## Combinatorial Representation Theory

Last update: 17 September 2013

## Appendix B

### B4. “Hecke algebras” of the groups $G\left(r,p,n\right)$

Let $q$ and ${u}_{0},{u}_{1},\dots ,{u}_{r-1}$ be indeterminates. Let ${H}_{r,1,n}$ be the algebra over the field $ℂ\left({u}_{0},{u}_{1},\dots ,{u}_{r-1},q\right)$ 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) $\left({T}_{1}-{u}_{0}\right)\left({T}_{1}-{u}_{1}\right)\cdots \left({T}_{1}-{u}_{r-1}\right)=0,$ (5) $\left({T}_{i}-q\right)\left({T}_{i}+{q}^{-1}\right)=0,$ for $2\le i\le n\text{.}$
Upon setting $q=1$ and ${u}_{i-1}={\xi }^{i-1},$ where $\xi$ is a primitive $r\text{th}$ root of unity, one obtains the group algebra $ℂG\left(r,1,n\right)\text{.}$ In the special case where $r=1$ and ${u}_{0}=1,$ we have ${T}_{1}=1,$ and ${H}_{1,1,n}$ is isomorphic to an Iwahori-Hecke algebra of type ${A}_{n-1}\text{.}$ The case ${H}_{2,1,n}$ when $r=2,$ ${u}_{0}=p,$ and ${u}_{1}={p}^{-1},$ is isomorphic to an Iwahori-Hecke algebra of type ${B}_{n}\text{.}$

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 $uℓd+kp+1= εℓxk1/p,$ where the subscripts on the ${u}_{i}$ are taken mod $r\text{.}$ The “Hecke algebra” ${H}_{r,p,n}$ corresponding to the group $G\left(r,p,n\right)$ is the subalgebra of ${H}_{r,1,n}$ generated by the elements $a0=T1p, a1=T1-1T2T1, andai=Ti, 2≤i≤n.$ 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 $ℂG\left(r,p,n\right)\text{.}$ Thus ${H}_{r,p,n}$ is a $\text{“}q\text{-analogue”}$ of the group algebra of the group $G\left(r,p,n\right)\text{.}$

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.

## Notes and references

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..