Learn · Representation Theory of Finite Groups
Lecture 51 of 56Week 11 Eigenvalues of Cayley graph, combinatorics in symmetric group and introduction to representation theory of…
Cayley graph and its eigenvalues
Prof. R. Venkatesh · IISc Bangalore
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- Representation Theory of Finite Groups, IISc Bangalore (at NPTEL)
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All the lectures
- Basics of vector spaces
- Inner product spaces
- Diagonalization
- Generalized eigen spaces
- Decomposition theorems
- Definition of representation
- Isomorphism theorems
- Indecomposable representations
- Schur's Lemma
- Filtration of representations
- Construction of new representations by extensions
- Basics of representation theory of finite groups
- Classification of irreducible representations of finite cyclic group and Symmteric group
- Irreducibility via Cyclic Vectors
- Construction of representations of group generated by a set with defining relations
- Complete reducibility of unitary representations
- Complete reducibility of finite dimensional representations of finite groups
- Projection map from representation of G to its fixed subspace
- Character Theory: Basic properties
- Orthogonality relations of irreducible characters of finite group
- Orthonormal subset of space of class functions and Orthogonality of irreducible characters
- Schur's orthogonality relations and its applications
- Applications of character theory
- Regular representation, decomposition of regular representation
- Orthonormal basis of space of class functions and computation of character tables
- Orthogonality of the columns of the character table
- Representations of finite abeilian group
- Algebraic numbers and equivalent conditions for algebraic integers
- Dimension theorem
- Applications of dimension theorem
- Character table computation of groups: Q_8, D_8, A_4
- Burnside Theorem and its proof
- Proof of key result used in Burnside Theorem
- Notion of Tensor product
- Basic properties of Tensor product
- Symmetric tensor product
- Alternating tensor product
- G-action on Symmetric and exterior tensor product
- Graded character of Symmetric tensor algebra and exterior algebra
- Alternate proof of orthogonality of irreducible characters
- Density theorem
- Frobenius theorem
- Induced representation
- Frobenius reciprocity
- Mackey's restrcition formula
- Mackey's irreducibility criterion
- Fourier analysis on finite groups
- Fourier inversion
- Fourier analysis on nonabeilian groups
- Applications of Fourier analysis on finite groups
- Cayley graph and its eigenvalues
- Notations and combinatorics in representation theory of symmetric group
- Irreducible representations of symmetric group
- Permutation modules
- Constructing irreducibles of symmetric group
- Classification of irreducible representations of symmetric group