UM Math Graduate Students Seminar

Dr. Drew Armstrong
University of Miami

will present

What is... a Root System?

Friday, September 23, 2011, 4:00pm
Ungar Building Room 402

Abstract: Fact 1: Every isometry \(\varphi:\mathbb{R}^n\to\mathbb{R}^n\) is affine. That is, we can write \(\varphi(x)=Ax+b\) for some matrix \(A\) and vector \(b\). Fact 2: (Cartan-Dieudonné) Every othogonal matrix \(A\in O(n)\) is the product of at most \(n\) reflections. Conclusion 1: The study of euclidean geometry is the study of the orthogonal group \(O(n)\). Conclusion 2: The orthogonal group is generated by (infinitely many) reflections. Question 1: What can be said about groups generated by finitely many reflections? Answer 1: Quite a lot, actually.

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