82 Cyclotomic Hecke algebras

The cyclotomic Hecke algebras (Hecke algebras for complex reflection groups) are deformations of the group algebras, generalizing those for real reflection groups (see the next chapter on Iwahori-Hecke algebras).

Their general definition is as a quotient of the algebra of the braid group. We assume now that W is a finite reflection group in the complex vector space V since the theory for infinite groups has not yet been investigated in sufficient generality. The braid group associated is the fundamental group Π1 of the space (V-H∈\cal H H)/W, where \cal H is the set of reflection hyperplanes of W. This group is generated by generators of the monodromy around the hyperplanes, elements which by the natural map from the braid group to the reflection group project to generating (complex) reflections. All generators of the monodromy around all the hyperplanes in a given W-orbit are conjugate. For each such orbit \cal O, let e be the order of the one-dimensional reflection subgroup of W which fixes one of the hyperplanes in \cal O, and let u0 O ,...,ue-1 O be indeterminates. The generic Hecke algebra is the Z[ui O ]i,\cal O-algebra quotient of the braid group algebra by the relations (T-u0 O )...(T-ue-1 O )=0 for a generator T of the monodromy around an hyperplane in \cal O, and in general an Hecke algebra is any algebra obtained from this generic algebra by specializing some of the parameters.

Let s be the reflection which is the image of T in W and let E(e) be its non-trivial eigenvalue. Then the quotient of the Hecke algebra obtained by specializing u0 O ,...,ue-1 O to 1,E(e),E(e)2,...,E(e)e-1 is isomorphic to the group algebra of W. It is actually conjectured that over a suitable ring (such as the algebraic closure of the field of fractions Q(ui O )i,\cal O) the Hecke algebra is itself isomorphic to the group algebra of W over the same ring (this conjecture has been proven in BMR98 for imprimitive groups and exceptional groups G(4), G(5), G(8) and G(25); in addition it is well known to hold for real reflection groups; in the missing cases the ingredient lacking is to show that the dimension of the Hecke algebra does not exceed the cardinality of W).

The cyclotomic Hecke algebras can also been defined in terms of presentations. The braid group is presented by homogeneous relations, called braid relations, described in BMR98 (for the 6 groups G24, G27, G29, G31, G33, G34 the presentations conjectured in BMR98 have either been confirmed or confirmed in a slightly modified form by work of Bessis and Michel using the VKCURVE package of GAP they have written; the case of G31 is still partly conjectural). Further, these relations are such that the reflection group is presented by the same relations, plus relations describing the order of the generators, called the order relations. This allows to define the Hecke algebra by the same presentation as W, with the order relations replaced by a deformed version. Specifically, for each orbit \cal O of reflection hyperplanes of W, let us chose one reflection s of W around one of these hyperplanes with a non-trivial eigenvalue of minimal argument (i.e., of the form E(e) where e is the order of s; then any reflection around an hyperplane of \cal O is a conjugate of a power of s). Let then u0 O ,...,ue-1 O be indeterminates. The generic Hecke algebra is the Z[ui O ]i,\cal O-algebra H with generators Ts in bijection with the generators of W, presented by the braid relations and the deformed order relations (Ts-u0 O )...(Ts-ue-1 O )=0 for each s as above.

Ariki, Koike and Malle have computed character tables for some of these algebras, including all those for 2-dimensional reflection groups, see BM93 and Mal96; CHEVIE contains those for real reflection groups, for types G(e,1,r), G(e,2,2) and for the primitive reflection groups G(4) to G(26), excepted G(24).

A refinement of the conjecture that H has the same dimension as W is that there exists a set {bw}w∈ W of elements of the Braid group such that b1=1 and bw maps to w by the natural quotient map, such that their images Tw form a basis of the Hecke algebra. It is further conjectured that these can be chosen such that the linear form t defined by t(Tw)=0 if w ≠ 1 and t(1)=1 is a symmetrizing form for the symmetric algebra H. This is well known for all real reflection groups and has been proved by Malle and Mathas for imprimitive reflection groups. Then for each irreducible character χ of H we define the Schur element Sχ associated to χ by the condition that for any element T of H we have t(T)=∑χ χ(T)/Sχ. It can be shown that the Schur elements are Laurent polynomials, and they do not depend on the choice of a basis having the above property. Malle has the computed the Schur elements for all algebras associated to irreducible finite reflection groups (where the algebras are defined by their presentations, and modulo the conjecture that some suitable basis satisfies the above conjecture). CHEVIE contains them for all complex reflection groups.

