1) Let G = <x, y| x^5, y^4, yxy^-1x^-1>. Determine the character table of G. 2) Prove that the standard three-dimensional representation of the tetrahedral group T is irreducible as a complex representation. 3) Let G be a cyclic group of order 3. The matrix A = [ -1 -1/ 1 0] has order 3. Use the averaging process to produce a G-invariantfrom the standard Hermitian product X*Y on C^2. 4) Suppose given a representation of the symmetric group S3 on a vector space V. a. Let u be a nonzero vector in V. Let v = u + ux + ux^2 and w = u + yu. By analyzing the G-orbits of v, w, show that V contains a nonzero invariant subspace of dimension at most 2. b. Prove that all irreducible two-dimensional representations of G are isomorphic, and determine all irreducible representations of G.

Subject | Mathematics |

Due By (Pacific Time) | 04/14/2013 12:00 am |

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