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There are not too many symmetries in two dimensions, or put more
mathematically, there are only three crystallographic groups of rank 2: the
finite groups A_2, B_2 and G_2. But they have a remarkable amount of
structure associated to them. I will talk about some polynomials -- the
Kazhdan-Lusztig polynomials discovered (by guess who) in 1979 -- which arise
in the study of reflection groups like these. Some beautiful geometric patterns
emerge quite unexpectedly even in just two dimensions. If only I could draw the
three dimensional case too...
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