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I am reading the paper Cherednik Algebras, Macdonald Polynomials, and Combinatorics by Mark Haiman.

The definition (2.7) of the braid group $\mathcal{B}(W)$ seems to be the same as the definition of the Weyl group except that the relations $T_i^2 = 1$ are no longer included. Is there anything else that makes it distinct?

What is the motivation for introducing a new construction so similar to the Weyl group? Any intuition would be appreciated.

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Do you know the usual braid group? It corresponds to the case where the Weyl group is simply the symmetric group. en.wikipedia.org/wiki/Braid_theory The omission of the relations T^2 = 1 really changes the structure of the group... –  PseudoNeo May 1 '13 at 18:31
    
Thanks PseudoNeo! I just wanted to make sure that was the only difference in definition. –  braid May 1 '13 at 18:48

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Are you familiar with the braid groups associated to the symmetric groups (often just called the braid groups)? They are not so similar to the symmetric groups! For one thing, they're much prettier. The braid groups have many applications; for example, they occur as mapping class groups and as the fundamental groups of configuration spaces. Moreover they can be used to write down knot invariants. See Kassel and Turaev's Braid Groups for much more.

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I'm not too familiar with braid groups -- thanks for the references! –  braid May 1 '13 at 18:49

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