Let W,S be a finite Coxeter system and H = H(W, R, {us,i}s∈ S,i∈{0,1}) 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 we know a presentation of H makes it easy to check that a list of matrices Ms ∈ Rd × d for s∈ S gives rise to a representation: there is a (unique) representation ρ:H → Rd × d such that ρ(Ts)=Ms for all s∈ S, if and only if the matrices Ms satisfy the same relations as those for the generators Ts of H.
A general approach for the construction of representations is in terms of
W-graphs, see KL79, p.165. Any such W-graph carries a
representation of H. Note that in this approach, it is necessary to know
the square roots of the parameters of H. The simplest example, the
standard W-graph defined in KL79, Ex.~6.2 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=ℚ(vs)s∈ S where the parameters of the corresponding algebra HK satisfy -us,0/us,1=vs2 for all s.
By Tits' Deformation Theorem (see CR87, Sec.~68, for example), the algebra HK is (abstractly) isomorphic to the group algebra of W over K. Moreover, we have a bijection between the irreducible characters of HK and W, given as follows. Let χ be an irreducible character of HK. Then we have χ(Tw) ∈ A where A=ℤ[vs]s∈ S and ℤ denotes the ring of algebraic integers in ℚ. We can find a ring homomorphism f: A → ℚ such that f(a)=a for all a ∈ ℤ and f(vs)=1 for s∈ S. Then it turns out that the function χf : w → f(χ(Tw)) is an irreducible character of W, and the assignment χ → χf defines a bijection between the irreducible characters of HK 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 HK on basis elements Tw lie in the ring ℤ[vs]s∈ S.
The character table of HK is defined to be the square matrix (χ(Tw)) where χ runs over the irreducible characters of HK and w runs over a set of representatives of 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 I2(m), H3, H4 can be constructed from the explicit matrix representations given in CR87, Sec.~67C, 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 GAP3 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 HK by specializing
the indeterminates vi to the variables in rootParameter
.
HeckeReflectionRepresentation( W )
returns a list of matrices which give the reflection representation of
the Iwahori-Hecke algebra corresponding to the Coxeter group W. The
function Hecke
must have been applied to the record W.
gap> v:= X( Rationals );; v.name := "v";; gap> H := Hecke(CoxeterGroup( "B", 2) , v^2, v); Hecke(B2,v^2,v) gap> ref:= HeckeReflectionRepresentation( H ); [ [ [ -v^0, 0*v^0 ], [ -v^2, v^2 ] ], [ [ v^2, -2*v^0 ], [ 0*v^0, -v^0 ] ] ]
gap> H := Hecke( CoxeterGroup( "H", 3 ));; gap> HeckeReflectionRepresentation( H ); [ [ [ -1, 0, 0 ], [ -1, 1, 0 ], [ 0, 0, 1 ] ], [ [ 1, E(5)+2*E(5)^2+2*E(5)^3+E(5)^4, 0 ], [ 0, -1, 0 ], [ 0, -1, 1 ] ], [ [ 1, 0, 0 ], [ 0, 1, -1 ], [ 0, 0, -1 ] ] ]
92.2 CheckHeckeDefiningRelations
CheckHeckeDefiningRelations( H , t )
returns true or false, according to whether a given set t of elements corresponding to the standard generators of Group(H) defines a representation of the Hecke algebra H or not.
gap> H := Hecke(CoxeterGroup( "F", 4 ));; gap> r := HeckeReflectionRepresentation( H );; gap> CheckHeckeDefiningRelations( H, r ); true gap> CheckHeckeDefiningRelations(H,List([1..4],i->Basis(H,"T")(i))); true
92.3 CharTable for Hecke algebras
CharTable
returns the character table record of the Iwahori-Hecke
algebra H. This is basically the same as the character table of a
Coxeter group described earlier with the exception that the component
irreducibles
contains the matrix of the values of the irreducible
characters of the generic Iwahori-Hecke algebra specialized at the
parameters in the component parameter
of H. Thus, if all these
parameters are equal to 1 ∈ ℚ then the component irreducibles
just contains the ordinary character table of the underlying Coxeter
group.
The function CharTable
first recognizes the type of H, then calls
special functions for each type involved in H and finally forms the
direct product of all these tables.
gap> W := CoxeterGroup( "G", 2 );; gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> u := u * v^0;; gap> H := Hecke( W, [ u^2, v^2 ], [ u, v ] ); Hecke(G2,[u^2,v^2],[u,v]) gap> Display( CharTable( H ) ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 A0 ~A1 A1 G2 A2 A1+~A1 2P A0 A0 A0 A2 A2 A0 3P A0 ~A1 A1 A1+~A1 A0 A1+~A1 phi{1,0} 1 v^2 u^2 u^2v^2 u^4v^4 u^6v^6 phi{1,6} 1 -1 -1 1 1 1 phi{1,3}' 1 v^2 -1 -v^2 v^4 -v^6 phi{1,3}'' 1 -1 u^2 -u^2 u^4 -u^6 phi{2,1} 2 v^2-1 u^2-1 -uv -u^2v^2 2u^3v^3 phi{2,2} 2 v^2-1 u^2-1 uv -u^2v^2 -2u^3v^3
As mentioned before, the record components classparam
, classnames
and irredinfo
contain canonical labels and parameters for the classes
and the characters (see the introduction to chapter Classes and
representations for reflection groups and also ChevieCharInfo). For
direct products, sequences of such canonical labels of the individual
types are given.
