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`

.

- HeckeReflectionRepresentation
- CheckHeckeDefiningRelations
- CharTable for Hecke algebras
- HeckeCharValues
- HeckeClassPolynomials
- PoincarePolynomial
- SchurElements
- SchurElement
- GenericDegrees
- HeckeCentralMonomials
- HeckeCharValuesGood

`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(CoxeterGroup("B", 2),[ v^2, v^2 ],[ v, 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 ] ] ]

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

`CheckHeckeDefiningRelations( `

returns true or false, according to whether a given set `H` , `t` )`t` of matrices
corresponding to the standard generators of the Coxeter group `Group(H)`
defines a representation of the Iwahori-Hecke algebra `H` or not.

gap> H := Hecke(CoxeterGroup( "F", 4 ));; gap> r := HeckeReflectionRepresentation( H );; gap> CheckHeckeDefiningRelations( H, r ); true

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

`CharTable( `

`H` )

`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 in QQ* 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(CoxeterGroup("G", 2),[ u^2*v^0, v^2 ],[ u*v^0, v ]) gap> Display( CharTable( H ) ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 ~A_1 A_1 G_2 A_2 A_1 + ~A_1 2P A_2 A_2 3P ~A_1 A_1 A_1 + ~A_1 A_1 + ~A_1 phi_{1,0} 1 v^2 (u^2) (u^2)v^2 (u^4)v^4 (u^6)v^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) (u)v (-u^2)v^2 (-2u^3)v^3 phi_{2,2} (2) v^2+(-1) (u^2-1) (-u)v (-u^2)v^2 (2u^3)v^3

As mentioned before, the record components `classparam`

, `classnames`

and
`irredinfo`

contain canonical labels and parameters for the classes and
Character tables for Coxeter 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(CoxeterGroup("G", 2),[ E(3), E(3)^2 ],[ -E(3)^2, E(3) ]) gap> ct := CharTable( H1 ); CharTable( "H(G2)" ) gap> Display( ct ); H(G2) 2 2 2 2 1 1 2 3 1 . . 1 1 1 ~A_1 A_1 G_2 A_2 A_1 + ~A_1 2P A_2 A_2 3P ~A_1 A_1 A_1 + ~A_1 A_1 + ~A_1 phi_{1,0} 1 A /A 1 1 1 phi_{1,6} 1 -1 -1 1 1 1 phi_{1,3}' 1 A -1 -A /A -1 phi_{1,3}'' 1 -1 /A -/A A -1 phi_{2,1} 2 B /B -1 -1 2 phi_{2,2} 2 B /B 1 -1 -2 A = E(3)^2 = (-1-ER(-3))/2 = -1-b3 B = E(3)+2*E(3)^2 = (-3-ER(-3))/2 = -2-b3 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.

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

`HeckeCharValues( `

`T` [,`irreds`])

`T` is an element of an Iwahori-Hecke algebra (expressed in any basis)
and `irreds` is a set of irreducible characters of the algebra (given as
vectors). `HeckeCharValues`

returns the values of `irreds` on the
element `T` (the method used is to convert to the `T`

basis, and then use
`HeckeClassPolynomials`

). If `irreds` is not given, all character values
are returned.

gap> q := X( Rationals );; q.name := "q";; gap> H := Hecke( CoxeterGroup( "B", 2 ), q ^ 2, q );; gap> HeckeCharValues( Basis( H, "C'" )( 1, 2, 1 ) ); [ 0*q^0, q + q^(-1), 0*q^0, q^3 + 2*q + 2*q^(-1) + q^(-3), -q - q^(-1) ]

See also HeckeClassPolynomials.

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

`HeckeClassPolynomials( `

`h` [, `reps`] )

returns the class polynomials of the Hecke element `h` of the Hecke
algebra *H* with respect to representatives `reps` of minimal length in
the conjugacy classes of the Coxeter group `Group(H)`.

If absent, `reps` is taken as `CoxeterConjugacyClasses(Group(H))`

. These
polynomials have the following property. Given the class polynomials `p`
corresponding to `h` and the matrix *X* of the values of the irreducible
characters of the Iwahori-Hecke algebra on *T_w* (for *w* in `reps`),
then the product `X*p`

is the list of values of the irreducible
characters on the element `h` of the Iwahori-Hecke algebra.

gap> u := X( Rationals );; u.name := "u";; gap> W := CoxeterGroup( "A", 3 ); CoxeterGroup("A", 3) gap> H := Hecke( W, u );; gap> h := Basis( H, "T" )( LongestCoxeterElement( W ) ); T(1,2,1,3,2,1) gap> cp := HeckeClassPolynomials( h ); [ 0*u^0, 0*u^0, u^2, u^3 - 2*u^2 + u, u^3 - u^2 + u - 1 ] gap> CharTable( H ).irreducibles * cp; [ u^0, -u^2, 2*u^3, -u^4, u^6 ]

So, the entries in this list are the values of the irreducible characters on the basis element corresponding to the longest element in the Coxeter group.