Subsections

  1. Hecke
  2. Operations for cyclotomic Hecke algebras
  3. SchurElements
  4. SchurElement
  5. HeckeCentralMonomials
  6. Representations for cyclotomic Hecke algebras

82.1 Hecke

Hecke( G, para )

Hecke( rec )

returns the cyclotomic Hecke algebra corresponding to the complex reflection group G (see the introduction). The following forms are accepted for para: if para is a single value, it is replicated to become a list of same length as the number of generators of W. Otherwise, para should be a list of the same length as the number of generators of W, with possibly unbound entries (which means it can also be a list of lesser length). There should be at least one entry bound for each orbit of reflections, and if several entries are bound for one orbit, they should all be identical. Now again, an entry for a reflection of order e can be either a single value or a list of length e. If it is a list, it is interpreted as the list [u0,...,ue-1] of parameters for that reflection. If it is a single value q, it is interpreted as the partly specialized list of parameters [q,E(e),...,E(e-1)] (thus the convention is upwardly compatible with that for Coxeter groups, and Hecke(G,1) is the group algebra of G over the cyclotomic field Q({E(e)}e) where e runs over the orders of the generating reflections).

    gap> G := ComplexReflectionGroup(4);
    ComplexReflectionGroup(4)
    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> CH := Hecke( G, v );
    Hecke(ComplexReflectionGroup(4),v)
    gap> CH.parameter;
    [ [ v, E(3), E(3)^2 ], [ v, E(3), E(3)^2 ] ]

Here the single parameter v is interpreted as [v,v] which is in turn interpreted according to the above rules as [[v,E(3),E(3)^2],[v,E(3),E(3)^2]].

The second form of the function Hecke takes as an argument a record which has a field hecke and returns the value of this field. This is used to return the Hecke algebra of objects derived from Hecke algebras, such as Hecke elements in various bases.

This function requires the package "chevie" (see RequirePackage).

82.2 Operations for cyclotomic Hecke algebras

Group:

returns the complex reflection group from which the cyclotomic Hecke algebra was generated.

Print:

prints the cyclotomic Hecke algebra in a form which can be read back into GAP.

CharTable:

returns the character table for some types of cyclotomic Hecke algebras, namely those of imprimitive type and the primitive reflection groups numbered G(4) to G(26) in the Shephard-Todd classification. This is a record with exactly the same components as for the corresponding complex reflection group but where the component irreducibles contains the values of the irreducible characters of the algebra on certain basis elements Tw where w runs over the elements in the component classtext. Thus, the values are now polynomials in the parameters of the algebra.

    gap> G := ComplexReflectionGroup( 4 );
    ComplexReflectionGroup(4)
    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> CH := Hecke( G, v );
    Hecke(ComplexReflectionGroup(4),v)
    gap> Display( CharTable( CH ) );
    H(G4)

              2 3     3   2     1        1      1            1
              3 1     1   .     1        1      1            1

                .     z 212    12      z12      1           1z
             2P .     .   z     1        1    z12          z12
             3P .     z 212     z        .      .            z
             5P .     z 212    1z        1    z12           12

    phi{1,0}    1   v^6 v^3   v^2      v^8      v          v^7
    phi{1,4}    1     1   1  E3^2     E3^2     E3           E3
    phi{1,8}    1     1   1    E3       E3   E3^2         E3^2
    phi{2,5}    2    -2   .     1       -1     -1            1
    phi{2,3}    2 -2v^3   . E3^2v -E3^2v^4 v+E3^2 -v^4-E3^2v^3
    phi{2,1}    2 -2v^3   .   E3v   -E3v^4   v+E3   -v^4-E3v^3
    phi{3,2}    3  3v^2  -v     .        .    v-1      v^3-v^2

This function requires the package "chevie" (see RequirePackage).

82.3 SchurElements

SchurElements( H )

returns the list Schur elements for the (cyclotomic) Hecke algebra H (see the introduction for their definition).

      gap> v:=X(Cyclotomics);;v.name:="v";;
      gap> H:=Hecke(ComplexReflectionGroup(4),v);
      Hecke(ComplexReflectionGroup(4),v)
      gap> SchurElements(H);
      [ v^8 + 2*v^7 + 3*v^6 + 4*v^5 + 4*v^4 + 4*v^3 + 3*v^2 + 2*v + 1,
        (2*E(3)-2*E(3)^2) + (-2*E(3)-10*E(3)^2)*v^(-1) + 12*v^(-2) + 
        (-10*E(3)-2*E(3)^2)*v^(-3) + (-2*E(3)+2*E(3)^2)*v^(-4),
        (-2*E(3)+2*E(3)^2) + (-10*E(3)-2*E(3)^2)*v^(-1) + 12*v^(-2) + 
        (-2*E(3)-10*E(3)^2)*v^(-3) + (2*E(3)-2*E(3)^2)*v^(-4),
        2 + 2*v^(-1) + 4*v^(-2) + 2*v^(-3) + 2*v^(-4),
        (-2*E(3)-E(3)^2)*v^3 + (-4*E(3)-2*E(3)^2)*v^2 + 3*v + 
        (-2*E(3)-4*E(3)^2) + (-E(3)-2*E(3)^2)*v^(-1),
        (-E(3)-2*E(3)^2)*v^3 + (-2*E(3)-4*E(3)^2)*v^2 + 3*v + 
        (-4*E(3)-2*E(3)^2) + (-2*E(3)-E(3)^2)*v^(-1),
        v^2 + 2*v + 2 + 2*v^(-1) + v^(-2) ]
      gap> List(last,CycPol);
      [ P2^2P3P4P6, 2ER(-3)v^-4P2^2P'3P'6, -2ER(-3)v^-4P2^2P"3P"6, 
      2v^-4P3P4, ((3-ER(-3))/2)v^-1P2^2P'3P"6, 
      ((3+ER(-3))/2)v^-1P2^2P"3P'6, v^-2P2^2P4 ]

