In mathematics, an affine braid group is a
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
associated to an affine
Coxeter system
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
. Their
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
s have quotients called
affine Hecke algebra
In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.
Definition
Let V be a Euclidean space of a finite dimension and \ ...
s. They are subgroups of
double affine braid group In mathematics, a double affine braid group is a group containing the braid group of an affine Weyl group. Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quoti ...
s.
Definition
References
*
*
Macdonald, I. G. ''Affine Hecke Algebras and Orthogonal Polynomials.'' Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, Eng., 2003. x+175 pp.
Braid groups
Representation theory
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