REPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012

Size: px
Start display at page:

Download "REPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012"

Transcription

1 REPRESENTATION THEORY NOTES FOR MATH 4108 SPRING 2012 JOSEPHINE YU This note will cover introductory material on representation theory, mostly of finite groups. The main references are the books of Serre [4] and Artin [1]. I will roughly follow Fulton [3] for the part on symmetric groups. 1. Basic notions 1.1. Definitions and first examples. Let G be a group with identity 1. A matrix representation of G is a homomorphism ρ : G GL n where GL n (C) denotes the general linear group of n n invertible matrices over the complex numbers C. For g G, we will denote ρ(g) as R g. For any g,h G, we have R gh = R g R h. Moreover, R 1 is the identity automorphism and R g 1 = Rg 1. Let V be an n-dimensional vector space over the field C. The general linear group GL(V ) is the group of automorphisms of V, i.e. one-to-one and onto linear transformations from V to itself. A linear representation of G on V is a homomorphism ρ : G GL(V ). If we choose a basis e 1,e 2,...,e n of V, then each element in GL(V ) is represented by an invertible n n matrix. Hence GL n (C) is a matrix representation of the group GL(V ). The dimension of a representation ρ is the dimension of the vector space V. By abuse of notation, we may sometimes call V as a representation of G. If we choose a basis {e 1,e 2,...,e n } of V, then any linear representation gives rise to a matrix representation. A representation is called faithful if ρ is injective. Example 1. (1) The map that sends every g G to the identity element in GL(V ) is a representation, called the trivial representation or the unit representation. Date: April 25,

2 2 REPRESENTATION THEORY NOTES FOR MATH 4108 (2) Let G = C m, the cyclic group of order m with a generator a. Let V = C. Then ρ : G GL(V ) given by ρ a k(v) = ω k v where ω is a primitive mth root of unity is a faithful representation. In other words, GL(V ) is represented by GL 1 = C. Then ρ gives a matrix representation R : a k ω k. Two representations ρ : G V and ρ : G V are called isomorphic (or similar) if there is an onto isomorphism τ : V V such that τ ρ g = ρ g τ for all g G. If R and R are two matrix representations, then they are similar if there is an invertible matrix T such that TR g T 1 = R g for all g G. In other words, two matrix representations are isomorphic means that they are conjugates. We can think of a representation of G on V as a linear group action of G on V. Example 2. (1) Let A be a finite set with the group G acting on it, i.e. there is a map α : G A A, where α(g,a) A is denoted g a, such that 1 a = a and gh a = g (h a). Let V be a vector space with basis {e a : a A} indexed by elements of A. It is isomorphic to C A. We can extend the action of G on A linearly to all of V and get a representation. In other words, for g G, let ρ g : V V be the linear transformation a A c ae a a A c ae g a. In basis {e a : a A}, ρ is represented by A A permutation matrices. (2) Let S n act on {1, 2,...,n} by permutation. Then each σ S n is represented by the n n permutation matrix of σ. (3) Let G be finite and consider the action of G on itself (on the left) defined by g h = gh. The resulting representation is called the regular representation of G. Write down the matrices for regular representation of C 2 C 2. Example 3. The quaternion group H = {±1, ±i, ±j, ±k} with i 1 = j 2 = k 2 = ijk = 1 is represented by R : H GL 2 (C) where ( ) ( ) ( ) ( ) 1 0 i i R 1 = R 0 1 i =,R 0 i j =,R 1 0 k =. i 0 Example 4. Let G = S n and V = C. Then the sign representation is given by σ 1 if σ is an even permutation and σ 1 if σ is an odd permutation. The main problem that will concern us for most of our investigations is the following:

3 Problem 5. Classify all representations of a given group. What are the building blocks? From now on, we will assume that the group G is finite Subrepresentations and Irreducible Representations. Let ρ : G GL(V ) be a linear representation and W V be a subspace of V. We say that W is stable under the action of G or G-invariant if for any w W and g G, ρ g w W. Note that w does not need to be fixed by ρ g. For g G, let ρ W g : W V be the restriction of ρ g to W. Since W is G-invariant, ρ W g maps W to W. Since ker(ρ g ) = 0, ker(ρ W g ) = 0 as well. But linear transformations that are one-to-one are also onto, so ρ W g W is in GL(W). Let ρ W : G GL(W) be defined by g ρ W g. For g,h G, we have ρ g ρ h = ρ gh, so ρ W g ρ W h = ρw gh. Thus ρ W is a representation of G in W. It is called a subrepresentation of ρ or V. Example 6. In the regular representation, the subspace spanned by g G e g is invariant under G. A representation ρ : G GL(V ) is called irreducible if it has no proper subrepresentations. That is, there is no proper G-invariant subspace W V. An irreducible representation is often called an irrep. The next question we want to answer is: Question 7. Can all representations be decomposed into irreducible representations? Theorem 8. Let ρ : G GL(V ) be a representation and W V be a G-invariant subspace. Then there exists a complement W 0 of W that is also G-invariant. Proof. Let W be any complement of W, not necessarily G-invariant. Let p : V W be the projection associated to the decomposition V = W W. Let p 0 be the average of p over G: p 0 = 1 h G ρ h pρ 1 h. Then p 0 is also a projection from V to W. Let W 0 = ker(p 0 ). We wish to show that W 0 is G-invariant. For g G, we have ρ g p 0 ρ 1 g = 1 ρ g ρ h pρ 1 h ρ 1 g = 1 ρ ghpρ (gh) 1 = p 0 h G as the sum runs over every element of G. Hence p 0 ρ g = ρ g p 0 for all g G. Let w W 0. Then p 0 w = 0, so p 0 ρ g w = ρ g p 0 w = 0. Thus ρ g w W 0. This shows that W 0 is G-invariant. 3

