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12 Mar 2007: JS07. Representation Theory Sheet 4. Unless otherwise stated, all vector spaces are finite dimensional over C. ... Q.8 (a) Let G be a compact group. Show that there is a group homomorphismρ : G On(R) if and only if G has an n-dimensional representation

25 Oct 2007: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

21 Nov 2007: 1. The square matrices A and B over the field F are congruent if B = P T AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

5 Feb 2007: b) Determine which of the following polynomials are irreducible over Q: X3X 1, X3X 2,X3 X 3. ... 2,. 7) over Q. You may assume that G = {1,α,β,αβ}where.

3 Jun 2007: c) Zω1 Zω2 = {nω1 mω2 : n, m Z} for some ω1, ω2 C which are linearly independent over R. ... Let this be ω2. Suppose that ω1, ω2 were not linearly independent over R.

7 Nov 2007: 13. Let V be a vector space of finite dimension over a field F. ... Verify that (DS) = SD and (SD) = DS. 16. For A an n m and B an m n matrix over the field F , let τA(B) denote trAB.Show that, for

3 Jun 2007: log |zn| is still convergent when we sum over all of the zeros. ... B(z) = ωzk |zn|. zn. (z zn1 znz. )where ω C is of modulus 1, 0 occurs k times in the sequence (zn), and the product is over the non-zeroelements

26 Nov 2007: 3. The Heisenberg group. We continue to work over a field k of characteristic 0. ... case n = 2. 6. Computations. Let E be an elliptic curve over Q.

5 Apr 2007: Taking the infimum over T , we obtain in particular. ||T1 T2|| ||T1|| ||T2||. ... 2.9 The dual space V. Definition 2.10. Let V be a normed vector space over R (C, respectively).

19 May 2007: This is an expectation over pairs (x, x′) of terms that are real and non-negative. ... 2. Now let us take the expectation over y of both sides.