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.

Subsections

  1. HeckeReflectionRepresentation
  2. CheckHeckeDefiningRelations
  3. CharTable for Hecke algebras
  4. HeckeCharValues
  5. HeckeClassPolynomials
  6. PoincarePolynomial
  7. SchurElements
  8. SchurElement
  9. GenericDegrees
  10. HeckeCentralMonomials
  11. HeckeCharValuesGood
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GAP 3.4.4
April 1997