- Character Theory of Finite Groups
- Representation Theory of Finite Groups
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- Representation theory of finite groups - Wikipedia
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Algebraic Integers. Valuations Padic Numbers. Norms of Ideals Ideal Classes.

## Character Theory of Finite Groups

Cyclotomic Fields. Modules over Dedekind Domains. Semisimple Rings and Group Algebras. The Radical of a Ring with Minimum Condition. Semisimple Rings and Completely Reducible Modules. The Structure of Simple Rings. Theorems of Burnside Frobenius and Schur. Irreducible Representations of the Symmetric Group.

Extension of the Ground Field. Group Characters. Orthogonality Relations. Simple Applications of the Orthogonality Relations.

Central Idempotents. Burnsides Criterion for Solvable Groups. Units in a Group Ring. Induced Characters. Rational Characters.

## Representation Theory of Finite Groups

Brauers Theorem on Induced Characters. The Generalized Induction Theorem.

Induced Representations. Induced Representations and Modules. Irreducibility and Equivalence of Induced Modules. NonSemiSimple Rings. Projective Modules. Injective Modules. QuasiFrobenius Rings. Modules over QuasiFrobenius Rings. Frobenius Algebras. Frobenius and QuasiFrobenius Algebras. Projective and Injective Modules for a Frobenius Algebra.

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Group Algebras of Finite Representation Type. The Vertex and Source of an Indecomposable Module. Centralizers of Modules over Symmetric Algebras. Splitting Fields and Separable Algebras. Separable Extensions of the Base Field. The Schur Index.

## Representation theory of finite groups - Wikipedia

Separable Algebras. We also define characters and state the corresponding SORs.

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Lecture 13 Play Video RT8. 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. 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.

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Techniques include convolution and weighted averages of representations. At the end, we state and prove the Plancherel Formula Parseval's Identity using irreducibles. 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 Play Video RT9.

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. As an application, we use irreducible characters to decompose a tensor product.

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We note that A5 is isomorphic to the group of rigid motions of an icosahedron. Subject: Mathematics. All rights reserved. Representation Theory. Lecture 1 Play Video. Lecture 2 Play Video. RT2: Unitary Representations Representation Theory: We explain unitarity and invariant inner products for representations of finite groups. Lecture 3 Play Video. Lecture 4 Play Video. Lecture 5 Play Video. Lecture 6 Play Video. Lecture 7 Play Video. Lecture 8 Play Video.

Lecture 9 Play Video. Lecture 10 Play Video. Lecture 11 Play Video.