1 Representations, Maschke's Theorem, and Semisimplicity
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REPRESENTATIONS of FINITE GROUPS 1. Definition And
REPRESENTATIONS OF FINITE GROUPS 1. Definition and Introduction 1.1. Definitions. Let V be a vector space over the field C of complex numbers and let GL(V ) be the group of isomorphisms of V onto itself. An element a of GL(V ) is a linear mapping of V into V which has a linear inverse a−1. When V has dimension n and has n a finite basis (ei)i=1, each map a : V ! V is defined by a square matrix (aij) of order n. The coefficients aij are complex numbers. They are obtained by expressing the image a(ej) in terms of the basis (ei): X a(ej) = aijei: i Remark 1.1. Saying that a is an isomorphism is equivalent to saying that the determinant det(a) = det(aij) of a is not zero. The group GL(V ) is thus identified with the group of invertible square matrices of order n. Suppose G is a finite group with identity element 1. A representation of a finite group G on a finite-dimensional complex vector space V is a homomorphism ρ : G ! GL(V ) of G to the group of automorphisms of V . We say such a map gives V the structure of a G-module. The dimension V is called the degree of ρ. Remark 1.2. When there is little ambiguity of the map ρ, we sometimes call V itself a representation of G. In this vein we often write gv_ or gv for ρ(g)v. Remark 1.3. Since ρ is a homomorphism, we have equality ρ(st) = ρ(s)ρ(t) for s; t 2 G: In particular, we have ρ(1) = 1; ; ρ(s−1) = ρ(s)−1: A map ' between two representations V and W of G is a vector space map ' : V ! W such that ' V −−−−! W ? ? g? ?g (1.1) y y V −−−−! W ' commutes for all g 2 G. -
Solution to Homework 5
Solution to Homework 5 Sec. 5.4 1 0 2. (e) No. Note that A = is in W , but 0 2 0 1 1 0 0 2 T (A) = = 1 0 0 2 1 0 is not in W . Hence, W is not a T -invariant subspace of V . 3. Note that T : V ! V is a linear operator on V . To check that W is a T -invariant subspace of V , we need to know if T (w) 2 W for any w 2 W . (a) Since we have T (0) = 0 2 f0g and T (v) 2 V; so both of f0g and V to be T -invariant subspaces of V . (b) Note that 0 2 N(T ). For any u 2 N(T ), we have T (u) = 0 2 N(T ): Hence, N(T ) is a T -invariant subspace of V . For any v 2 R(T ), as R(T ) ⊂ V , we have v 2 V . So, by definition, T (v) 2 R(T ): Hence, R(T ) is also a T -invariant subspace of V . (c) Note that for any v 2 Eλ, λv is a scalar multiple of v, so λv 2 Eλ as Eλ is a subspace. So we have T (v) = λv 2 Eλ: Hence, Eλ is a T -invariant subspace of V . 4. For any w in W , we know that T (w) is in W as W is a T -invariant subspace of V . Then, by induction, we know that T k(w) is also in W for any k. k Suppose g(T ) = akT + ··· + a1T + a0, we have k g(T )(w) = akT (w) + ··· + a1T (w) + a0(w) 2 W because it is just a linear combination of elements in W . -
Spectral Coupling for Hermitian Matrices
Spectral coupling for Hermitian matrices Franc¸oise Chatelin and M. Monserrat Rincon-Camacho Technical Report TR/PA/16/240 Publications of the Parallel Algorithms Team http://www.cerfacs.fr/algor/publications/ SPECTRAL COUPLING FOR HERMITIAN MATRICES FRANC¸OISE CHATELIN (1);(2) AND M. MONSERRAT RINCON-CAMACHO (1) Cerfacs Tech. Rep. TR/PA/16/240 Abstract. The report presents the information processing that can be performed by a general hermitian matrix when two of its distinct eigenvalues are coupled, such as λ < λ0, instead of λ+λ0 considering only one eigenvalue as traditional spectral theory does. Setting a = 2 = 0 and λ0 λ 6 e = 2− > 0, the information is delivered in geometric form, both metric and trigonometric, associated with various right-angled triangles exhibiting optimality properties quantified as ratios or product of a and e. The potential optimisation has a triple nature which offers two j j e possibilities: in the case λλ0 > 0 they are characterised by a and a e and in the case λλ0 < 0 a j j j j by j j and a e. This nature is revealed by a key generalisation to indefinite matrices over R or e j j C of Gustafson's operator trigonometry. Keywords: Spectral coupling, indefinite symmetric or hermitian matrix, spectral plane, invariant plane, catchvector, antieigenvector, midvector, local optimisation, Euler equation, balance equation, torus in 3D, angle between complex lines. 1. Spectral coupling 1.1. Introduction. In the work we present below, we focus our attention on the coupling of any two distinct real eigenvalues λ < λ0 of a general hermitian or symmetric matrix A, a coupling called spectral coupling. -