We can also have character tables for algebras where the parameters are not necessarily indeterminates:
gap> H1 := Hecke( W, [ E(6)^2, E(6)^4 ],[ E(6), E(6)^2 ] ); Hecke(G2,[E3,E3^2],[-E3^2,E3]) gap> ct := CharTable( H1 ); CharTable( "H(G2)" ) gap> Display( ct ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 A0 ~A1 A1 G2 A2 A1+~A1 2P A0 A0 A0 A2 A2 A0 3P A0 ~A1 A1 A1+~A1 A0 A1+~A1 phi{1,0} 1 E3^2 E3 1 1 1 phi{1,6} 1 -1 -1 1 1 1 phi{1,3}' 1 E3^2 -1 -E3^2 E3 -1 phi{1,3}'' 1 -1 E3 -E3 E3^2 -1 phi{2,1} 2 (-3-ER(-3))/2 (-3+ER(-3))/2 1 -1 -2 phi{2,2} 2 (-3-ER(-3))/2 (-3+ER(-3))/2 -1 -1 2 gap> RankMat( ct.irreducibles ); 5
The last result tells us that the specialized character table is no more invertible.
Character tables of Iwahori--Hecke algebras were introduced in GP93; see also the introduction to this chapter for further information about the origin of the various tables.
92.4 Representations for Hecke algebras
This function returns the list of representations of the Iwahori-Hecke algebra H. Each representation is returned as a list of the matrix images of the generators Ts.
If there is a second argument, it can be a list of indices (or a single integer) and only the representations with these indices (or that index) in the list of all representations are returned.
gap> W:=CoxeterGroup("I",2,5); CoxeterGroup("I",2,5) gap> q:=X(Cyclotomics);;q.name:="q";; gap> H:=Hecke(W,q); Hecke(I2(5),q) gap> Representations(H); [ [ [ [ q ] ], [ [ q ] ] ], [ [ [ -q^0 ] ], [ [ -q^0 ] ] ], [ [ [ -q^0, q^0 ], [ 0*q^0, q ] ], [ [ q, 0*q^0 ], [ (-E(5)-2*E(5)^2-2*E(5)^3-E(5)^4)*q, -q^0 ] ] ], [ [ [ -q^0, q^0 ], [ 0*q^0, q ] ], [ [ q, 0*q^0 ], [ (-2*E(5)-E(5)^2 -E(5)^3-2*E(5)^4)*q, -q^0 ] ] ] ]
The models implemented for types Bn and Dn involve rational
fractions, thus work only with algebras whose parameters are Mvp
s.
gap> W:=CoxeterGroup("B",3); CoxeterGroup("B",3) gap> H:=Hecke(W,Mvp("x")); Hecke(B3,x) gap> Representations(H,2); [ [ [ -1, 0, 0 ], [ 0, x, 0 ], [ 0, 0, x ] ], [ [ (-x+x^2)/(1+x), (1+x^2)/(1+x), 0 ], [ 2x/(1+x), (-1+x)/(1+x), 0 ], [ 0, 0, -1 ] ], [ [ -1, 0, 0 ], [ 0, -1/2+1/2x, 1/2+1/2x ], [ 0, 1/2+1/2x, -1/2+1/2x ] ] ]
PoincarePolynomial( H )
The Poincaré polynomial of the Hecke algebra H, which is
equal to SchurElements(H)[ind]
where ind is the position of the
1-dimensional index representation in the character table of H, that is,
the representation which maps Ts to the corresponding parameter
us,0.
gap> q := X( Rationals );; q.name := "q";; gap> W := CoxeterGroup( "G", 2 );; H := Hecke( W, q ); Hecke(G2,q) gap> PoincarePolynomial( H ); q^6 + 2*q^5 + 2*q^4 + 2*q^3 + 2*q^2 + 2*q + 1
92.6 SchurElements for Iwahori-Hecke algebras
SchurElements( H )
returns the list of constants Sχ arising from the Schur relations for the irreducible characters χ of the Iwahori-Hecke algebra H, that is δw,1=∑χ χ(Tw)/Sχ where δ is the Kronecker symbol.
The element Sχ also equal to P/Dχ where P is the
Poincare polynomial and Dχ is the generic degree of χ.
