By Robert S. Doran, Calvin C. Moore, Robert J. Zimmer

George Mackey used to be a unprecedented mathematician of serious strength and imaginative and prescient. His profound contributions to illustration idea, harmonic research, ergodic idea, and mathematical physics left a wealthy legacy for researchers that maintains at the present time. This e-book is predicated on lectures provided at an AMS certain consultation held in January 2007 in New Orleans devoted to his reminiscence. The papers, written particularly for this quantity through internationally-known mathematicians and mathematical physicists, diversity from expository and old surveys to unique high-level examine articles. The effect of Mackey's basic rules is obvious all through. The introductory article comprises reminiscences from former scholars, pals, colleagues, and relations in addition to a biography describing his exclusive profession as a mathematician at Harvard, the place he held the Landon D. Clay Professorship of arithmetic.

**Read Online or Download Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey PDF**

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**Additional info for Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey**

**Example text**

Write down all the elements of Aut(C2 × C2 ). To which well-known group is it isomorphic? 13. Calculate Inn(G) when G = D8 . Show that Aut(G) ∼ = D8 . 14. Show that Aut(Q8 ) ∼ = S4 , where Q8 = i, j | i2 = j 2 = (ij)2 is the quaternion group of order 8. 15. Show that if p is an odd prime then Aut(Cpn ) ∼ = Cpn −pn−1 ∼ = Cpn−1 × Cp−1 . 16. Prove that Aut(A4 ) ∼ = S4 and Aut(A5 ) ∼ = S5 . 17. 2 to show that if n on a set of n + 1 points. 18. 3 to show that any automorphism of A6 which maps 3-cycles to 3-cycles is realised by an element of S6 .

This group, sometimes denoted AGLk (p), and called the aﬃne general linear group, acts as permutations of the vectors, so is a subgroup of Sn where n = pk . The translations form a normal subgroup isomorphic to the additive group of the vector space, which is isomorphic to a direct product of k copies of the cyclic group Cp . In other words it is an elementary abelian group of order pk , which we denote Epk , or simply pk . With this notation, AGLk (p) ∼ = pk :GLk (p). ∼ An example of an aﬃne group is the group AGL3 (2) = 23 :GL3 (2), which acts as a permutation group on the 8 vectors of F2 3 , and so embeds in S8 .

If the block system is non-trivial, since H = KN and KN/N = with l blocks of size k, say, and l > 1, then N ∼ = T kl and N ∩ K ∼ = T l so (k−1)l . Thus we see that H lies inside Sr Sl , in its product action, n = |T | where r = |T |k−1 . This is case (iii) of the theorem again. Otherwise, the block system is trivial, so K ∩ N is a diagonal copy of T inside T1 × · · · × Tm , and we can choose our notation such that it consists of the elements (g, g, . . , g) for all g ∈ T . Also n = |T |m−1 , and the n points can be identiﬁed with the n conjugates of K ∩ N by elements of N .