflections 3 3 The orthogonal group 7 4 Finite subgroups in two dimensions 9 5 Finite subgroups in three
More informationIntroduction to Lie Groups
Introduction to Lie Groups MAT 4144/5158 Winter 2015 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons AttributionShareAlike
More informationLinear algebra 2. Yoav Zemel. March 1, 2012
Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at zamsh7@gmail.com.
More informationExercises Lie groups
Exercises Lie groups E.P. van den Ban Spring 2012 Exercise 1. Let G be a group, equipped with the structure of a C manifold. Let µ : G G G, (x, y) xy be the multiplication map. We assume that µ is smooth,
More informationREPRESENTATION THEORY. WEEK 4
REPRESENTATION THEORY. WEEK 4 VERA SERANOVA 1. uced modules Let B A be rings and M be a Bmodule. Then one can construct induced module A B M = A B M as the quotient of a free abelian group with generators
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationThe Spectral Theorem for normal linear maps
MAT067 University of California, Davis Winter 2007 The Spectral Theorem for normal linear maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 14, 2007) In this section we come back to the question
More informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More informationProblems 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
More informationREPRESENTATION THEORY FOR FINITE GROUPS
REPRESENTATION THEORY FOR FINITE GROUPS SHAUN TAN Abstract. We cover some of the foundational results of representation theory including Maschke s Theorem, Schur s Lemma, and the Schur Orthogonality Relations.
More informationSupplementary Notes on Linear Algebra
Supplementary Notes on Linear Algebra Mariusz Wodzicki May 3, 2015 1 Vector spaces 1.1 Coordinatization of a vector space 1.1.1 Given a basis B = {b 1,..., b n } in a vector space V, any vector v V can
More informationRepresentations 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
More informationReview of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem
Review of Linear Algebra Definitions, Change of Basis, Trace, Spectral Theorem Steven J. Miller June 19, 2004 Abstract Matrices can be thought of as rectangular (often square) arrays of numbers, or as
More informationHomework set 4  Solutions
Homework set 4  Solutions Math 407 Renato Feres 1. Exercise 4.1, page 49 of notes. Let W := T0 m V and denote by GLW the general linear group of W, defined as the group of all linear isomorphisms of W
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationWeyl Group Representations and Unitarity of Spherical Representations.
Weyl Group Representations and Unitarity of Spherical Representations. Alessandra Pantano, University of California, Irvine Windsor, October 23, 2008 β ν 1 = ν 2 S α S β ν S β ν S α ν S α S β S α S β ν
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationEigenvalues and Eigenvectors
Chapter 1 Eigenvalues and Eigenvectors Among problems in numerical linear algebra, the determination of the eigenvalues and eigenvectors of matrices is second in importance only to the solution of linear
More informationPeter Hochs. Strings JC, 11 June, C algebras and Ktheory. Peter Hochs. Introduction. C algebras. Group. C algebras.
and of and Strings JC, 11 June, 2013 and of 1 2 3 4 5 of and of and Idea of 1 Study locally compact Hausdorff topological spaces through their algebras of continuous functions. The product on this algebra
More informationThe Weyl Character Formula
he Weyl Character Formula Math 4344, Spring 202 Characters We have seen that irreducible representations of a compact Lie group can be constructed starting from a highest weight space and applying negative
More informationOptimization Theory. A Concise Introduction. Jiongmin Yong
October 11, 017 16:5 wsbook9x6 Book Title Optimization Theory 01708Lecture Notes page 1 1 Optimization Theory A Concise Introduction Jiongmin Yong Optimization Theory 01708Lecture Notes page Optimization
More informationNOTES ON THE NUMERICAL RANGE
NOTES ON THE NUMERICAL RANGE JOEL H. SHAPIRO Abstract. This is an introduction to the notion of numerical range for bounded linear operators on Hilbert space. The main results are: determination of the
More informationREPRESENTATIONS OF S n AND GL(n, C)
REPRESENTATIONS OF S n AND GL(n, C) SEAN MCAFEE 1 outline For a given finite group G, we have that the number of irreducible representations of G is equal to the number of conjugacy classes of G Although
More informationOPERATOR THEORY ON HILBERT SPACE. Class notes. John Petrovic
OPERATOR THEORY ON HILBERT SPACE Class notes John Petrovic Contents Chapter 1. Hilbert space 1 1.1. Definition and Properties 1 1.2. Orthogonality 3 1.3. Subspaces 7 1.4. Weak topology 9 Chapter 2. Operators
More informationREPRESENTATION THEORY. WEEKS 10 11
REPRESENTATION THEORY. WEEKS 10 11 1. Representations of quivers I follow here CrawleyBoevey lectures trying to give more details concerning extensions and exact sequences. A quiver is an oriented graph.