Note, however, that this only works if Dχ ≠ 0. (We can have
Dχ=0 if the parameters of H are suitably chosen roots of
unity, for example.) The ordering of the Schur elements corresponds to
the ordering of the characters as returned by the function CharTable
.
gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> W := CoxeterGroup("G",2);; gap> schur := SchurElements( Hecke( W, [ u ^ 2, v ^ 2 ])); #warning: u*v chosen as 2nd root of (u\^2)*v\^2 [ (u^6 + u^4)*v^6 + (u^6 + 2*u^4 + u^2)*v^4 + (u^4 + 2*u^2 + 1)*v^ 2 + (u^2 + 1), (1 + u^(-2)) + (1 + 2*u^(-2) + u^(-4))*v^( -2) + (u^(-2) + 2*u^(-4) + u^(-6))*v^(-4) + (u^(-4) + u^(-6))*v^( -6), (u^(-4) + u^(-6))*v^6 + (u^(-2) + 2*u^(-4) + u^(-6))*v^4 + ( 1 + 2*u^(-2) + u^(-4))*v^2 + (1 + u^(-2)), (u^2 + 1) + (u^4 + 2*u^2 + 1)*v^(-2) + (u^6 + 2*u^4 + u^2)*v^( -4) + (u^6 + u^4)*v^(-6), (2*u^0)*v^2 + (-2*u + 2*u^(-1))*v + ( 2*u^2 - 2 + 2*u^(-2)) + (2*u - 2*u^(-1))*v^(-1) + (2*u^0)*v^(-2), (2*u^0)*v^2 + (2*u - 2*u^(-1))*v + (2*u^2 - 2 + 2*u^(-2)) + (-2*u + 2*u^(-1))*v^(-1) + (2*u^0)*v^(-2) ]
The Poincaré polynomial is just the Schur element corresponding to the trivial (or index) representation:
gap> schur[PositionId(W)]; (u^6 + u^4)*v^6 + (u^6 + 2*u^4 + u^2)*v^4 + (u^4 + 2*u^2 + 1)*v^ 2 + (u^2 + 1)
(note that the trivial character is not always the first character,
which is why we use PositionId
) For further information about generic
degrees and connections with the representation theory of finite groups
of Lie type, see BC72 and Car85.
92.7 SchurElement for Iwahori-Hecke algebras
SchurElement( H, phi )
returns the Schur element (see Schur Elements for Iwahori-Hecke
algebras
) of the Iwahori-Hecke algebra H for the irreducible
character of H of parameter phi (see CharParams
in section
CHEVIE utility functions).
gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> H := Hecke( CoxeterGroup( "G", 2 ), [ u , v]); Hecke(G2,[u,v]) gap> SchurElement( H, [ [ 1, 3, 1] ] ); (u^(-2) + u^(-3))*v^3 + (u^(-1) + 2*u^(-2) + u^(-3))*v^2 + (1 + 2*u^( -1) + u^(-2))*v + (1 + u^(-1))
We do not have a function for the generic degrees of an Iwahori-Hecke algebra since they are not always defined (for example, if the parameters of the algebra are roots of unity). If they are defined, they can be computed with the command:
List( SchurElements( H ), x -> PoincarePolynomial( H ) / x );
(See PoincarePolynomial and SchurElement.)
92.9 LowestPowerGenericDegrees for Hecke algebras
LowestPowerGenericDegrees( H )
H should be an Iwahori-Hecke algebra all of whose parameters are
monomials in the same indeterminate. LowestPowerGenericDegrees
returns a
list holding the a-function for all irreducible characters of this
algebra, that is, for each character χ, the valuation of the Schur
element of χ. The ordering of the result corresponds to the ordering
of the characters in CharTable(H)
. One should note that this function
first computes explicitly the Schur elements, so for a one-parameter
algebra, LowestPowerGenericDegrees(Group(H))
may be much faster.
gap> q:=X(Rationals);;q.name:="q";; gap> H:=Hecke(CoxeterGroup("B",4),[q^2,q]); Hecke(B4,[q^2,q,q,q]) gap> LowestPowerGenericDegrees(H); [ 7, 6, 7, 12, 20, 3, 5, 3, 7, 6, 13, 2, 3, 10, 1, 4, 2, 7, 0, 3 ]
HeckeCharValuesGood( H, w )
Let H be a Hecke algebra for the Coxeter group CoxeterGroup(H)
,
let w be a good element of CoxeterGroup(H)
in the sense
of GM97 (the representatives of conjugacy classes stored in
CHEVIE are such elements), and let d be the order of w.
HeckeCharValuesGood
computes the values of the irreducible characters
of the Iwahori-Hecke algebra HW on Twd. The point is that the
character table of the Hecke algebra is not used, and that all the
eigenvalues of Twd are monomials in H.parameters
, so this can be
used to find the absolute value of the eigenvalues of Tw, a step
towards computing the character table of the Hecke algebra.
gap> q:=X(Rationals);;q.name:="q";; gap> H:=Hecke(CoxeterGroup("B",4),[q^2,q]); Hecke(B4,[q^2,q,q,q]) gap> HeckeCharValuesGood( H, [ 1, 2, 3 ] ); [ q^12, 4*q^12, 3*q^12 + 3*q^8, 3*q^8 + 1, q^0, 2*q^18 + q^12, 6*q^12, 2*q^18 + 3*q^16 + 3*q^12, 3*q^12 + 3*q^8 + 2*q^6, 3*q^16 + 3*q^8, 2*q^6 + 1, 2*q^18, 3*q^16 + 3*q^12, 2*q^6, q^24 + 2*q^18, 4*q^12, q^24 + 3*q^16, q^12 + 2*q^6, q^24, q^12 ]
gap3-jm