92 Representations of Iwahori-Hecke algebras

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=(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=[vs]s∈ S and denotes the ring of algebraic integers in . We can find a ring homomorphism f: A → such that f(a)=a for all a ∈ 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`.

92.1 HeckeReflectionRepresentation

`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 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```

92.3 CharTable for Hecke algebras

`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 ∈ 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

`Representations( H, l )`

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 ] ] ]```

92.5 PoincarePolynomial

`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))```

92.8 GenericDegrees

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 ]```

92.10 HeckeCharValuesGood

`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
11 Mar 2019