Last update: 17 September 2013
Let $q$ and ${u}_{0},{u}_{1},\dots ,{u}_{r-1}$ be indeterminates. Let ${H}_{r,1,n}$ be the algebra over the field $\u2102({u}_{0},{u}_{1},\dots ,{u}_{r-1},q)$ given by generators ${T}_{1},{T}_{2},\dots ,{T}_{n}$ and relations
(1) | ${T}_{i}{T}_{j}={T}_{j}{T}_{i},$ | for $|i-j|>1,$ |
(2) | ${T}_{i}{T}_{i+1}{T}_{i}={T}_{i+1}{T}_{i}{T}_{i+1},$ | for $2\le i\le n-1,$ |
(3) | ${T}_{1}{T}_{2}{T}_{1}{T}_{2}={T}_{2}{T}_{1}{T}_{2}{T}_{1},$ | |
(4) | $({T}_{1}-{u}_{0})({T}_{1}-{u}_{1})\cdots ({T}_{1}-{u}_{r-1})=0,$ | |
(5) | $({T}_{i}-q)({T}_{i}+{q}^{-1})=0,$ | for $2\le i\le n\text{.}$ |
Now suppose that $p$ and $d$ are positive integers such that $pd=r\text{.}$ Let ${x}_{0}^{1/p},\dots ,{x}_{d-1}^{1/p}$ be indeterminates, let $\epsilon ={e}^{2\pi i/p}$ be a primitive $p\text{th}$ root of unity and specialize the variables ${u}_{0},\dots ,{u}_{r-1}$ according to the relation $${u}_{\ell d+kp+1}={\epsilon}^{\ell}{x}_{k}^{1/p},$$ where the subscripts on the ${u}_{i}$ are taken mod $r\text{.}$ The “Hecke algebra” ${H}_{r,p,n}$ corresponding to the group $G(r,p,n)$ is the subalgebra of ${H}_{r,1,n}$ generated by the elements $${a}_{0}={T}_{1}^{p},\phantom{\rule{1em}{0ex}}{a}_{1}={T}_{1}^{-1}{T}_{2}{T}_{1},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{a}_{i}={T}_{i},\phantom{\rule{1em}{0ex}}2\le i\le n\text{.}$$ Upon specializing ${x}_{k}^{1/p}={\xi}^{kp},$ where $\xi $ is a primitive $r\text{th}$ root of unity, ${H}_{r,p,n}$ becomes the group algebra $\u2102G(r,p,n)\text{.}$ Thus ${H}_{r,p,n}$ is a $\text{\u201c}q\text{-analogue\u201d}$ of the group algebra of the group $G(r,p,n)\text{.}$
References
The algebras ${H}_{r,1,n}$ were first constructed by Ariki and Koike [AKo1994], and they were classified as cyclotomic Hecke algebras of type Bn by Broué and Malle [BMa1993] and the representation theory of ${H}_{r,p,n}$ was studied by Ariki [Ari1995]. See [HRa1998] for information about the characters of these algebras.
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..