4 4 REPRESENTATION THEORY NOTES FOR MATH 4108 Corollary 9 (Maschke s Theorem). Every representation of a finite group is a direct sum of irreducible representations. Proof. We will proceed by induction on dim(v ). If dim(v ) = 0, then the representation is irreducible, so we are done. Suppose dim(v ) 1. If V is irreducible, then we are done. Suppose not. Then there is a G-invariant subspace W of V such that 0 < dim(w) < dim(v ). By Theorem 8, there is a G-invariant subspace W 0 in V such that V = W W 0. Since dim(w) < dim(v ) and dim(w 0 ) < dim(v ), by the inductive hypothesis, W and W are direct sums of irreducible representations, and so is V. Remark 10. The decomposition need not be unique. Consider the representation R : G GL n (C) where R g is the identity for all g G. Then every subspace of C n is G-invariant, and there are infinitely many decomposition of C n to a direct sum of lines. However, we will see later that the number of components isomorphic to a given irreducible representation does not depend on the choice of decomposition Unitary Representations. An n n complex matrix U is called unitary if UU = I where U = U T denotes the adjoint of a complex matrix. Given a choice of basis for a vector space V of dimension n, we get a scalar product on V given by u,v = [u] [v] where [u], [v] C n denote the coordinate vectors of u,v V respectively. An n n matrix A is unitary if and only if x,y = Ax,Ay for every x,y C n and scalar product is with respect to the standard basis in C n. A matrix representation R : G GL n (C) is called unitary if the matrix R g is unitary for all g G. The following result shows why unitary matrices are desirable. For a subspace W C n, the orthogonal complement W is defined as the space {v C n : v w = 0 for all w W }. Lemma 11. If a representation R : G GL n (C) is unitary and W C n is a G-invariant subspace of C n, then the orthogonal complement W is also G-invariant. Proof. Let v W and g G. Since R g is unitary, for any w W, we have (R g v) w = v (R g w) = 0. Hence R g v W. It is natural to ask whether all linear representations are unitary with respect to some choice of basis. Theorem 12. Let G be a finite group and ρ : G GL(V ) be a linear representation. Then there is a basis of V such that the associated matrix representation is unitary.

5 Proof. Let (, ) be a scalar product on V with respect to some basis. Define another scalar product, on V by u,v = 1 (ρ h u,ρ h v). h G Then for any g G and u,v V we have ρ g u,ρ g v = u,v. Choose a basis of V orthonormal with respect to, using Gram-Schmidt process, and let R be the matrix representation associated to ρ with this basis. For any g G, ρ g (u),ρ g (v) = u,v for all u,v V, so (R g x) (R g y) = x y for all x,y C n ; thus R g is a unitary matrix. We get some nice consequences of this theorem. Corollary 13. Let G be a finite group and R : G GL n (C) be a matrix representation. Then for any g G the matrix R g is diagonalizable. Proof. By the previous theorem, there is an invertible matrix P such that PR g P 1 is unitary for all g G. Since unitary matrices are diagonalizable, each R g is also diagonalizable. Corollary 14. If a matrix A GL n (C) has finite order, i.e. A m = I for some integer m > 0, then A is diagonalizable. Proof. Let G be the cyclic subgroup of GL n (C) generated by A. Then the inclusion G GL n (C) is a matrix representation, so the previous corollary applies Tensor products. Tensor product gives as a way to construct new representations from old ones. Direct sum is another. Let V 1 and V 2 be vector spaces with bases {v 1,v 2,...,v n } and {w 1, w 2,...,w n } respectively. The tensor product V 1 V 2 is a vector space with basis {v i w j : 1 i n, 1 j m}. Note that dim(v 1 V 2 ) = dim(v 1 ) dim(v 2 ). For v = n i=1 c iv i V 1 and w = m j=1 d jw j V 2, let v w := c i d j (v i w j ). 1 i n 1 j m However, not every element of V 1 V 2 is of the form v w. Let ρ 1 : G V 1 and ρ 2 : G V 2 be representations of a finite group G. We can define a map ρ : G GL(V 1 V 2 ), where rho g : V 1 V 2 V 1 V 2 is the linear transformation given by linearly extending the following map on the basis elements ρ g (v i w i ) = ρ 1 g(v i ) ρ 2 g(w i ). It is easy to check that ρ is a linear representation. 5

6 6 REPRESENTATION THEORY NOTES FOR MATH 4108 Let us fix bases {v 1,v 2,...,v n } of V 1, {w 1, w 2,...,w n } of V 2, and {v 1 w 1,v 1 w 2,...,v n w m 1,v n w m } of V 1 V 2. If A and B are matrices associated to ρ 1 g and ρ 2 g, then the matrix A B associated to ρ g is the block matrix whose i,j block is a ij B where a ij is the ij-entry of A. 2. Character Theory Let ρ : G GL(V ) be a representation. The character χ of ρ is a function χ : G C defined by χ(g) = Tr(ρ g ) where Tr denotes the trace of with respect to a choice of basis. Since Tr(AB) = Tr() for any n n matrices A and B, the character does not depend on the choice of basis. We will see that it characterizes the representation. Proposition 15. Let χ be the character of a representation ρ : G GL(V ). Then (1) χ(1) = dim(v ), (2) χ(g 1 ) = χ(g) for any g G, (3) χ(hgh 1 ) = χ(g) for any g,h G. Proof. Let R be a matrix representation associated to ρ with respect to some basis of V. Then χ(1) = Tr(R 1 ) = Tr(I) = dim(v ). For g,h G, χ(hgh 1 = Tr(R hgh 1) = Tr(R h R g R h 1) = Tr(R g R h R h 1) = Tr(R g ) = χ(g). To see the remaining part, let g G. Since G is a finite group, (R g ) k = I for some integer k, which can be taken to be o(g). Suppose λ 1,...,λ n be eigenvalues of R g. Then λ k i = 1 for all i, so λ i = 1. Thus 1/λ i = λ i for all i = 1, 2,...,n. Then the eigenvalues of Rg 1 are λ 1,...,λ n. The claim then follows, using the fact that the trace of a matrix is the sum of its eigenvalues. Complex valued functions on G that are constant on each conjugacy class are called class functions. They form a sub-vector space of the space C G of all complex valued functions on G. The last statement of the proposition above shows that characters are class functions. Let us write a character as a vector of length where the entries are indexed by elements of G. Example 16. Let G = S 3 and order the elements as follows: {e, (12), (13), (23), (123), (132)}. Then the unit representation has character (1, 1, 1, 1, 1, 1) and sign representation sending an even permutation to 1 and odd permutation to 1 is a representation with character (1, 1, 1, 1, 1, 1). The premutation representation sending

