Last update: 3 March 2013

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

- As braids in a (slightly thickened) cylinder,
- As braids in a (slightly thickened) annulus,
- 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*
${\stackrel{\sim}{\mathcal{B}}}_{n}$ 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:

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 ${\stackrel{\sim}{\mathcal{B}}}_{n}\text{.}$ The
*braid group* is the subgroup ${\mathcal{B}}_{n}$ 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 $X\subseteq {\stackrel{\sim}{\mathcal{B}}}_{n}\text{.}$ If
$\gamma =({\gamma}_{1},{\gamma}_{2},\dots ,{\gamma}_{n})\in {\mathbb{Z}}^{n}$
define

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

$$w{x}^{\gamma}={x}^{w\gamma},\phantom{\rule{2em}{0ex}}\text{for}\hspace{0.17em}w\in {S}_{n},\gamma \in {\mathbb{Z}}^{n}\text{.}$$
**The affine Hecke algebra.** Fix an element $q\in {\u2102}^{*}$ which is not a root
of unity. The *affine Hecke algebra* ${\stackrel{\sim}{H}}_{n}$ is the quotient of the group algebra
$\u2102{\stackrel{\sim}{\mathcal{B}}}_{n}$ by the relations

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 $\u2102\left[X\right]=\u2102[{x}_{1}^{\pm 1},\dots ,{x}_{n}^{\pm 1}]$ 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

$$\begin{array}{cc}{x}_{i+1}{T}_{i}={T}_{i}{x}_{i}+(q-{q}^{-1}){x}_{i+1},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{x}_{i}{T}_{i}={T}_{i}{x}_{i+1}-(q-{q}^{-1}){x}_{i+1}\text{.}& \text{(1.4)}\end{array}$$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

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].

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.