Groups for Which It Is Easy to Detect Graphical Regular Representations
CC Creative Commons ADAM Logo Also available at http://adam-journal.eu The Art of Discrete and Applied Mathematics nn (year) 1–x Groups for which it is easy to detect graphical regular representations Dave Witte Morris , Joy Morris Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta. T1K 3M4, Canada Gabriel Verret Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Received day month year, accepted xx xx xx, published online xx xx xx Abstract We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G; S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism ' of G such that '(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product Zp o Zp is not DRR-detecting, for every odd prime p. We also show that if G1 and G2 are nontrivial groups that admit a digraphical regular representation and either gcd jG1j; jG2j = 1, or G2 is not DRR-detecting, then the direct product G1 × G2 is not DRR-detecting. Some of these results also have analogues for graphical regular representations. Keywords: Cayley graph, GRR, DRR, automorphism group, normalizer Math. Subj. Class.: 05C25,20B05 1 Introduction All groups and graphs in this paper are finite. Recall [1] that a digraph Γ is said to be a digraphical regular representation (DRR) of a group G if the automorphism group of Γ is isomorphic to G and acts regularly on the vertex set of Γ. -
Section 18.1-2. in the Next 2-3 Lectures We Will Have a Lightning Introduction to Representations of finite Groups
Section 18.1-2. In the next 2-3 lectures we will have a lightning introduction to representations of finite groups. For any vector space V over a field F , denote by L(V ) the algebra of all linear operators on V , and by GL(V ) the group of invertible linear operators. Note that if dim(V ) = n, then L(V ) = Matn(F ) is isomorphic to the algebra of n×n matrices over F , and GL(V ) = GLn(F ) to its multiplicative group. Definition 1. A linear representation of a set X in a vector space V is a map φ: X ! L(V ), where L(V ) is the set of linear operators on V , the space V is called the space of the rep- resentation. We will often denote the representation of X by (V; φ) or simply by V if the homomorphism φ is clear from the context. If X has any additional structures, we require that the map φ is a homomorphism. For example, a linear representation of a group G is a homomorphism φ: G ! GL(V ). Definition 2. A morphism of representations φ: X ! L(V ) and : X ! L(U) is a linear map T : V ! U, such that (x) ◦ T = T ◦ φ(x) for all x 2 X. In other words, T makes the following diagram commutative φ(x) V / V T T (x) U / U An invertible morphism of two representation is called an isomorphism, and two representa- tions are called isomorphic (or equivalent) if there exists an isomorphism between them. Example. (1) A representation of a one-element set in a vector space V is simply a linear operator on V . -
The Conjugate Dimension of Algebraic Numbers
THE CONJUGATE DIMENSION OF ALGEBRAIC NUMBERS NEIL BERRY, ARTURAS¯ DUBICKAS, NOAM D. ELKIES, BJORN POONEN, AND CHRIS SMYTH Abstract. We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q-dimension of the space spanned by its conjugates. For all but seven nonnegative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(!`) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number. 1. Introduction Let Q be an algebraic closure of the field Q of rational numbers, and let α 2 Q. Let α1; : : : ; αd 2 Q be the conjugates of α over Q, with α1 = α. Then d equals the degree d(α) := [Q(α): Q], the dimension of the Q-vector space spanned by the powers of α. In contrast, we define the conjugate dimension n = n(α) of α as the dimension of the Q-vector space spanned by fα1; : : : ; αdg. -
Notes on Representations of Finite Groups
NOTES ON REPRESENTATIONS OF FINITE GROUPS AARON LANDESMAN CONTENTS 1. Introduction 3 1.1. Acknowledgements 3 1.2. A first definition 3 1.3. Examples 4 1.4. Characters 7 1.5. Character Tables and strange coincidences 8 2. Basic Properties of Representations 11 2.1. Irreducible representations 12 2.2. Direct sums 14 3. Desiderata and problems 16 3.1. Desiderata 16 3.2. Applications 17 3.3. Dihedral Groups 17 3.4. The Quaternion group 18 3.5. Representations of A4 18 3.6. Representations of S4 19 3.7. Representations of A5 19 3.8. Groups of order p3 20 3.9. Further Challenge exercises 22 4. Complete Reducibility of Complex Representations 24 5. Schur’s Lemma 30 6. Isotypic Decomposition 32 6.1. Proving uniqueness of isotypic decomposition 32 7. Homs and duals and