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Unformatted text preview: Problem Set #5 Solutions Chapter VI.8. 8.1.1. Write out the permutation matrix representation of the dihedral group D 4 corresponding to its action on the vertices of a square. Let S = { 1 , 2 , 3 , 4 } be the vertices of a square. In terms of generators and relations we have D 4 = σ, τ  σ 4 = τ 2 = 1 , στ = τσ 1 . Thus ˆ T (1) = 1 1 1 1 , ˆ T ( τ ) = 1 1 1 1 , ˆ T ( σ ) = 1 1 1 1 , ˆ T ( τσ ) = 1 1 1 1 , ˆ T ( σ 2 ) = 1 1 1 1 , ˆ T ( τσ 2 ) = 1 1 1 1 , ˆ T ( σ 3 ) = 1 1 1 1 , ˆ T ( τσ 3 ) = 1 1 1 1 . 8.1.2. Write out the left regular matrix representation if G = Z 4 , Z 2 × Z 2 , S 3 , or D 4 . We do a couple examples for each group. First let G = Z 4 = a  a 4 = 1 and let V be the vector space with basis { 1 , a, a 2 , a 3 } . Then ˆ T ( a ) = 1 1 1 1 , ˆ T ( a 2 ) = 1 1 1 1 . Let G = Z 2 × Z 2 and let V be the vector space with basis { (1 , 1) , (1 , a ) , ( a, 1) , ( a, a ) } . Then ˆ T (( a, 1)) = 1 1 1 1 , ˆ T (( a, a )) = 1 1 1 1 . 1 Let G = S 3 and let V be the vector space with basis { 1 , (123) , (132) , (12) , (13) , (23) } . Then ˆ T ((123)) = 1 1 1 1 1 1 , ˆ T ((23)) = 1 1 1 1 1 1 . Finally let G = D 4 be as above and let V be the vector space with basis { 1 , σ, σ 2 , σ 3 , τ, τσ, τσ 2 , τσ 3 } . Then ˆ T ( τσ ) = 1 1 1 1 1 1 1 , ˆ T ( τ ) = 1 1 1 1 1 1 1 1 . 8.2. Suppose T 1 , T 2 , . . . , T k are mutually inequivalent irreducible representations of G , with ˆ T m ( x ) = ( t ( m ) ij ( x )) , 1 ≤ m ≤ k . Use proposition 8.4 to show that the functions t ( m ) ij , all m, i, j , are Clinearly independent. If n m = deg T m...
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This note was uploaded on 08/06/2008 for the course MATH 220 taught by Professor Morrison during the Spring '08 term at UCSB.
 Spring '08
 MORRISON
 Algebra

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