7 σ S 3 to the 3 3 permutation matrix representing σ has character (3, 1, 1, 1, 0, 0). Lemma 17. Let ρ 1 : G GL(V 1 ) and ρ 2 : G GL(V 2 ) be representations of characters χ 1 and χ 2 respectively. Then (1) the character of V 1 V 2 is χ 1 + χ 2, and (2) the character of V 1 V 2 is χ 1 χ 2. Proof. The statements follow from the definition of characters and by considering the matrices of the direct sum and tensor products Orthogonality. We will see a very powerful theorem about characters. Characters of irreducible representations are called irreducible characters. Theorem 18. Let G be a finite group. (1) If χ is an irreducible character, then χ,χ = 1. (2) If χ and χ are characters of non-isomorphic irreducible representations, then χ,χ = 0. We will prove this theorem in the next section. Let us first look at nice consequences. Corollary 19. The number of non-isomorphic irreducible representations of G is at most the number of conjugacy classes of G. Proof. The dimension of the vector space of class functions on G is equal to the number of conjugacy classes of G. The characters of nonisomorphic irreducible representations are orthonormal, hence linearly independent. Corollary 20. Let V be a representation of G with character φ and W be an irrecucible representation of G with character χ. Then in any decomposition of V into irreducible representations, the number of components isomorphic to W is φ,χ. Proof. Suppose V = W 1 W 2 W k where each W i is an irreducible representation. Let χ 1,χ 2,...,χ k be the characters of W 1,W 2,...,W k respectively. Then φ = χ 1 + χ χ k. Then χ,χ i is equal to 1 if W is isomorphic to W i and 0 otherwise. Corollary 21. Two representations with the same character are isomorphic. Proof. They contain each irreducible representation the same number of times. 7

8 8 REPRESENTATION THEORY NOTES FOR MATH 4108 Corollary 22. Let V be a representation of G with character φ. Suppose W 1,W 2,...,W r are irreducible representations of G and V contains W i m i times, then r φ,φ = m 2 i. Example 23. Let G = S 4 be the symmetric group of order 4!. The character table is i=1 identity (12) (12)(34) (123) (1234) χ χ χ χ χ The unit and sign representations give the characters χ 1 and χ 2 respectively. The representation of S 4 by 4 4 permutation matrices is a direct sum of the unit representation and another irreducible representation ρ 3. See Exercise (5). The permutation representation has character (4, 2, 0, 1, 0) by Exercise (4) so subtracting χ 1 from this gives us the character χ 3 of ρ 3. Tensoring ρ 3 with the sign representation gives another representation ρ 4 with character χ 4. We can easily check that χ 4,χ 4 = 1, so ρ 4 is irreducible. We will now construct one more irreducible representation of S 4. Let A be the collection of 2-element subsets of {1, 2, 3, 4}. The group S 4 acts on A naturally by σ {a,b} = {σ(a),σ(b)} for σ S 4 and {a,b} A. By Exercise (4), we see that the character of the corresponding representation is φ = (6, 2, 2, 0, 0). We can compute that φ,χ 1 = 1, φ,χ 2 = 0, φ,χ 3 = 1, and φ,χ 4 = 0, so removing the unit representation and ρ 3 from this representation, we get a representation with character χ 5, which is seen to be irreducible because χ 5,χ 5 = Schur s Lemma. In this section, we will use Schur s Lemma to prove the orthogonality relation of characters. As before let G be a finite group. Let ρ 1 : G GL(V 1 ) and ρ 2 : G GL(V 2 ) be representations of G. A linear transformation f : V 1 V 2 is called G-invariant if f ρ 1 g = ρ 2 g f for every g G. A G-invariant linear transformation is also called a homomorphism of representations. Theorem 24. (1) If ρ 1 and ρ 2 are irreducible, then f is either an isomorphism or zero.

