Last update: 3 March 2013
Affine brads. There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:
See Figure 1. The multiplication is by placing one cylinder on top of another, placing one annulus inside another, or placing one flagpole braid on top of another. These are equivalent formulations: an annulus can be made into a cylinder by turning up the edges, and a cylindrical braid can be made into a flagpole braid by putting a flagpole down the middle of the cylinder and pushing the pole over to the left so that the strings begin and end to its right.
The group formed by the affine braids with strands is the affine braid group of type A. Let for and for be as given in Figure 2. The following identities can be checked by drawing pictures:
where the indices on the elements are taken modulo The elements and generate The braid group is the subgroup generated by the The elements generate an abelian group If define
The symmetric group acts on by permuting the coordinates. This action induces an action on by
The affine Hecke algebra. Fix an element which is not a root of unity. The affine Hecke algebra is the quotient of the group algebra by the relations
The images of and in are again denoted by and The Laurent polynomial ring is a (large) commutative subalgebra of
The relations and can be used to derive the identities
More generally, if then
where is the simple transposition The right hand term in this expression can always be written as a Laurent polynomial in This important relation is due to Bernstein, Zelevinsky and Lusztig [Lus1989]. The affine Hecke algebra can be defined as the algebra generated by and subject to the relations in (1.1a), (1.1b), (1.1f), (1.3) and (1.5).
The symmetric group. The simple transpositions are the elements in A reduced word for a permutation is an expression of minimal length. This minimal length is called the length of The symmetric group is partially ordered by the Bruhat-Chevalley order: if a reduced expression for has a subword which is equal to in
The Iwahori-Hecke algebra. The Iwahori-Hecke algebra is the subalgebra of generated by the elements For each let
where is a reduced word for Since the satisfy the braid relations (1.1a,b), the element is independent of the choice of the reduced word of The elements are a basis of [Bou1968, IV §2 Ex. 23].
This is an excerpt of the preprint entitled Skew shape representations are irreducible authored by Arun Ram in 1998.
Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.