More informationis an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent
Lecture 4. GModules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of Gmodules, mostly for finite groups, and a recipe for finding every irreducible Gmodule of a
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationMath 25a Practice Final #1 Solutions
Math 25a Practice Final #1 Solutions Problem 1. Suppose U and W are subspaces of V such that V = U W. Suppose also that u 1,..., u m is a basis of U and w 1,..., w n is a basis of W. Prove that is a basis
More informationM.6. Rational canonical form
book 2005/3/26 16:06 page 383 #397 M.6. RATIONAL CANONICAL FORM 383 M.6. Rational canonical form In this section we apply the theory of finitely generated modules of a principal ideal domain to study the
More information18.06 Problem Set 8  Solutions Due Wednesday, 14 November 2007 at 4 pm in
806 Problem Set 8  Solutions Due Wednesday, 4 November 2007 at 4 pm in 206 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state
More informationJordan normal form notes (version date: 11/21/07)
Jordan normal form notes (version date: /2/7) If A has an eigenbasis {u,, u n }, ie a basis made up of eigenvectors, so that Au j = λ j u j, then A is diagonal with respect to that basis To see this, let
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationDUALITY, CENTRAL CHARACTERS, AND REALVALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE
DUALITY, CENTRAL CHARACTERS, AND REALVALUED CHARACTERS OF FINITE GROUPS OF LIE TYPE C. RYAN VINROOT Abstract. We prove that the duality operator preserves the Frobenius Schur indicators of characters
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular pchains with coefficients in a field K. Furthermore, we can define the
More informationReal symmetric matrices/1. 1 Eigenvalues and eigenvectors
Real symmetric matrices 1 Eigenvalues and eigenvectors We use the convention that vectors are row vectors and matrices act on the right. Let A be a square matrix with entries in a field F; suppose that
More informationLECTURE VI: SELFADJOINT AND UNITARY OPERATORS MAT FALL 2006 PRINCETON UNIVERSITY
LECTURE VI: SELFADJOINT 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 ndimensional euclidean vector
More informationRepresentations and Linear Actions
Representations and Linear Actions Definition 0.1. Let G be an Sgroup. A representation of G is a morphism of Sgroups φ G GL(n, S) for some n. We say φ is faithful if it is a monomorphism (in the category
More information(VII.E) The Singular Value Decomposition (SVD)
(VII.E) The Singular Value Decomposition (SVD) In this section we describe a generalization of the Spectral Theorem to nonnormal operators, and even to transformations between different vector spaces.
More informationNATIONAL BOARD FOR HIGHER MATHEMATICS. M. A. and M.Sc. Scholarship Test. September 25, Time Allowed: 150 Minutes Maximum Marks: 30
NATIONAL BOARD FOR HIGHER MATHEMATICS M. A. and M.Sc. Scholarship Test September 25, 2010 Time Allowed: 150 Minutes Maximum Marks: 30 Please read, carefully, the instructions on the following page 1 INSTRUCTIONS
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More informationarxiv: v1 [math.sg] 6 Nov 2015
A CHIANGTYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiangtype lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the xcomponent of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationLecture 12: Diagonalization
Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors
More informationIntroduction to Lie Groups and Lie Algebras. Alexander Kirillov, Jr.
Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr. Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA Email address: kirillov@math.sunysb.edu URL: http://www.math.sunysb.edu/~kirillov/liegroups/
More informationIMPORTANT 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
More informationRepresentations of Matrix Lie Algebras
Representations of Matrix Lie Algebras Alex Turzillo REU Apprentice Program, University of Chicago aturzillo@uchicago.edu August 00 Abstract Building upon the concepts of the matrix Lie group and the matrix
More information1 Classifying Unitary Representations: A 1
Lie Theory Through Examples John Baez Lecture 4 1 Classifying Unitary Representations: A 1 Last time we saw how to classify unitary representations of a torus T using its weight lattice L : the dual of
More informationOctober 4, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
October 4, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationLinear Algebra: Graduate Level Problems and Solutions. Igor Yanovsky
Linear Algebra: Graduate Level Problems and Solutions Igor Yanovsky Linear Algebra Igor Yanovsky, 5 Disclaimer: This handbook is intended to assist graduate students with qualifying examination preparation.
