82 Representations of Iwahori-Hecke algebras

Let W be a finite Coxeter group with generators {s_1, ldots, s_n}, and H = H(W, R, {q_i}) a corresponding Iwahori-Hecke algebra over the ring R as defined in chapter Iwahori-Hecke algebras. We shall now describe functions for dealing with representations and characters of H.

The fact that the algebra H is given by a presentation makes it particularly easy to work with representations. Assume we are given any set of matrices M_1,ldots,M_n in R^{d times d}. The fact that H is given by generators and defining relations immediately implies that there is a (unique) representation rho:H rightarrow R^{d times d} such that rho(T_{s_i})=M_i for all i, if and only if the matrices M_i satisfy the same relations as those for the generators T_{s_i} of H.

A general approach for the construction of representations is in terms of W-graphs, see cite[p.165]KL79. Any such W-graph carries a representation of H. Note that, for these purposes, it is necessary to assume that the parameters of H are squares of some elements of the ground ring. The simplest example, the standard W-graph defined in cite[Ex.~6.2]KL79 yields a ``deformation of the natural reflection representation of W. This can be produced in CHEVIE using the function `HeckeReflectionRepresentation`.

Another possibility to construct W-graphs is by using the Kazhdan-Lusztig theory of left cells (see KL79); see the following chapter for more details.

In a similar way as the ordinary character table of the finite Coxeter group W is defined, one also has a character table for the Iwahori-Hecke algebra H in the case when the ground ring A is a field such H is split and semisimple. The generic choice for such a ground ring is the rational function field K=overline{QQ}(v_1, ldots,v_n) where the parameters of the corresponding algebra H_K are given by q_i=v_i^2 for all~i.

By Tits' Deformation Theorem (see cite[Sec.~68]CR87, for example), the algebra H_K is (abstractly) isomorphic to the group algebra of W over K. Moreover, we have a bijection between the irreducible characters of H_K and W, given as follows. Let chi be an irreducible character of H_K. Then we have chi(T_w) in A where A=overline{ZZ}[v_1,ldots,v_n] and overline{ZZ} denotes the ring of algebraic integers in overline{QQ}. We can find a ring homomorphism f colon A rightarrow overline{QQ} such that f(a)=a for all a in overline{ZZ} and f(v_i)=1 for i=1,ldots,n. Then it turns out that the function chi_f colon w mapsto f(chi(T_w)) is an irreducible character of W, and the assignment chi mapsto chi_f defines a bijection between the irreducible characters of H_K and W.

Now this bijection does depend on the choice of f. But one should keep in mind that this only plays a role in the case where W is a non-crystallographic Coxeter group. In all other cases, as is well-known, the character table of W is rational; moreover, the values of the irreducible characters of H_K at basis elements T_w lie in the ring {ZZ}[v_1,ldots,v_n].

The character table of H_K is defined to be the square matrix (chi(T_w)) where chi runs over the irreducible characters of H_K and w runs over a set of representatives of em minimal length in the conjugacy classes of W. The character tables of Iwahori-Hecke algebras (in this sense) are known for all types: the table for type A was first computed by Starkey (see the description of his work in Car86); then different descriptions with different proofs were given in Ram91 and Pfe94b. The tables for the non crystallographic types I_2(m), H_3, H_4 can be constructed from the explicit matrix representations given in cite[Sec.~67C]CR87, Lus81 and AL82, respectively. For the classical types B and D see HR94 and Pfe96. The tables for the remaining exceptional types were computed in Gec94, Gec95 and GM97.

If H is an Iwahori-Hecke algebra over an arbitrary ground ring R as above, then the GAP function `CharTable` applied to the corresponding record returns a character table record which is build up in exactly the same way as for the finite Coxeter group W itself but where the record component `irreducibles` contains the character values which are obtained from those of the generic multi-parameter algebra H_K by specializing the indeterminates v_i to the variables in `sqrtParameters`.

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GAP 3.4.4
April 1997