9 (2) If V = V and ρ 1 = ρ 2, then f is a scalar multiple of the identity map. Proof. If f is G-invariant, then the kernel ker(f) is a subrepresentation of V 1 and the image im(f) is a subrepresentation of V 2. See Exercise 1. Since V 1 and V 2 are irreducible, it follows that f is either 0 or an isomorphism. Suppose V 1 = V 2 = V and ρ 1 = ρ 2 = ρ. By the fundamental theorem of algebra, the characteristic polynomial of f has a root in C, so f has an eigenvalue λ in C. Let f = f λi where I : V V is the identity map. Then f is G-invariant, and ker(f ) {0}. Thus ker(f ) = V, so f = 0 and f = λi. Lemma 25. Let f : V 1 V 2 be any linear transformation and let f 0 = 1 h G (ρ2 h ) 1 fρ 1 h. Then f0 : V 1 V 2 is a G-invariant linear transformation. The proof is straightforward and left as an exercise. Corollary 26. (1) If ρ 1 and ρ 2 are non-isomorphic irreducible representations, then f 0 = 0. (2) If V 1 = V 2 and ρ 1 = ρ 2, then f 0 is λi where λ = 1 Tr(f). dim(v ) Proof. The first statement follows immediately from Schur s Lemma. In the second case, by Schur s Lemma, f 0 is λi where λ is an eigenvalue of f 0, so λ = Tr(f 0 )/ dim(v ). However Tr(f 0 ) = 1 Tr((ρ 1 h) 1 fρ 1 h) = Tr(f). h G We will interpret the previous corollary in terms of matrices. Corollary 27. (1) If R 1 and R 2 are non-isomorphic irreducible matrix representations, then for every i,j,k,l, 1 R 1 g 1 ij R2 g kl = 0 g G (2) If R is an irreducible matrix representation of dimension n, then 1 { 1/n if i = l and j = k R g 1 ij R gkl = 0 otherwise g G Proof. Let f denote the matrix with 1 in (j,k) entry and 0 everywhere else. The statements follow from applying Corollary 26 to this f and looking at the (i,l) entry of the matrix for f 0, using the matrix multiplication rule: (ABC) il = j,k A ijb jk C kl. 9

10 10 REPRESENTATION THEORY NOTES FOR MATH 4108 The orthogonality relations now follow from the previous corollary. Proof of Theorem 18. Let χ 1 and χ 2 be characters of non-isomorphic irreducible representations R 1 and R 2 respectively. Then using the first part of previous corollary with i = j and k = l, we get χ 1,χ 2 = 1 R 1 g 1 ii R2 g kk = 0. i,k g G If χ is the character of an n-dimensional irreducible representation R, then applying the second part of the previous corollary with i = j and k = l gives χ,χ = 1 i,k n 1 R g 1 ii R gkk = g G 1 i=k n 1 n = Regular Representation. A finite group G acts on itself by left multiplication. The representation associated to this action is called the regular representation of G and will be denoted ρ reg. More concretely, let V be the vector space with basis {e g : g G}. For g G, the linear transformation ρ reg g : V V is by its action on the basis: e h e gh. With this chosen basis, ρ reg is represented by some permutation matrices. Lemma 28. Let χ reg be the character of the regular representation. Then (1) χ reg (1) = (2) χ reg (g) = 0 for all g 1. Proof. The character value of g is the number of fixed points g. The identity element fixes every element, while a non-identity element fixes none, under the action of left-multiplication. Corollary 29. Let ρ 1,ρ 2...,ρ r be irreducible representations of G with dimensions d 1,d 2,...,d r respectively. Let χ 1,χ 2,...,χ r be the characters. Then χ reg = d 1 χ 1 + d 2 χ d r χ r. Hence (1) ρ reg = d1 ρ 1 d 2 ρ 2 d r ρ r where dρ denotes the direct sum of ρ with itself d times. In other words, each d-dimensional irreducible representation is contained in the regular representation d times.

11 Proof. From the lemma above, we see that χ reg,χ i = d i. Corollary 30. If d 1,d 2,...,d r are dimensions of irreducible representations of G, then d d d 2 r =. Proof. Evaluate both sides of equation (1) at 1 G. Corollary 31. If g 1 in G, then then d 1 χ 1 (g) + d 2 χ 2 (g) + + d r χ r (g) = 0. Proof. Evaluate both sides of equation (1) at g G. We saw in the previous section that the irreducible characters are orthonomal in the space of class functions. We will now show that they in fact form an orthonomal basis for the class functions. Theorem 32. The irreducible characters of a finite group G form an orthonormal basis for the space of class functions on G. Proof. By Theorem 18, the irreducible characters are orthonormal. We are left to show that they span the space of class functions. Let f : G C be a class function, and suppose f,χ = 0 for all irreducible character χ. We will show that f = 0. For any representation ρ : G GL(V ), let ρ f : V V be a linear transformation ρ f = g G f(g)ρ g. It is easy to see that ρ g is G-invariant. If ρ is irreducible with character χ, then by Schur s Lemma, ρ f = 0 or ρ f = λi where λ = Tr(ρ f )/ dim(v ). However, Tr(ρ f ) = g G f(g) Tr(ρ g ) = f,χ = 0 since f was assumed to be orthogonal to the irreducible characters. Hence λ = 0, and ρ f must be 0. Since every representation is a direct sum of irreducible representations, it follows that ρ f = 0 for every representation. Now consider the regular representation ρ reg : G V as above, where V is a vecor space with basis {e g : g G} and e g = ρ reg g 11 (e 1 ). Then ρ reg f : V V is the (e 1 ) = 0. Since the set {e g : g G} is zero map, so g G f(g)e g = ρ reg f linearly independent, we conclude that f(g) = 0 for every g G. 3. Representations of symmetric groups Coming soon... mainly based on Fulton [3].