More informationMATH JORDAN FORM
MATH 53 JORDAN FORM Let A,, A k be square matrices of size n,, n k, respectively with entries in a field F We define the matrix A A k of size n = n + + n k as the block matrix A 0 0 0 0 A 0 0 0 0 A k It
More informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10PHYBIPMA2) EXAM  Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10PHYBIPMA2 EXAM  Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
More informationGroup 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,
More informationNotes 2 for MAT4270 Connected components and universal covers π 0 and π 1.
Notes 2 for MAT4270 Connected components and universal covers π 0 and π 1. Version 0.00 with misprints, Connected components Recall thaty if X is a topological space X is said to be connected if is not
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationProblem 1A. Suppose that f is a continuous real function on [0, 1]. Prove that
Problem 1A. Suppose that f is a continuous real function on [, 1]. Prove that lim α α + x α 1 f(x)dx = f(). Solution: This is obvious for f a constant, so by subtracting f() from both sides we can assume
More informationKirillov Theory. TCU GAGA Seminar. Ruth Gornet. January University of Texas at Arlington
TCU GAGA Seminar University of Texas at Arlington January 2009 A representation of a Lie group G on a Hilbert space H is a homomorphism such that v H the map is continuous. π : G Aut(H) = GL(H) x π(x)v
More informationTRANSLATIONINVARIANT FUNCTION ALGEBRAS ON COMPACT GROUPS
PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 3, 1965 TRANSLATIONINVARIANT FUNCTION ALGEBRAS ON COMPACT GROUPS JOSEPH A. WOLF Let X be a compact group. $(X) denotes the Banach algebra (point multiplication,
More informationSeptember 26, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
September 26, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationMath 396. An application of GramSchmidt to prove connectedness
Math 396. An application of GramSchmidt to prove connectedness 1. Motivation and background Let V be an ndimensional vector space over R, and define GL(V ) to be the set of invertible linear maps V V
More informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More informationThe Jordan Canonical Form
The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finitedimensional vector space over an algebraically closed field. Here we develop
More informationIMPORTANT 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
More informationTHE 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
More informationM3/4/5P12 GROUP REPRESENTATION THEORY
M3/4/5P12 GROUP REPRESENTATION THEORY JAMES NEWTON Course Arrangements Send comments, questions, requests etc. to j.newton@imperial.ac.uk. The course homepage is http://wwwf.imperial.ac.uk/ jjmn07/m3p12.html.
More informationHonors 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
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam  Course Instructor : Prashanth L.A. Date : Sep24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationSolutions of exercise sheet 8
DMATH Algebra I HS 14 Prof. Emmanuel Kowalski Solutions of exercise sheet 8 1. In this exercise, we will give a characterization for solvable groups using commutator subgroups. See last semester s (Algebra
More informationHonors Linear Algebra, Spring Homework 8 solutions by Yifei Chen
.. Honors Linear Algebra, Spring 7. Homework 8 solutions by Yifei Chen 8... { { W {v R 4 v α v β } v x, x, x, x 4 x x + x 4 x + x x + x 4 } Solve the linear system above, we get a basis of W is {v,,,,
More informationLecture 9. Econ August 20
Lecture 9 Econ 2001 2015 August 20 Lecture 9 Outline 1 Linear Functions 2 Linear Representation of Matrices 3 Analytic Geometry in R n : Lines and Hyperplanes 4 Separating Hyperplane Theorems Back to vector
More informationLIE GROUPS, LIE ALGEBRAS, AND APPLICATIONS IN PHYSICS
LIE GROUPS, LIE ALGEBRAS, AND APPLICATIONS IN PHYSICS JOO HEON YOO Abstract. This paper introduces basic concepts from representation theory, Lie group, Lie algebra, and topology and their applications
More informationCHAPTER X THE SPECTRAL THEOREM OF GELFAND
CHAPTER X THE SPECTRAL THEOREM OF GELFAND DEFINITION A Banach algebra is a complex Banach space A on which there is defined an associative multiplication for which: (1) x (y + z) = x y + x z and (y + z)
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finitedimensional vector space V. Then the following conditions are equivalent:
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationFinitedimensional spaces. C n is the space of ntuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a prehilbert space, or a unitary space) if there is a mapping (, )
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationBott Periodicity. Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel
Bott Periodicity Anthony Bosman Senior Honors Thesis Department of Mathematics, Stanford University Adviser: Eleny Ionel Acknowledgements This paper is being written as a Senior honors thesis. I m indebted
More informationSingular Value Decomposition (SVD) and Polar Form
Chapter 2 Singular Value Decomposition (SVD) and Polar Form 2.1 Polar Form In this chapter, we assume that we are dealing with a real Euclidean space E. Let f: E E be any linear map. In general, it may
More information