Representation Theory of Finite Groups

Video Lectures

Displaying all 18 video lectures.
Lecture 1
RT1: Representation Theory Basics
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RT1: Representation Theory Basics
Representation Theory: We present basic concepts about the representation theory of finite groups. Representations are defined, as are notions of invariant subspace, irreducibility and full reducibility. For a general finite group G, we suggest an analogue to the finite abelian case, where every representation may be represented by diagonal matrices with unitary diagonal entries.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 2
RT2: Unitary Representations
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RT2: Unitary Representations
Representation Theory: We explain unitarity and invariant inner products for representations of finite groups. Full reducibility of such representations is derived.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 3
RT3. Equivalence and Examples (Expanded)
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RT3. Equivalence and Examples (Expanded)
Representation Theory: We define equivalence of representations and give examples of irreducible representations for groups of low order. Then we use the commutator subgroup to characterize all one dimensional representations of G (characters) in terms of the abelianization of G.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 4
RT4.1. Constructions from Linear Algebra (Expanded)
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RT4.1. Constructions from Linear Algebra (Expanded)
Representation Theory: We apply techniques from linear algebra to construct new representations from old ones. Constructions include direct sums, dual spaces, tensor products, and Hom spaces.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 5
RT4.1.1: Complex Conjugate Representations
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RT4.1.1: Complex Conjugate Representations
Representation Theory: We look at the complex conjugate of a representation in more detail. We present two equivalent formulations and show the conjugate is equivalent to the dual representation when pi is unitary.
Lecture 6
RT4.2. Schur's Lemma (Expanded)
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RT4.2. Schur's Lemma (Expanded)
Representation Theory: We introduce Schur's Lemma for irreducible representations and apply it to our previous constructions. In particular, we identify Hom(V,V) with invariant sesquilinear forms on V when (pi, V) is unitary.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 7
RT5. Mostly Exercises (Expanded)
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RT5. Mostly Exercises (Expanded)
Representation Theory: We collect some loose ends and noteworthy facts on representations as a set of exercises. Topics include Schur's Lemma, full reducibility, and tensor product representations.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 8
RT6. Representations on Function Spaces
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RT6. Representations on Function Spaces
Representation Theory: We note how to transfer a group action of a group G on a set X to a group action on F(X), the functions on X. Because F(X) is a vector space, we obtain a representation of the group, and we can apply previous techniques. In particular, the group acts on itself in several ways. We give an overview of features to expect from Fourier analysis on the circle group.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 9
RT7.1: Finite Abelian Groups: Character Orthogonality
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RT7.1: Finite Abelian Groups: Character Orthogonality
We establish an analogue of Fourier analysis for a finite abelian group G. A decomposition of L^2(G) is given in terms of characters. Versions of Schur Orthogonality Relations and the Peter-Weyl Theorem are given.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 10
RT7.2. Finite Abelian Groups: Fourier Analysis
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RT7.2. Finite Abelian Groups: Fourier Analysis
Representation Theory: With orthogonality of characters, we have an orthonormal basis of L^2(G). We note the basic philosophy behind the Fourier transform and apply it to the character basis. From this comes the definition of convolution, explored in 7.3.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 11
RT7.3. Finite Abelian Groups: Convolution
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RT7.3. Finite Abelian Groups: Convolution
Representation Theory: We define convolution of two functions on L^2(G) and note general properties. Three themes: convolution as an analogue of matrix multiplication, convolution with character as an orthogonal projection on L^2(G), and using using convolution to project onto irreducible types in representations.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 12
RT8.1. Schur Orthogonality Relations
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RT8.1. Schur Orthogonality Relations
Representation Theory of Finite Groups: As a first step to Fourier analysis on finite groups, we state and prove the Schur Orthogonality Relations. With these relations, we may form an orthonormal basis of matrix coefficients for L^(G), the set of functions on G. We also define characters and state the corresponding SORs.
Lecture 13
RT8.2. Finite Groups: Classification of Irreducibles
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RT8.2. Finite Groups: Classification of Irreducibles
Representation Theory: Using the Schur orthogonality relations, we obtain an orthonormal basis of L^2(G) using matrix coefficients of irreducible representations. This shows the sum of squares of dimensions of irreducibles equals |G|. We also obtain an orthonormal basis of Class(G) using irreducible characters, and from this we see that the number of irreducible classes equals the number of conjugacy classes in G. We also obtain character formulas for multiplicities.

Course materials, including problem sets and solutions, available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 14
RT8.3. Finite Groups: Projection to Irreducibles
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RT8.3. Finite Groups: Projection to Irreducibles
Representation Theory: Having classified irreducibles in terms of characters, we adapt the methods of the finite abelian case to define projection operators onto irreducible types. Techniques include convolution and weighted averages of representations. At the end, we state and prove the Plancherel Formula (Parseval's Identity using irreducibles).

Course materials, including problem sets with solutions, are available at http://mathdoctorbob.org/UR-RepTheory.html
Lecture 15
RT9. Basic Tensor Analysis
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RT9. Basic Tensor Analysis
Representation Theory: We apply character theory to tensor products. We obtain a character formula for general tensor products and, as special cases, alternating and symmetric 2-tensors. As an application, we compute the character table for S4, the symmetric group on 4 letters.
Lecture 16
RT9.1. Application of Tensors: Normal Modes
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RT9.1. Application of Tensors: Normal Modes
Representation Theory: As an application of tensor analysis, we consider normal modes of mass-spring systems. Cases include motion in a line and planar motion.
Lecture 17
Character Tables for S4 and A4
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Character Tables for S4 and A4
Representation Theory of Finite Groups: We build the character tables for S4 and A4 from scratch. As an application, we use irreducible characters to decompose a tensor product.
Lecture 18
Character Tables for S5 and A5
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Character Tables for S5 and A5
Representation Theory of Finite Groups: We compute the character tables of S5, the symmetric group on 5 letters, and A5, the subgroup of even permutations. We note that A5 is isomorphic to the group of rigid motions of an icosahedron.