The affine Hecke algebra of type A

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 3 March 2013

The affine Hecke algebra of type A

Affine brads. There are three common ways of depicting affine braids [Cri1997], [GLa1997], [Jon1994]:

  1. As braids in a (slightly thickened) cylinder,
  2. As braids in a (slightly thickened) annulus,
  3. As braids with a flagpole.

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 n strands is the affine braid group n of type A. Let ω,Ti for 0in-1, and xi for 1in, be as given in Figure 2. The following identities can be checked by drawing pictures:

(a) TiTj=Tj Ti, for i-j>1, (b) TiTi+1Ti= Ti+1Ti Ti+1, for 0in-1, (c) ωTiω-1= Ti-1, for 0in-1, (d) xiTj=Tj xi, if i-j>1, (e) xi+1=Ti xiTi, for 1in-1, (f) xixj= xjxi, for 1i,jn, (g) xnx1-1= T0Tn-1 T2T1T2 Tn-1, (h) xn=ωT1T2 Tn-1, (i) ωn=x1x2 xn, (1.1)

where the indices on the elements Ti are taken modulo n. The elements Ti,0in-1, and ω generate n. The braid group is the subgroup n generated by the Ti,1in-1. The elements xi,1in, generate an abelian group Xn. If γ=(γ1,γ2,,γn)n define

xγ= x1γ1 x2γ2 xnγn . (1.2)

The symmetric group Sn acts on n by permuting the coordinates. This action induces an action on X by

wxγ= xwγ,for wSn,γn.

The affine Hecke algebra. Fix an element q* which is not a root of unity. The affine Hecke algebra Hn is the quotient of the group algebra n by the relations

Ti2= (q-q-1)Ti +1,0in. (1.3)

The images of Ti,xi and ω in Hn are again denoted by Ti,xi and ω. The Laurent polynomial ring [X]= [ x1±1 ,, xn±1 ] is a (large) commutative subalgebra of Hn.

The relations Ti-1=Ti- (q-q-1) and xi+1=TixiTi can be used to derive the identities

xi+1Ti=Ti xi+(q-q-1) xi+1,and xiTi=Ti xi+1- (q-q-1) xi+1. (1.4)

More generally, if γ=(γ1,γ2,,γn)n then

xγTi=Ti xsiγ+ (q-q-1) (xγ-xsiγ) xi+1 xi+1-xi , (1.5)

where siSn is the simple transposition (i,i+1). The right hand term in this expression can always be written as a Laurent polynomial in x1,,xn. This important relation is due to Bernstein, Zelevinsky and Lusztig [Lus1989]. The affine Hecke algebra Hn can be defined as the algebra generated by Ti,1in, and xi,1in 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 si=(i,i+1), 1in-1, in Sn. A reduced word for a permutation wSn is an expression w=si1sip of minimal length. This minimal length is called the length (w) of w. The symmetric group Sn is partially ordered by the Bruhat-Chevalley order: vw if a reduced expression si1sip for w has a subword sik1 sik, 1k1<<kp which is equal to v in Sn.

The Iwahori-Hecke algebra. The Iwahori-Hecke algebra Hn is the subalgebra of Hn generated by the elements Ti,1in-1. For each wSn let

Tw=Ti1 Tip, (1.6)

where w=si1sip is a reduced word for w. Since the Ti satisfy the braid relations (1.1a,b), the element Tw is independent of the choice of the reduced word of w. The elements Tw,wSn, are a basis of Hn [Bou1968, IV §2 Ex. 23].

Notes and References

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.

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