hw1_913-SS11

# hw1_913-SS11 - Math 913 Spring 2011 – HW 1 Due Monday 21...

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Unformatted text preview: Math 913, Spring 2011 – HW 1 Due Monday, 21 March 2011 1. Prove that a subgroup of the symmetric group Sym(Ω), for Ω finite, that is generated by a set S of transpositions (2-cycles) is a direct sum L k i =1 Sym(Ω i ) for pairwise disjoint subsets Ω i of Ω. Hint: On Ω define a graph by letting a ∼ b if and only if ( a,b ) is one of the transpositions of S . Then the Ω i will be the connected components of this graph. Remark. The result remains true for infinite Ω with FSym(Ω i ) in place of Sym(Ω i ). 2. This problem will (among other things) show that the sign homomorphism on finite Sym(Ω) (and hence on arbitrary FSym(Ω)) is well-defined. Let F be a field and V the Sym(Ω) permutation module F Ω with basis { e ω | ω ∈ Ω } and action given by e ω .g = e ω.g . Let n ≥ 2 and set Ω = { 1 , 2 ,...,n } so that Sym(Ω) = Sym( n ). ( a ) Recall that for g ∈ GL F ( V ), we have defined [ V,g ] = V ( g- 1). Prove: ( i ) The subspace W of V is g-invariant with g trivial on V/W if and only if W ≥ [ V,g ]. ( ii ) [ V,gh ] ≤ [ V,g ] + [ V,h ]. ( b ) For g ∈ Sym( n ), let ‘ ( g ) be the smallest number of transpositions with product g . Prove that dim K [ V,g ] ≤ ‘ ( g ). ( c ) Prove that dim K [ V,g ] = ‘ ( g ) = n- n k = | supp( g ) | - c k where n k is the number of cycles in g (including cycles of length 1) and c k is the number of cycles in g of length greater than 1. ( Hint: First prove that if g is a k-cycle, then dim K [ V,g ] = ‘ ( g ) = k- 1.) ( d ) For t a transposition, prove that ‘ ( gt ) = ‘ ( g ) ± 1. ( e ) Prove that sgn: g 7→ (- 1) ‘ ( g ) = (- 1) dim k [ V,g ] is a homomorphism from Sym( n ) onto the multiplicative group ± 1. (This is the sign homomorphism.) 3. Let G = GL 2 ( R ), the group of 2 × 2 invertible matrices with entries from the ring with identity R . Set B = a b d a,d ∈ U ( R ) and t =- 1 1 ....
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hw1_913-SS11 - Math 913 Spring 2011 – HW 1 Due Monday 21...

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