Representations of Finite Groups

# Representations of Finite Groups - Southern Illinois...

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Unformatted text preview: Southern Illinois University Carbondale OpenSIUC Miscellaneous (presentations, translations, interviews, etc) Department of Mathematics 10-1-2010 Representations of Finite Groups Joseph Hundley Southern Illinois University Carbondale , [email protected] This Article is brought to you for free and open access by the Department of Mathematics at OpenSIUC. It has been accepted for inclusion in Miscellaneous (presentations, translations, interviews, etc) by an authorized administrator of OpenSIUC. For more information, please contact [email protected] . Recommended Citation Hundley, Joseph, "Representations of Finite Groups" (2010). Miscellaneous (presentations, translations, interviews, etc). Paper 46. http://opensiuc.lib.siu.edu/math_misc/46 REPRESENTATIONS OF FINITE GROUPS JOSEPH HUNDLEY Contents 1. Basic Definitions 1 1.1. G-Modules 3 1.2. A digression 3 2. Direct sum and tensor product 3 3. Irreducibles and complete reducibility 4 4. Irreducible representations of finite abelian groups 5 5. Irreducible Representations of S 3 5 6. Characters 6 References 9 1. Basic Definitions I’m following [F-H]. First let me fill in a little bit of background material. Definition 1.0.1. (field) A field is a set F equipped with two binary operations + and · such that ( F, +) and ( F \ { } , · ) are abelian groups, and x · ( y + z ) = x · y + x · z ( ∀ x,y,z, ∈ F ) . Examples 1.0.2. Q , R and C are fields, while Z is not. Definition 1.0.3. (Vector space) Let F be a field. A vector space over F is an abelian group V equipped with a function F × V → V called scalar multiplication and written ( α,v ) ∈ F × V 7→ αv ∈ V, such that α ( v + w ) = αv + αw ( ∀ α ∈ F, v,w, ∈ V ) , ( α + β ) v = αv + βv ( ∀ α,β ∈ F,v ∈ V ) , ( αβ ) v = α ( βv ) ( ∀ α,β ∈ F,v ∈ V ) , 1 v = v ( ∀ v ∈ V ) . Examples 1.0.4. Row vectors, column vectors, matrices, polynomials, functions. Definition 1.0.5. (linear function) Let F be a field and let V,W be vector spaces over F. A function L : V → W is linear if L ( αv + w ) = αL ( v ) + L ( w ) ( ∀ α ∈ F, v,w ∈ V ) . Definition 1.0.6. ( GL ( V ) ) ( GL ( V ) ) ( GL ( V ) ) Let F be a field and let V be a vector space over F. Then GL ( V ) GL ( V ) GL ( V ) is the set of bijective linear functions V → V, equipped with the binary operation ◦ (composition of functions). It is a group. Date : December 2, 2010. 1 Definition 1.0.7. ( GL n F GL n F GL n F ) Let F be a field and n be an integer. Then GL n F GL n F GL n F is the set of n × n invertible matrices with entries in F, equipped with matrix multiplication. It is a group. Further, the function A 7→ (multiplication by A ) is an isomorphism GL n F → GL ( F n ) , where F n is realized as column vectors....
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Representations of Finite Groups - Southern Illinois...

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