12 12 REPRESENTATION THEORY NOTES FOR MATH 4108 Exercises Let G be a finite group. (1) Let ρ 1 : G GL(V 1 ) and ρ 2 : G GL(V 2 ) be representations, and let f : V 1 V 2 be a G-invariant linear transformation, i.e. f ρ 1 g = ρ 2 g f for every g G. (a) Show that the kernel of f is a subrepresentation of V 1. (b) Show that the image of f is a subrepresentation of V 2. (c) Show the first part of Schur s Lemma: if ρ 1 and ρ 2 are irreducible, then f is either an isomorphism or 0. (2) (a) Let R : G GL n (C) be a representation. Show that det(r) is a one-dimensional representation. (b) Show that the map S n GL 1 (C) given by σ sign(σ) is a one-dimensional representation. (3) Show that there exists a finite-dimensional faithful representation of G. (4) Let A be a finite set on which G acts, let ρ be the corresponding permutation representation, and let χ be its character. (a) Show that for each g G, χ(g) is the number of elements of A fixed by g. (b) Show that the number of times ρ contains the unit representation is the number of orbits of G. (5) Show that the representation of the symmetric group S n by the n n permutation matrices is the direct sum of a unit representation and an irreducible representation. (6) Let R : G GL n (C) be a matrix representation with character χ. Find a representation R : G GL n (C) with character χ such that χ (g) = χ(g) for all g G. (7) Prove that G is abelian if and only if all irreducible representations are 1-dimensional. (8) (a) Prove that the characters of one-dimensional representations of G forms a group under multiplication, called the character group. (b) (extra credit) Suppose G is abelian. In this case, the character group is called the dual of G and denoted Ĝ. Prove that G is isomorphic to Ĝ, the dual of the dual of G. (9) (a) Find all irreducible representations of C n, the cyclic group of order n. (b) Is the dual Ĉn cyclic? (c) Verify orthogonality relations of characters explicitly.

13 References [1] Michael Artin, Algebra. Prentice Hall, Inc., Englewood Cliffs, NJ, [2] David S. Dummit and Richard M. Foote, Abstract algebra. Third edition. John Wiley & Sons, Inc., Hoboken, NJ, 2004 [3] William Fulton, Young tableaux. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, [4] Jean-Pierre Serre, Linear representations of finite groups. Translated from French by Leonard L. Scott. Graduate Texts in Mathematics, Vol. 42. Springer- Verlag, New York Heidelberg,

INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS

INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS HANMING ZHANG Abstract. In this paper, we will first build up a background for representation theory. We will then discuss some interesting topics in

More information

Representation Theory. Ricky Roy Math 434 University of Puget Sound

Representation Theory. Ricky Roy Math 434 University of Puget Sound Representation Theory Ricky Roy Math 434 University of Puget Sound May 2, 2010 Introduction In our study of group theory, we set out to classify all distinct groups of a given order up to isomorphism.

More information

is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent

is an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent Lecture 4. G-Modules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of G-modules, mostly for finite groups, and a recipe for finding every irreducible G-module of a

More information

MAT 445/ INTRODUCTION TO REPRESENTATION THEORY

MAT 445/ INTRODUCTION TO REPRESENTATION THEORY MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations

More information

Representation Theory

Representation Theory Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim

More information

PMATH 745: REPRESENTATION THEORY OF FINITE GROUPS

PMATH 745: REPRESENTATION THEORY OF FINITE GROUPS PMATH 745: REPRESENTATION THEORY OF FINITE GROUPS HEESUNG YANG 1. September 8: Review and quick definitions In this section we review classical linear algebra and introduce the notion of representation.

More information

18.702: Quiz 1 Solutions

18.702: Quiz 1 Solutions MIT MATHEMATICS 18.702: Quiz 1 Solutions February 28 2018 There are four problems on this quiz worth equal value. You may quote without proof any result stated in class or in the assigned reading, unless

More information

REPRESENTATION THEORY OF S n

REPRESENTATION THEORY OF S n REPRESENTATION THEORY OF S n EVAN JENKINS Abstract. These are notes from three lectures given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in November

More information

M3/4/5P12 GROUP REPRESENTATION THEORY

M3/4/5P12 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 information

Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =

Since G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg = Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for

More information

Math 594. Solutions 5

Math 594. Solutions 5 Math 594. Solutions 5 Book problems 6.1: 7. Prove that subgroups and quotient groups of nilpotent groups are nilpotent (your proof should work for infinite groups). Give an example of a group G which possesses

More information

Finite Group Representations. for the. Pure Mathematician

Finite Group Representations. for the. Pure Mathematician Finite Group Representations for the Pure Mathematician by Peter Webb Preface This book started as notes for courses given at the graduate level at the University of Minnesota. It is intended to be used

More information

REPRESENTATION THEORY FOR FINITE GROUPS

REPRESENTATION 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 information

Algebra Exam Topics. Updated August 2017

Algebra Exam Topics. Updated August 2017 Algebra Exam Topics Updated August 2017 Starting Fall 2017, the Masters Algebra Exam will have 14 questions. Of these students will answer the first 8 questions from Topics 1, 2, and 3. They then have

More information

The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph)

The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph) The Gelfand-Tsetlin Basis (Too Many Direct Sums, and Also a Graph) David Grabovsky June 13, 2018 Abstract The symmetric groups S n, consisting of all permutations on a set of n elements, naturally contain

More information

Representations. 1 Basic definitions

Representations. 1 Basic definitions Representations 1 Basic definitions If V is a k-vector space, we denote by Aut V the group of k-linear isomorphisms F : V V and by End V the k-vector space of k-linear maps F : V V. Thus, if V = k n, then

More information

Representation Theory

Representation Theory Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section II 19I 93 (a) Define the derived subgroup, G, of a finite group G. Show that if χ is a linear character

More information

A basic note on group representations and Schur s lemma

A basic note on group representations and Schur s lemma A basic note on group representations and Schur s lemma Alen Alexanderian Abstract Here we look at some basic results from group representation theory. Moreover, we discuss Schur s Lemma in the context

More information

(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for

(d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for Solutions to Homework #7 0. Prove that [S n, S n ] = A n for every n 2 (where A n is the alternating group). Solution: Since [f, g] = f 1 g 1 fg is an even permutation for all f, g S n and since A n is

More information

Math 210C. The representation ring

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 information

CHARACTERS OF FINITE GROUPS.