tensors, Oh My! 35 7.1. Homs of representations 35 7.2. Duals of representations 35 7.3. Tensors of representations 36 7.4. Relations among dual, tensor, and hom 38 8. Orthogonality of Characters 41 8.1. Reducing Theorem 8.1 to Proposition 8.6 41 8.2. Projection operators 43 1 2 AARON LANDESMAN 8.3. Proving Proposition 8.6 44 9. Orthogonality of character tables 46 10. The Sum of Squares Formula 48 10.1. The inner product on characters 48 10.2. The Regular Representation 50 11. The number of irreducible representations 52 11.1. Proving characters are independent 53 11.2. Proving characters form a basis for class functions 54 12. Dimensions of Irreps divide the order of the Group 57 Appendix A. -
Adiabatic Invariants of the Linear Hamiltonian Systems with Periodic Coefficients
JOURNAL OF DIFFERENTIAL EQUATIONS 42, 47-71 (1981) Adiabatic Invariants of the Linear Hamiltonian Systems with Periodic Coefficients MARK LEVI Department of Mathematics, Duke University, Durham, North Carolina 27706 Received November 19, 1980 0. HISTORICAL REMARKS In 1911 Lorentz and Einstein had offered an explanation of the fact that the ratio of energy to the frequency of radiation of an atom remains constant. During the long time intervals separating two quantum jumps an atom is exposed to varying surrounding electromagnetic fields which should supposedly change the ratio. Their explanation was based on the fact that the surrounding field varies extremely slowly with respect to the frequency of oscillations of the atom. The idea can be illstrated by a slightly simpler model-a pendulum (instead of Bohr’s model of an atom) with a slowly changing length (slowly with respect to the frequency of oscillations of the pendulum). As it turns out, the ratio of energy of the pendulum to its frequency changes very little if its length varies slowly enough from one constant value to another; the pendulum “remembers” the ratio, and the slower the change the better the memory. I had learned this interesting historical remark from Wasow [15]. Those parameters of a system which remain nearly constant during a slow change of a system were named by the physicists the adiabatic invariants, the word “adiabatic” indicating that one parameter changes slowly with respect to another-e.g., the length of the pendulum with respect to the phaze of oscillations. Precise definition of the concept will be given in Section 1. -
The Origin of Representation Theory
THE ORIGIN OF REPRESENTATION THEORY KEITH CONRAD Abstract. Representation theory was created by Frobenius about 100 years ago. We describe the background that led to the problem which motivated Frobenius to define characters of a finite group and show how representation theory solves the problem. The first results about representation theory in characteristic p are also discussed. 1. Introduction Characters of finite abelian groups have been used since Gauss in the beginning of the 19th century, but it was only near the end of that century, in 1896, that Frobenius extended this concept to finite nonabelian groups [21]. This approach of Frobenius to group characters is not the one in common use today, although some of his ideas that were overlooked for a long period have recently been revived [30]. Here we trace the development of the problem whose solution led Frobenius to introduce characters of finite groups, show how this problem can be solved using classical representa- tion theory of finite groups, and indicate some relations between this problem and modular representations. Other surveys on the origins of representation theory are by Curtis [7], Hawkins [24, 25, 26, 27], Lam [32], and Ledermann [35]. While Curtis describes the development of modular representation theory focusing on the work of Brauer, we examine the earlier work in characteristic p of Dickson. 2. Circulants For a positive integer n, consider an n × n matrix where each row is obtained from the previous one by a cyclic shift one step to the right. That is, we look at a matrix of the form 0 1 X0 X1 X2 :::Xn−1 B Xn−1 X0 X1 :::Xn−2 C B C B . -
Admissible Vectors for the Regular Representation 2961
