Last update: 27 February 2013
In this section we provide the mechanism for obtaining the representation theory of ${H}_{\infty ,p,n}$ from ${H}_{\infty ,1,n}$ and for obtaining the representation theory of cyclotomic Hecke algebras from affine Hecke algebras. In order to obtain the representation theory of ${H}_{\infty ,p,n}$ from ${H}_{\infty ,1,n}$ we identify ${H}_{\infty ,p,n}$ as the set of fixed points of a certain group $G$ acting on ${H}_{\infty ,1,n}$ by automorphisms. Once this is done, the extended version of Clifford theory given in the Appendix allows one to construct the representations of ${H}_{\infty ,p,n}$ from those of ${H}_{\infty ,1,n}\text{.}$ The same technique can be applied to obtain the representations of the braid groups ${\mathcal{B}}_{\infty ,p,n}$ from those of ${\mathcal{B}}_{\infty ,1,n},$ of the complex reflection groups $G(r,p,n)$ from those of $G(r,1,n),$ and of the Weyl groups $W{D}_{n}$ from those of the Weyl groups $W{B}_{n}\text{.}$
The following result is what is needed to apply the Clifford theory developed in the Appendix to derive the representation theory of the algebras ${H}_{\infty ,p,n}$ from that of ${H}_{\infty ,1,n}\text{.}$
Theorem 2.2. Let $\xi $ be a primitive $p\text{th}$ root of unity. The algebra automorphism $g:{H}_{\infty ,1,n}\to {H}_{\infty ,1,n}$ defined by
$$g\left({X}^{{\epsilon}_{1}}\right)=\xi {X}^{{\epsilon}_{1}},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}g\left({T}_{i}\right)={T}_{i},\phantom{\rule{1em}{0ex}}2\le i\le n,$$gives rise to an action of the group $\mathbb{Z}/p\mathbb{Z}=\{1,g,\dots ,{g}^{p-1}\}$ on ${H}_{\infty ,1,n}$ by algebra automorphisms and
$${H}_{\infty ,p,n}={\left({H}_{\infty ,1,n}\right)}^{\mathbb{Z}/p\mathbb{Z}}$$is the set of fixed points of the $\mathbb{Z}/p\mathbb{Z}\text{-action.}$
Proof. | |
Immediate from the definitions of ${H}_{\infty ,1,n},{H}_{\infty ,p,n}$ and (1.6). $\square $ |
The action of $\mathbb{Z}/p\mathbb{Z}$ on ${H}_{\infty ,1,n}$ which is given in Theorem 2.2 induces an action of $\mathbb{Z}/p\mathbb{Z}$ on the simple ${H}_{\infty ,1,n}\text{-modules,}$ see (A.1) in the Appendix. The stabilizer $K$ of the action of $\mathbb{Z}/p\mathbb{Z}$ on a simple ${H}_{\infty ,1,n}\text{-module}$ $M$ is the inertia group $K$ of $M\text{.}$ The action of $K$ commutes with the action of ${H}_{\infty ,p,n}$ on $M$ and we have a decomposition
$$\begin{array}{cc}M\cong \underset{j=0}{\overset{\mid K\mid -1}{\u2a01}}{M}^{\left(j\right)}\otimes {K}^{\left(j\right)},& \text{(2.3)}\end{array}$$where ${K}^{\left(j\right)},$ $1\le j\le \mid K\mid -1,$ are the simple $K\text{-modules}$ and ${M}^{\left(j\right)}$ are ${H}_{\infty ,p,n}\text{-modules.}$ Theorem A.13 of the Appendix shows that the ${M}^{\left(j\right)}$ are simple ${H}_{\infty ,p,n}\text{-modules}$ and that all simple ${H}_{\infty ,p,n}\text{-modules}$ are constructed in this way. In Theorem 3.15 we show that this method gives a combinatorial construction of the module ${M}^{\left(j\right)}$ in any case when the Young tableau theory is available. This is a generalization of the method used in [Ari1995] and [HRa1998].
The homomorphisms $\Phi $ and ${\Phi}_{p}$ described below are the primary tools for transferring results from the affine Hecke algebras to cyclotomic Hecke algebras. Many results are easier to prove for affine Hecke algebras because of the large commutative subalgebra $\u2102\left[X\right]$ which is available in the affine Hecke algebra. The homomorphism $\Phi $ has also been used by Cherednik [Che1991], Ariki [Ari1996] and many others.
Proposition 2.5. Let ${H}_{\infty ,1,n}$ be the affine Hecke algebra of type A defined in (1.9) and let ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q)$ denote the cyclotomic Hecke algebra of (1.1).
Proof. | |
The result follows directly from the definitions of the affine Hecke algebras ${H}_{\infty ,1,n}$ and ${H}_{\infty ,p,n}$ (see (1.9)) and the cyclotomic Hecke algebras ${H}_{r,1,n}$ and ${H}_{r,p,n}$ (see (1.1) and (1.2)). $\square $ |
Let ${L}_{p}$ be the lattice defined in (1.8) and define ${C}_{r}=\{\lambda ={\sum}_{i=1}^{n}{\lambda}_{i}{\epsilon}_{i}\hspace{0.17em}\mid \hspace{0.17em}0\le {\lambda}_{i}\le r\}\text{.}$ Ariki and Koike [AKo1994] and Ariki [Ari1995] have shown that the sets
$$\begin{array}{cc}\begin{array}{c}\{\Phi \left({X}^{\lambda}{T}_{w}\right)\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {C}_{r},w\in {S}_{n}\}\phantom{\rule{2em}{0ex}}\text{and}\\ \{{\Phi}_{p}\left({X}^{\lambda}{T}_{w}\right)\hspace{0.17em}\mid \hspace{0.17em}\lambda \in {C}_{r}\cap {L}_{p},w\in {S}_{n}\}\end{array}& \text{(2.6)}\end{array}$$are bases of ${H}_{r,1,n}$ and ${H}_{r,p,n},$ respectively.
The representation theory of the affine Hecke algebra ${H}_{\infty ,1,n}$ is equivalent to the representation theory of the cyclotomic Hecke algebras ${H}_{r,1,n}$ (considering all possible ${u}_{1},\dots ,{u}_{r}\in \u2102\text{).}$ The elementary constructions in the following theorem allow us to make ${H}_{\infty ,1,n}\text{-modules}$ into ${H}_{r,1,n}\text{-modules}$ and vice versa.
Theorem 2.8. Let ${H}_{\infty ,1,n}$ be the affine Hecke algebra of type A defined in (1.9) and let ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q)$ be the cyclotomic Hecke algebra of (1.1).
Proof. | |
The Theorem follows directly from the definitions of ${H}_{r,1,n}({u}_{1},\dots ,{u}_{r};q)$ and ${H}_{\infty ,1,n},$ and the construction of the surjective homomorphism $\Phi \text{.}$ $\square $ |
Remark 2.9. The same translations work for arbitrary finite dimensional modules; in particular, they work for indecomposable modules and preserve composition series.
This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.
Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).