Learn · Representation Theory of Finite Groups
Lecture 24 of 56Week 5 Schur orthogonality relations, regular representation, and its decomposition
Regular representation, decomposition of regular representation
Prof. R. Venkatesh · IISc Bangalore
Credits and licence
- From
- Representation Theory of Finite Groups, IISc Bangalore (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
- Who can watch
- Tweens profiles and older. How courses are placed
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