CHARACTERS OF FINITE GROUPS. CHARACTERS OF FINITE GROUPS. ANDREI YAFAEV As usual we consider a finite group G and the ground field F = C. Let U be a C[G]-module and let g G. Then g is represented by a matrix [g] in a certain basis.

More information

A REPRESENTATION THEORETIC APPROACH TO SYNCHRONIZING AUTOMATA

A REPRESENTATION THEORETIC APPROACH TO SYNCHRONIZING AUTOMATA A REPRESENTATION THEORETIC APPROACH TO SYNCHRONIZING AUTOMATA FREDRICK ARNOLD AND BENJAMIN STEINBERG Abstract. This paper is a first attempt to apply the techniques of representation theory to synchronizing

More information

3 Representations of finite groups: basic results

3 Representations of finite groups: basic results 3 Representations of finite groups: basic results Recall that a representation of a group G over a field k is a k-vector space V together with a group homomorphism δ : G GL(V ). As we have explained above,

More information

Math 250: Higher Algebra Representations of finite groups

Math 250: Higher Algebra Representations of finite groups Math 250: Higher Algebra Representations of finite groups 1 Basic definitions Representations. A representation of a group G over a field k is a k-vector space V together with an action of G on V by linear

More information

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

More information

A PROOF OF BURNSIDE S p a q b THEOREM

A PROOF OF BURNSIDE S p a q b THEOREM A PROOF OF BURNSIDE S p a q b THEOREM OBOB Abstract. We prove that if p and q are prime, then any group of order p a q b is solvable. Throughout this note, denote by A the set of algebraic numbers. We

More information

REPRESENTATIONS AND CHARACTERS OF FINITE GROUPS

REPRESENTATIONS AND CHARACTERS OF FINITE GROUPS SUMMER PROJECT REPRESENTATIONS AND CHARACTERS OF FINITE GROUPS September 29, 2017 Miriam Norris School of Mathematics Contents 0.1 Introduction........................................ 2 0.2 Representations

More information

Group Representation Theory

Group Representation Theory Group Representation Theory Ed Segal based on notes latexed by Fatema Daya and Zach Smith 2014 This course will cover the representation theory of finite groups over C. We assume the reader knows the basic

More information

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

More information

Real representations

Real representations Real representations 1 Definition of a real representation Definition 1.1. Let V R be a finite dimensional real vector space. A real representation of a group G is a homomorphism ρ VR : G Aut V R, where

More information

Weeks 6 and 7. November 24, 2013

Weeks 6 and 7. November 24, 2013 Weeks 6 and 7 November 4, 03 We start by calculating the irreducible representation of S 4 over C. S 4 has 5 congruency classes: {, (, ), (,, 3), (, )(3, 4), (,, 3, 4)}, so we have 5 irreducible representations.

More information

SCHUR-WEYL DUALITY FOR U(n)

SCHUR-WEYL DUALITY FOR U(n) SCHUR-WEYL DUALITY FOR U(n) EVAN JENKINS Abstract. These are notes from a lecture given in MATH 26700, Introduction to Representation Theory of Finite Groups, at the University of Chicago in December 2009.

More information

SOME ELEMENTARY RESULTS IN REPRESENTATION THEORY

SOME ELEMENTARY RESULTS IN REPRESENTATION THEORY SOME ELEMENTARY RESULTS IN REPRESENTATION THEORY ISAAC OTTONI WILHELM Abstract. This paper will prove that given a finite group G, the associated irreducible characters form an orthonormal basis for the

More information

Boolean Inner-Product Spaces and Boolean Matrices

Boolean Inner-Product Spaces and Boolean Matrices Boolean Inner-Product Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver

More information

Math 306 Topics in Algebra, Spring 2013 Homework 7 Solutions

Math 306 Topics in Algebra, Spring 2013 Homework 7 Solutions Math 306 Topics in Algebra, Spring 203 Homework 7 Solutions () (5 pts) Let G be a finite group. Show that the function defines an inner product on C[G]. We have Also Lastly, we have C[G] C[G] C c f + c

More information

Bibliography. Groups and Fields. Matrix Theory. Determinants

Bibliography. Groups and Fields. Matrix Theory. Determinants Bibliography Groups and Fields Alperin, J. L.; Bell, Rowen B. Groups and representations. Graduate Texts in Mathematics, 162. Springer-Verlag, New York, 1995. Artin, Michael Algebra. Prentice Hall, Inc.,

More information

Linear Algebra 2 Spectral Notes

Linear Algebra 2 Spectral Notes Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex

More information

HOMEWORK Graduate Abstract Algebra I May 2, 2004

HOMEWORK Graduate Abstract Algebra I May 2, 2004 Math 5331 Sec 121 Spring 2004, UT Arlington HOMEWORK Graduate Abstract Algebra I May 2, 2004 The required text is Algebra, by Thomas W. Hungerford, Graduate Texts in Mathematics, Vol 73, Springer. (it

More information

Spectra of Semidirect Products of Cyclic Groups

Spectra of Semidirect Products of Cyclic Groups Spectra of Semidirect Products of Cyclic Groups Nathan Fox 1 University of Minnesota-Twin Cities Abstract The spectrum of a graph is the set of eigenvalues of its adjacency matrix A group, together with

More information

REPRESENTATIONS OF S n AND GL(n, C)

REPRESENTATIONS 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 information

Solutions of exercise sheet 8

Solutions of exercise sheet 8 D-MATH 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 information