This function requires the package "chevie" (see RequirePackage).

82.4 SchurElement

SchurElement( H, phi )

returns the Schur element (see SchurElements) of the Cyclotomic Hecke algebra H for the irreducible character of H of parameter phi (see CharParams in section Appendix -- utility functions of CHEVIE);

    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> W:=ComplexReflectionGroup(4);;
    gap> H := Hecke( W, v);
    Hecke(ComplexReflectionGroup(4),v)
    gap> SchurElement( H, [ [ 2, 5] ] );
    2 + 2*v^(-1) + 4*v^(-2) + 2*v^(-3) + 2*v^(-4)

This function requires the package "chevie" (see RequirePackage).

82.5 HeckeCentralMonomials

HeckeCentralMonomials( HW )

Returns the scalars by which the central element Tπ acts on irreducible representations of HW. Here, for an irreducible group, π is the generator of the center of the pure braid group, which is also z|Z| where z is the generator of the center of the braid group and |Z| the order of the center of W. In the case of an Iwahori-Hecke algebra, Tπ is thus Tw02.

    gap> v := X( Cyclotomics );; v.name := "v";;
    gap> H := Hecke( CoxeterGroup( "H", 3 ),  v );;
    gap> HeckeCentralMonomials( H );
    [ v^0, v^30, v^12, v^18, v^10, v^10, v^20, v^20, v^15, v^15 ]

This function requires the package "chevie" (see RequirePackage).

82.6 Representations for cyclotomic Hecke algebras

Representations( H, l )

This function returns the list of representations of the algebra algebra H. Each representation is returned as a list of the matrix images of the generators. For the moment this function is only implemented for algebras whose irreducible components are of type G4 to G23 or imprimitive or of Coxeter type. A partial list is given also for types G24 to G34.

If there is a second argument, it must be a list of indices, and only the representations with these indices in the list of all representations are returned.

    gap> W:=ComplexReflectionGroup(4);;
    gap> q:=X(Cyclotomics);;q.name:="q";;
    gap> H:=Hecke(W,q);
    Hecke(ComplexReflectionGroup(4),q)
    gap> Representations(H);
    [ [ [ [ q ] ], [ [ q ] ] ], [ [ [ E(3)*q^0 ] ], [ [ E(3)*q^0 ] ] ], 
      [ [ [ E(3)^2*q^0 ] ], [ [ E(3)^2*q^0 ] ] ], 
      [ [ [ E(3)*q^0, 0*q^0 ], [ -E(3)*q^0, E(3)^2*q^0 ] ], 
	  [ [ E(3)^2*q^0, E(3)^2*q^0 ], [ 0*q^0, E(3)*q^0 ] ] ], 
      [ [ [ q, 0*q^0 ], [ -q, E(3)^2*q^0 ] ], 
	  [ [ E(3)^2*q^0, E(3)^2*q^0 ], [ 0*q^0, q ] ] ], 
      [ [ [ q, 0*q^0 ], [ -q, E(3)*q^0 ] ], 
	  [ [ E(3)*q^0, E(3)*q^0 ], [ 0*q^0, q ] ] ], 
      [ [ [ E(3)^2*q^0, 0*q^0, 0*q^0 ], 
	      [ (E(3)^2)*q + (E(3)^2), E(3)*q^0, 0*q^0 ], 
	      [ E(3)*q^0, q^0, q ] ], 
	  [ [ q, -q^0, E(3)*q^0 ], [ 0*q^0, E(3)*q^0, 
		  (-E(3)^2)*q + (-E(3)^2) ], 
	      [ 0*q^0, 0*q^0, E(3)^2*q^0 ] ] ] ]

This function requires the package "chevie" (see RequirePackage).

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GAP 3.4.4
April 1997