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 10, Pages 2959{2970 S 0002-9939(02)06433-X Article electronically published on March 12, 2002 ADMISSIBLE VECTORS FORTHEREGULARREPRESENTATION HARTMUT FUHR¨ (Communicated by David R. Larson) Abstract. It is well known that for irreducible, square-integrable representa- tions of a locally compact group, there exist so-called admissible vectors which allow the construction of generalized continuous wavelet transforms. In this paper we discuss when the irreducibility requirement can be dropped, using a connection between generalized wavelet transforms and Plancherel theory. For unimodular groups with type I regular representation, the existence of admis- sible vectors is equivalent to a finite measure condition. The main result of this paper states that this restriction disappears in the nonunimodular case: Given a nondiscrete, second countable group G with type I regular representation λG, we show that λG itself (and hence every subrepresentation thereof) has an admissible vector in the sense of wavelet theory iff G is nonunimodular. Introduction This paper deals with the group-theoretic approach to the construction of con- tinuous wavelet transforms. Shortly after the continuous wavelet transform of univariate functions had been introduced, Grossmann, Morlet and Paul estab- lished the connection to the representation theory of locally compact groups [12], which was then used by several authors to construct higher-dimensional analogues [19, 5, 4, 10, 14]. Let us roughly sketch the general group-theoretic formalism for the construction of wavelet transforms. Given a unitary representation π of the locally compact group G on the Hilbert space π, and given a vector η π (the representation H 2H space of π), we can study the mapping Vη, which maps each φ π to the bounded 2H continuous function Vηφ on G, defined by Vηφ(x):= φ, π(x)η : h i 2 Whenever this operator Vη is an isometry of π into L (G), we call it a continuous wavelet transform,andη is called a waveletH or admissible vector. -
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, a representation of a group G ! over k is the same thing as a representation of its group algebra k[G]. In this section, we begin a systematic development of representation theory of finite groups. 3.1 Maschke's Theorem Theorem 3.1. (Maschke) Let G be a finite group and k a field whose characteristic does not divide G . Then: j j (i) The algebra k[G] is semisimple. (ii) There is an isomorphism of algebras : k[G] EndV defined by g g , where V ! �i i 7! �i jVi i are the irreducible representations of G. In particular, this is an isomorphism of representations of G (where G acts on both sides by left multiplication). Hence, the regular representation k[G] decomposes into irreducibles as dim(V )V , and one has �i i i G = dim(V )2 : j j i Xi (the “sum of squares formula”). Proof. By Proposition 2.16, (i) implies (ii), and to prove (i), it is sufficient to show that if V is a finite-dimensional representation of G and W V is any subrepresentation, then there exists a → subrepresentation W V such that V = W W as representations. 0 → � 0 Choose any complement Wˆ of W in V . (Thus V = W Wˆ as vector spaces, but not necessarily � as representations.) Let P be the projection along Wˆ onto W , i.e., the operator on V defined by P W = Id and P W^ = 0. -
A Criterion for the Existence of Common Invariant Subspaces Of
Linear Algebra and its Applications 322 (2001) 51–59 www.elsevier.com/locate/laa A criterion for the existence of common invariant subspaces of matricesୋ Michael Tsatsomeros∗ Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Received 4 May 1999; accepted 22 June 2000 Submitted by R.A. Brualdi Abstract It is shown that square matrices A and B have a common invariant subspace W of di- mension k > 1 if and only if for some scalar s, A + sI and B + sI are invertible and their kth compounds have a common eigenvector, which is a Grassmann representative for W.The applicability of this criterion and its ability to yield a basis for the common invariant subspace are investigated. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 15A75; 47A15 Keywords: Invariant subspace; Compound matrix; Decomposable vector; Grassmann space 1. Introduction The lattice of invariant subspaces of a matrix is a well-studied concept with sev- eral applications. In particular, there is substantial information known about matrices having the same invariant subspaces, or having chains of common invariant subspac- es (see [7]). The question of existence of a non-trivial invariant subspace common to two or more matrices is also of interest. It arises, among other places, in the study of irreducibility of decomposable positive linear maps on operator algebras [3]. It also comes up in what might be called Burnside’s theorem, namely that if two square ୋ Research partially supported by NSERC research grant. ∗ Tel.: +306-585-4771; fax: +306-585-4020. E-mail address: [email protected] (M.