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers

Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop. Eric Sommers Notes on nilpotent orbits Computational Theory of Real Reductive Groups Workshop Eric Sommers 17 July 2009 2 Contents 1 Background 5 1.1 Linear algebra......................................... 5 1.1.1

More information

LECTURE NOTES AMRITANSHU PRASAD

LECTURE NOTES AMRITANSHU PRASAD LECTURE NOTES AMRITANSHU PRASAD Let K be a field. 1. Basic definitions Definition 1.1. A K-algebra is a K-vector space together with an associative product A A A which is K-linear, with respect to which

More information

4 Group representations

4 Group representations Physics 9b Lecture 6 Caltech, /4/9 4 Group representations 4. Examples Example : D represented as real matrices. ( ( D(e =, D(c = ( ( D(b =, D(b =, D(c = Example : Circle group as rotation of D real vector

More information

Supplementary Notes March 23, The subgroup Ω for orthogonal groups

Supplementary Notes March 23, The subgroup Ω for orthogonal groups The subgroup Ω for orthogonal groups 18.704 Supplementary Notes March 23, 2005 In the case of the linear group, it is shown in the text that P SL(n, F ) (that is, the group SL(n) of determinant one matrices,

More information

REPRESENTATION THEORY WEEK 7

REPRESENTATION THEORY WEEK 7 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

More information

Parameterizing orbits in flag varieties

Parameterizing orbits in flag varieties Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.

More information

Math 108b: Notes on the Spectral Theorem

Math 108b: Notes on the Spectral Theorem Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator

More information

LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS

LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in

More information

TENSOR PRODUCTS, RESTRICTION AND INDUCTION.

TENSOR PRODUCTS, RESTRICTION AND INDUCTION. TENSOR PRODUCTS, RESTRICTION AND INDUCTION. ANDREI YAFAEV Our first aim in this chapter is to give meaning to the notion of product of characters. Let V and W be two finite dimensional vector spaces over

More information

Artin s and Brauer s Theorems on Induced. Characters

Artin s and Brauer s Theorems on Induced. Characters Artin s and Brauer s Theorems on Induced Characters János Kramár December 14, 2005 1 Preliminaries Let G be a finite group. Every representation of G defines a unique left C[G]- module where C[G] is the

More information

INTRODUCTION TO COMMUTATIVE ALGEBRA MAT6608. References

INTRODUCTION TO COMMUTATIVE ALGEBRA MAT6608. References INTRODUCTION TO COMMUTATIVE ALGEBRA MAT6608 ABRAHAM BROER References [1] Atiyah, M. F.; Macdonald, I. G. Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills,

More information

Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem

Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem Topics in Representation Theory: Fourier Analysis and the Peter Weyl Theorem 1 Fourier Analysis, a review We ll begin with a short review of simple facts about Fourier analysis, before going on to interpret

More information

Radon Transforms and the Finite General Linear Groups

Radon Transforms and the Finite General Linear Groups Claremont Colleges Scholarship @ Claremont All HMC Faculty Publications and Research HMC Faculty Scholarship 1-1-2004 Radon Transforms and the Finite General Linear Groups Michael E. Orrison Harvey Mudd

More information

Categories and Quantum Informatics: Hilbert spaces

Categories and Quantum Informatics: Hilbert spaces Categories and Quantum Informatics: Hilbert spaces Chris Heunen Spring 2018 We introduce our main example category Hilb by recalling in some detail the mathematical formalism that underlies quantum theory:

More information

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.

Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces. Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,

More information

Elementary linear algebra

Elementary linear algebra Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The

More information

Topics in Representation Theory: Roots and Weights

Topics in Representation Theory: Roots and Weights Topics in Representation Theory: Roots and Weights 1 The Representation Ring Last time we defined the maximal torus T and Weyl group W (G, T ) for a compact, connected Lie group G and explained that our

More information

THE REPRESENTATIONS OF THE SYMMETRIC GROUP. Contents

THE REPRESENTATIONS OF THE SYMMETRIC GROUP. Contents THE REPRESENTATIONS OF THE SYMMETRIC GROUP JE-OK CHOI Abstract. Young tableau is a combinatorial object which provides a convenient way to describe the group representations of the symmetric group, S n.

More information

Character tables for some small groups

Character tables for some small groups Character tables for some small groups P R Hewitt U of Toledo 12 Feb 07 References: 1. P Neumann, On a lemma which is not Burnside s, Mathematical Scientist 4 (1979), 133-141. 2. JH Conway et al., Atlas

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE 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 information

Some notes on linear algebra

Some notes on linear algebra Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and, that

More information

FFTs in Graphics and Vision. Groups and Representations

FFTs in Graphics and Vision. Groups and Representations FFTs in Graphics and Vision Groups and Representations Outline Groups Representations Schur s Lemma Correlation Groups A group is a set of elements G with a binary operation (often denoted ) such that

More information

Groups and Representations

Groups and Representations Groups and Representations Madeleine Whybrow Imperial College London These notes are based on the course Groups and Representations taught by Prof. A.A. Ivanov at Imperial College London during the Autumn

More information

MIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???

MIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:??? MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present

More information

IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents

IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS. Contents IRREDUCIBLE REPRESENTATIONS OF SEMISIMPLE LIE ALGEBRAS NEEL PATEL Abstract. The goal of this paper is to study the irreducible representations of semisimple Lie algebras. We will begin by considering two

More information

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM

LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator

More information

Homework set 5 - Solutions

Homework set 5 - Solutions Homework set 5 - Solutions Math 469 Renato Feres 1. Hall s textbook, Exercise 4.9.2, page 105. Show the following: (a) The adjoint representation and the standard representation are isomorphic representations

More information

Algebra Exam Syllabus

Algebra 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 information

Real, Complex, and Quarternionic Representations

Real, Complex, and Quarternionic Representations Real, Complex, and Quarternionic Representations JWR 10 March 2006 1 Group Representations 1. Throughout R denotes the real numbers, C denotes the complex numbers, H denotes the quaternions, and G denotes

More information

Topics in linear algebra

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

More information

REPRESENTATION THEORY WEEK 5. B : V V k

REPRESENTATION 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 information

Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

More information

REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 2001

REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 2001 9 REPRESENTATIONS OF U(N) CLASSIFICATION BY HIGHEST WEIGHTS NOTES FOR MATH 261, FALL 21 ALLEN KNUTSON 1 WEIGHT DIAGRAMS OF -REPRESENTATIONS Let be an -dimensional torus, ie a group isomorphic to The we

More information

Additional notes on group representations Hjalmar Rosengren, 30 September 2015

Additional notes on group representations Hjalmar Rosengren, 30 September 2015 Additional notes on group representations Hjalmar Rosengren, 30 September 2015 Throughout, group means finite group and representation means finitedimensional representation over C. Interpreting the character

More information

M3/4/5P12 PROBLEM SHEET 1

M3/4/5P12 PROBLEM SHEET 1 M3/4/5P12 PROBLEM SHEET 1 Please send any corrections or queries to jnewton@imperialacuk Exercise 1 (1) Let G C 4 C 2 s, t : s 4 t 2 e, st ts Let V C 2 with the stard basis Consider the linear transformations

More information

Problems in Linear Algebra and Representation Theory

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

More information

Algebra SEP Solutions

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

More information

Introduction to representation theory of finite groups

Introduction to representation theory of finite groups Introduction to representation theory of finite groups Alex Bartel 9th February 2017 Contents 1 Group representations the first encounter 2 1.1 Historical introduction........................ 2 1.2 First

More information

Exercises on chapter 1

Exercises on chapter 1 Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

More information

Representation theory and quantum mechanics tutorial Spin and the hydrogen atom

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

More information

ALGEBRA QUALIFYING EXAM PROBLEMS

ALGEBRA QUALIFYING EXAM PROBLEMS ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General

More information

A linear algebra proof of the fundamental theorem of algebra

A 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 information

0.2 Vector spaces. J.A.Beachy 1

0.2 Vector spaces. J.A.Beachy 1 J.A.Beachy 1 0.2 Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a

More information

A PRIMER ON SESQUILINEAR FORMS

A PRIMER ON SESQUILINEAR FORMS A PRIMER ON SESQUILINEAR FORMS BRIAN OSSERMAN This is an alternative presentation of most of the material from 8., 8.2, 8.3, 8.4, 8.5 and 8.8 of Artin s book. Any terminology (such as sesquilinear form

More information

A linear algebra proof of the fundamental theorem of algebra

A 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 information

NOTES FOR 128: COMBINATORIAL REPRESENTATION THEORY OF COMPLEX LIE ALGEBRAS AND RELATED TOPICS

NOTES FOR 128: COMBINATORIAL REPRESENTATION THEORY OF COMPLEX LIE ALGEBRAS AND RELATED TOPICS NOTES FOR 128: COMBINATORIAL REPRESENTATION THEORY OF COMPLEX LIE ALGEBRAS AND RELATED TOPICS (FIRST COUPLE LECTURES MORE ONLINE AS WE GO) Recommended reading [Bou] N. Bourbaki, Elements of Mathematics:

More information

MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003

MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space

More information

INVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction

INVARIANT PROBABILITIES ON PROJECTIVE SPACES. 1. Introduction INVARIANT PROBABILITIES ON PROJECTIVE SPACES YVES DE CORNULIER Abstract. Let K be a local field. We classify the linear groups G GL(V ) that preserve an probability on the Borel subsets of the projective

More information

1 Invariant subspaces

1 Invariant subspaces MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another

More information

October 25, 2013 INNER PRODUCT SPACES

October 25, 2013 INNER PRODUCT SPACES October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal

More information

Math 121 Homework 5: Notes on Selected Problems

Math 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 information

Linear Algebra. Workbook

Linear 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 information

(Refer Slide Time: 2:04)

(Refer Slide Time: 2:04) Linear Algebra By Professor K. C. Sivakumar Department of Mathematics Indian Institute of Technology, Madras Module 1 Lecture 1 Introduction to the Course Contents Good morning, let me welcome you to this

More information

Generalized eigenspaces

Generalized eigenspaces Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction

More information

Math 291-2: Lecture Notes Northwestern University, Winter 2016

Math 291-2: Lecture Notes Northwestern University, Winter 2016 Math 291-2: Lecture Notes Northwestern University, Winter 2016 Written by Santiago Cañez These are lecture notes for Math 291-2, the second quarter of MENU: Intensive Linear Algebra and Multivariable Calculus,

More information

implies that if we fix a basis v of V and let M and M be the associated invertible symmetric matrices computing, and, then M = (L L)M and the

implies that if we fix a basis v of V and let M and M be the associated invertible symmetric matrices computing, and, then M = (L L)M and the Math 395. Geometric approach to signature For the amusement of the reader who knows a tiny bit about groups (enough to know the meaning of a transitive group action on a set), we now provide an alternative

More information

A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

More information

NONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction

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

More information

YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP

YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP YOUNG TABLEAUX AND THE REPRESENTATIONS OF THE SYMMETRIC GROUP YUFEI ZHAO ABSTRACT We explore an intimate connection between Young tableaux and representations of the symmetric group We describe the construction

More information