The affine Hecke algebra of type A

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 ${\stackrel{\sim }{ℬ}}_n$ of type A. Let $\omega ,T_i$ for $0\le i\le n-1,$ and $x_i$ for $1\le i\le n,$ be as given in Figure 2. The following identities can be checked by drawing pictures:

$$\begin{array}{cc}\begin{array}{ccc}\text{(a)}& T_i{T}_j=T_j{T}_i,& \text{for}\hspace{0.17em}\mid i-j\mid >1,\\ \text{(b)}& T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},& \text{for}\hspace{0.17em}0\le i\le n-1,\\ \text{(c)}& \omega T_i{\omega }^{-1}=T_{i-1},& \text{for}\hspace{0.17em}0\le i\le n-1,\\ \text{(d)}& x_i{T}_j=T_j{x}_i,& \text{if}\hspace{0.17em}\mid i-j\mid >1,\\ \text{(e)}& x_{i+1}=T_i{x}_i{T}_i,& \text{for}\hspace{0.17em}1\le i\le n-1,\\ \text{(f)}& x_i{x}_j=x_j{x}_i,& \text{for}\hspace{0.17em}1\le i,j\le n,\\ \text{(g)}& x_n{x}_1^{-1}=T_0{T}_{n-1}\dots T_2{T}_1{T}_2\dots T_{n-1},\\ \text{(h)}& x_n=\omega T_1{T}_2\dots T_{n-1},\\ \text{(i)}& {\omega }^n=x_1{x}_2\dots x_n,\end{array}& \text{(1.1)}\end{array}$$

where the indices on the elements $T_i$ are taken modulo $n\text{.}$ The elements $T_i,0\le i\le n-1,$ and $\omega$ generate ${\stackrel{\sim }{ℬ}}_n\text{.}$ The braid group is the subgroup ${ℬ}_n$ generated by the $T_i,1\le i\le n-1\text{.}$ The elements $x_i,1\le i\le n,$ generate an abelian group $X\subseteq {\stackrel{\sim }{ℬ}}_n\text{.}$ If $\gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)\in {\mathbb{Z}}^n$ define

$$\begin{array}{cc}{x}^{\gamma }=x_1^{{\gamma }_1}{x}_2^{{\gamma }_2}\dots x_n^{{\gamma }_n}\text{.}& \text{(1.2)}\end{array}$$

The symmetric group $S_n$ acts on ${\mathbb{Z}}^n$ by permuting the coordinates. This action induces an action on $X$ by

$$w{x}^{\gamma }=x^{w\gamma },\text{for}\hspace{0.17em}w\in S_n,\gamma \in {\mathbb{Z}}^n\text{.}$$

The affine Hecke algebra. Fix an element $q\in {ℂ}^{*}$ which is not a root of unity. The affine Hecke algebra ${\stackrel{\sim }{H}}_n$ is the quotient of the group algebra $ℂ{\stackrel{\sim }{ℬ}}_n$ by the relations

$$\begin{array}{cc}{T}_i^2=(q-q^{-1})T_i+1,0\le i\le n\text{.}& \text{(1.3)}\end{array}$$

The images of $T_i,x_i$ and $\omega$ in ${\stackrel{\sim }{H}}_n$ are again denoted by $T_i,x_i$ and $\omega \text{.}$ The Laurent polynomial ring $ℂ\left[X\right]=ℂ[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ is a (large) commutative subalgebra of ${\stackrel{\sim }{H}}_n\text{.}$

The relations $T_i^{-1}=T_i-(q-q^{-1})$ and $x_{i+1}=T_i{x}_i{T}_i$ can be used to derive the identities

$$\begin{array}{cc}{x}_{i+1}{T}_i=T_i{x}_i+(q-q^{-1})x_{i+1},\text{and}{x}_i{T}_i=T_i{x}_{i+1}-(q-q^{-1})x_{i+1}\text{.}& \text{(1.4)}\end{array}$$

More generally, if $\gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)\in {\mathbb{Z}}^n$ then

$$\begin{array}{cc}{x}^{\gamma }{T}_i=T_i{x}^{s_i\gamma }+(q-q^{-1})\frac{(x^{\gamma }-x^{s_i\gamma })x_{i+1}}{x_{i+1}-x_i},& \text{(1.5)}\end{array}$$

where $s_i\in S_n$ is the simple transposition $(i,i+1)\text{.}$ The right hand term in this expression can always be written as a Laurent polynomial in $x_1,\dots ,x_n\text{.}$ This important relation is due to Bernstein, Zelevinsky and Lusztig [Lus1989]. The affine Hecke algebra ${\stackrel{\sim }{H}}_n$ can be defined as the algebra generated by $T_i,1\le i\le n,$ and $x_i,1\le i\le n$ 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 $s_i=(i,i+1),$ $1\le i\le n-1,$ in $S_n\text{.}$ A reduced word for a permutation $w\in S_n$ is an expression $w=s_{i_1}\dots s_{i_p}$ of minimal length. This minimal length is called the length $\ell \left(w\right)$ of $w\text{.}$ The symmetric group $S_n$ is partially ordered by the Bruhat-Chevalley order: $v\le w$ if a reduced expression $s_{i_1}\dots s_{i_p}$ for $w$ has a subword $s_{i_{k_1}}\dots s_{i_{k_{\ell }}},$ $1\le k_1<\dots <k_{\ell }\le p$ which is equal to $v$ in $S_n\text{.}$

The Iwahori-Hecke algebra. The Iwahori-Hecke algebra $H_n$ is the subalgebra of ${\stackrel{\sim }{H}}_n$ generated by the elements $T_i,1\le i\le n-1\text{.}$ For each $w\in S_n$ let

$$\begin{array}{cc}{T}_w=T_{i_1}\dots T_{i_p},& \text{(1.6)}\end{array}$$

where $w=s_{i_1}\dots s_{i_p}$ is a reduced word for $w\text{.}$ Since the $T_i$ satisfy the braid relations (1.1a,b), the element $T_w$ is independent of the choice of the reduced word of $w\text{.}$ The elements $T_w,w\in S_n,$ are a basis of $H_n$ [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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