The class polynomials were introduced in GP93.

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

`PoincarePolynomial( `

`H` )

The Poincaraccent19 e polynomial of the Hecke algebra `H`, which is
equal to `SchurElements(`

where `H`)[`ind`]`ind` is the position of the
1-dimensional index representation in the character table of `H`, that
is, the representation which maps *T_{s_i}* to the corresponding
parameter *q_i*.

gap> q := X( Rationals );; q.name := "q";; gap> W := CoxeterGroup( "G", 2 );; H := Hecke( W, q ); Hecke(CoxeterGroup("G", 2),[ q, q ],[ ]) gap> PoincarePolynomial( H ); q^6 + 2*q^5 + 2*q^4 + 2*q^3 + 2*q^2 + 2*q + 1

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

`SchurElements( `

`H` )

returns the list of constants arising from the Schur relations for the
irreducible characters of the Iwahori-Hecke algebra `H`.

The Schur element corresponding to an irreducible character *chi* is
also equal to *P/D_{chi}* where *P* is the Poincare polynomial and
*D_{chi}* is the generic degree of *chi*. Note, however, that this only
works if *D_{chi} neq 0*. (We can have *D_{chi}=0* if the parameters
of *H* are suitably chosen roots of unity, for example.) The ordering of
the degrees corresponds to the ordering of the characters as returned by
the function `CharTable`

. Note that the Schur element corresponding to
the *ind*-character is the Poincare polynomial *P*.

gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> schur := SchurElements( Hecke( CoxeterGroup( "G", 2 ), > [ u ^ 2, v ^ 2 ], [ u, v ] ) ); [ (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 Poincaraccent19 e polynomial is just the Schur element corresponding to the trivial (or index) representation:

gap> schur[1]; (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)

(But note that the trivial character is not always the first character!) For further information about generic degrees and connections with the representation theory of finite groups of Lie type, see BC72 and Car85.

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

`SchurElement( `

`H`, `phi` )

returns the constants arising from the Schur relations for the
irreducible character `phi` of the Iwahori-Hecke algebra `H`. `phi`
should be specified by its parameter (see `CharParams`

in section
Operations and functions for Coxeter groups) .

gap> u := X( Rationals );; u.name := "u";; gap> v := X( LaurentPolynomialRing( Rationals ) );; v.name := "v";; gap> H := Hecke( CoxeterGroup( "G", 2 ), [ u ^ 2, v ^ 2 ], [ u, v ] ); Hecke(CoxeterGroup("G", 2),[ u^2, v^2 ],[ u, v ]) gap> SchurElement( H, [ [ 1, 3, "'" ] ] ); (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))

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

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 we have a generic multi-parameter
Iwahori-Hecke algebra `H` then the generic degrees are certainly defined,
and they can be computed with the command:

` List( SchurElements( `

`H` ), x - PoincarePolynomial( `H` ) / x );

(See PoincarePolynomial and SchurElement.)

`HeckeCentralMonomials( `

`HW` )

Returns the scalars by which the central element *T_{w_0}^2* acts on
irreducible representations of `HW`.

gap> v := X( Cyclotomics );; v.name := "v";; gap> H := Hecke( CoxeterGroup( "H", 3 ), v ^ 2, v );; gap> HeckeCentralMonomials( H ); [ v^0, v^60, v^24, v^36, v^20, v^20, v^40, v^40, v^30, v^30 ]

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

`HeckeCharValuesGood( `

`HW`, `w` )

Let `HW` be a Hecke algebra for the Coxeter group `CoxeterGroup(`

,
let `HW`)`w` be a good element of `CoxeterGroup(`

in the sense of
GM97, and let `HW`)*d* be the order of *w*.

`HeckeCharValuesGood`

computes the values of the irreducible characters
of the Iwahori-Hecke algebra `HW` on *T_w^d*. The point is that the
character table of the Hecke algebra is not used, and that all the
eigenvalues of *T_w^d* are monomials in `HW.parameters`

, so this can be
used to find the absolute value of the eigenvalues of *T_w*, a step
towards computing the character table of the Hecke algebra.

We continue the example in `HeckeCentralMonomial`

:

gap> HeckeCharValuesGood( H, [ 1, 2, 3 ] ); [ v^0, v^60, 5*v^24, 5*v^36, 3*v^20, 3*v^20, 3*v^40, 3*v^40, 4*v^30, 4*v^30 ]

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

GAP 3.4.4

April 1997