Representation theory transfer

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 27 February 2013

Representation theory transfer

In this section we provide the mechanism for obtaining the representation theory of H,p,n from H,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,p,n from H,1,n we identify H,p,n as the set of fixed points of a certain group G acting on H,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,p,n from those of H,1,n. The same technique can be applied to obtain the representations of the braid groups ,p,n from those of ,1,n, of the complex reflection groups G(r,p,n) from those of G(r,1,n), and of the Weyl groups WDn from those of the Weyl groups WBn.

2.1 Obtaining H,p,n-modules from H,1,n-modules

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,p,n from that of H,1,n.

Theorem 2.2. Let ξ be a primitive pth root of unity. The algebra automorphism g:H,1,n H,1,n defined by

g(Xε1)= ξXε1,and g(Ti)= Ti,2in,

gives rise to an action of the group /p= {1,g,,gp-1} on H,1,n by algebra automorphisms and

H,p,n= (H,1,n) /p

is the set of fixed points of the /p-action.


Immediate from the definitions of H,1,n, H,p,n and (1.6).

The action of /p on H,1,n which is given in Theorem 2.2 induces an action of /p on the simple H,1,n-modules, see (A.1) in the Appendix. The stabilizer K of the action of /p on a simple H,1,n-module M is the inertia group K of M. The action of K commutes with the action of H,p,n on M and we have a decomposition

Mj=0K-1 M(j) K(j), (2.3)

where K(j), 1jK-1, are the simple K-modules and M(j) are H,p,n-modules. Theorem A.13 of the Appendix shows that the M(j) are simple H,p,n-modules and that all simple H,p,n-modules are constructed in this way. In Theorem 3.15 we show that this method gives a combinatorial construction of the module M(j) in any case when the Young tableau theory is available. This is a generalization of the method used in [Ari1995] and [HRa1998].

2.4 The surjective algebra homomorphisms Φ and Φp

The homomorphisms Φ and Φp 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 [X] which is available in the affine Hecke algebra. The homomorphism Φ has also been used by Cherednik [Che1991], Ariki [Ari1996] and many others.

Proposition 2.5. Let H,1,n be the affine Hecke algebra of type A defined in (1.9) and let Hr,1,n (u1,,ur;q) denote the cyclotomic Hecke algebra of (1.1).

  1. Fix u1,,ur, q*. There is a surjective algebra homomorphism given by
Φ: H,1,n Hr,1,n (u1,,ur;q) Ti Ti, Xε1 T1. 2in,
  1. Restricting the homomorphism Φ to H,p,n yields a surjective homomorphism defined by
Φ: H,p,n Hr,p,n (x0,,xd-1;q) Ti ai, 2in, Xpε1 a0, Xε2-ε1 a1a2.


The result follows directly from the definitions of the affine Hecke algebras H,1,n and H,p,n (see (1.9)) and the cyclotomic Hecke algebras Hr,1,n and Hr,p,n (see (1.1) and (1.2)).

Let Lp be the lattice defined in (1.8) and define Cr= { λ=i=1n λiεi 0λir } . Ariki and Koike [AKo1994] and Ariki [Ari1995] have shown that the sets

{ Φ(XλTw) λCr, wSn } and { Φp (XλTw) λCr Lp,wSn } (2.6)

are bases of Hr,1,n and Hr,p,n, respectively.

2.7 Relating H,1,n-modules and Hr,1,n-modules

The representation theory of the affine Hecke algebra H,1,n is equivalent to the representation theory of the cyclotomic Hecke algebras Hr,1,n (considering all possible u1,,ur). The elementary constructions in the following theorem allow us to make H,1,n-modules into Hr,1,n-modules and vice versa.

Theorem 2.8. Let H,1,n be the affine Hecke algebra of type A defined in (1.9) and let Hr,1,n (u1,,ur;q) be the cyclotomic Hecke algebra of (1.1).

  1. Fix u1,,ur q*, and let Φ be the surjective homomorphism of Proposition 2.5. If M is a simple Hr,1,n (u1,,ur;q) -module then defining hm=Φ(h) m,for allh H,1,n and allmM, makes M into a simple H,1,n-module.
  2. Let M be a simple H,1,n-module and let ρ:H,1,n End(M) be the corresponding representation. Let u1,,ur be such that the minimal polynomial p(t) of the matrix ρ(Xε1) divides the polynomial (t-u1) (t-ur). Define an action of Hr,1,n (u1,,ur;q) on M by T1m=Xε1m and Tim=Tim,2 in, for all mM. Then M is a simple Hr,1,n (u1,,ur;q) -module.


The Theorem follows directly from the definitions of Hr,1,n (u1,,ur;q) and H,1,n, and the construction of the surjective homomorphism Φ.

Remark 2.9. The same translations work for arbitrary finite dimensional modules; in particular, they work for indecomposable modules and preserve composition series.

Notes and References

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).

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