# Braid Group of a Weyl Group

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. – user67771 May 1 '13 at 18:48