93 Kazhdan-Lusztig polynomials and bases

Let H be the Iwahori-Hecke algebra of a Coxeter system (W,S), with quadratic relations (Ts-us,0)(Ts-us,1)=0 for s∈ S. If -us,0/us,1 has a square root vs, we can scale the basis Ts to get a new basis ts=Ts/(-vs us,1) with quadratic relations (ts-vs)(ts+vs-1)=0. The most general case when Kazhdan-Lusztig bases and polynomials can be defined is when the parameters vs belong to a totally ordered abelian group Γ (where the group law is multiplication of parameters), see Lus83. We set Γ+= {γ∈Γ| γ>0} and Γ-= {γ-1 |γ∈Γ+} ={γ∈Γ| γ<0}.

Thus we assume H defined over the ring ℤ[Γ], the group algebra of Γ over , and the quadratic relations of H associate to each s∈ S a vs∈Γ+ such that (ts-vs)(ts+vs-1)=0. We also set qs=vs2 and define the basis Ts=vs ts with quadratic relations (Ts-qs)(Ts+1)=0; we extend the notation to define an element qw∈Γ+ by setting qw=qs1... qsn if w=s1... sn is a reduced expression for some w∈ W, and denote qw(1/2)=vs1... vsn.

We define the bar involution on H by linearity: on ℤ[Γ] we define it by γ∈Γaγγ= ∑γ∈Γ aγ γ-1 and we extend it to H by Ts=Ts-1. Then the Kazhdan-Lusztig basis C'w is defined as the only basis of H stable by the bar involution and congruent to tw modulo w∈ WΓ- tw.

The basis C'w can be computed as follows. We define elements Rx,y of ℤ[Γ] by Ty-1=∑x Rx,y-1 qx-1Tx. We then define inductively the Kazhdan-Lusztig polynomials (in this general context we should say the Kazhdan-Lusztig elements of ℤ[Γ], which belong to the subalgebra of ℤ[Γ] generated by the qs) by Px,w ≤(qw/qx)1/2 (∑x<y ≤ wRx,yPy,w) where τ is the truncation : τ ≤νγ∈Γ aγγ= ∑γ ≤νaγγ; the induction is thus on decreasing x for the Bruhat order and starts at Pw,w=1. We have then C'w=∑y qw-1/2 Py,wTy.

CHEVIE can compute Kazhdan-Lusztig polynomials, left cells, and the various Kazhdan-Lusztig bases of Iwahori-Hecke algebras (see KL79). More facilities are implemented for the one-parameter case when all vs have a common value v.

From a computational point of view, Kazhdan-Lusztig polynomials are quite a challenge, even in the one-parameter case. For this case it seems that the best approach still is by using the recursion formula in the original article KL79 (which deals with the one-parameter case, where the above recursion simplifies). One can first run a number of standard checks on a given pair of elements to see if the computation of the corresponding polynomial can be reduced to a similar computation for elements of smaller length, for example. One such check involves the notion of critical pairs (cf. Alv87): We say that a pair of elements w1, w2 ∈ W such that w1 ≤ w2 is critical if L(w2) ⊆ L(w1) and R(w2) ⊆ R(w1), where L and R denote the left and right descent set, respectively. Now if y,w ∈ W are arbitrary elements with y ≤ w then there always exists a critical pair (z,w) with y ≤ z ≤ w such that the Kazhdan-Lusztig polynomials Py,w and Pz,w are equal. Given two elements y and w, such a critical pair is found by the function CriticalPair. The CHEVIE programs for computing Kazhdan-Lusztig polynomials are organized in such a way that whenever the polynomial corresponding to a critical pair is computed then this pair and the polynomial are stored in the component criticalPairs of the record of the underlying Coxeter group.

A good example to see how long the programs will take for computations in big Coxeter groups is the following:

    gap> W:=CoxeterGroup("D",5);;
    gap> LeftCells(W);;

which takes 10 seconds cpu time on 3Ghz computer. The computation of all Kazhdan-Lusztig polynomials for type F4 takes a bit more than 1 minute. Computing the Bruhat order is a bottleneck for these computations; they can be speeded up by a factor of two if one does:

    gap> ReadChv("contr/brbase");
    gap> BaseBruhat(W);;

after which the computation of the Bruhat order will be speeded up by a large factor.

However, Alvis' computation of the Kazhdan--Lusztig polynomials of the Coxeter group of type H4 in a computer algebra system like GAP3 would still take many hours. For such applications, it is probably more efficient to use a special purpose program like the one provided by F. DuCloux DuC91.

The code for the Kazhdan-Lusztig bases C, D and their primed versions has been written by Andrew Mathas, who also contributed to the initial implementation and to the design of the programs dealing with Kazhdan-Lusztig bases. He also implemented some other bases, such as the Murphy basis which can be found in the contributions directory (see also his Specht package). The extension to the unequal parameters case has been written by F.Digne and J.Michel.

The other Kazhdan-Lusztig bases are computed in CHEVIE in terms of the C' basis.

CHEVIE is able to define automatically the bar and truncation operations on ℤ(Γ) when all parameters are powers of the same indeterminate q, with total order on Γ by the power of q, or when the parameters are monomials in some Mvps, with the lexicographic order. The bar involution is evaluating a Laurent polynomial at the inverse of the variables, and truncation is keeping terms of smaller degree than that of ν. It is possible to use arbitrary groups Γ by doing the following steps: first, define the Hecke algebra H. Then, before defining any of the Kazhdan-Lusztig bases, write functions H.Bar(p), H.PositivePart(p) and H.NegativePart(p) which perform the operations respectively γ∈Γ aγγ→ ∑γ∈Γ aγγ-1, γ∈Γ aγγ→ ∑γ ≥ 1 aγγ and γ∈Γ aγγ→ ∑γ ≤ 1 aγγ on elements p of ℤ[Γ]. It is then possible to define Kahzdan-Lusztig bases and the operations above will be used internally by CHEVIE to compute them.

Subsections

  1. KazhdanLusztigPolynomial
  2. CriticalPair
  3. KazhdanLusztigCoefficient
  4. KazhdanLusztigMue
  5. LeftCells
  6. LeftCell
  7. Functions for LeftCells
  8. W-Graphs
  9. WGraph
  10. WGraphToRepresentation
  11. Hecke elements of the C basis
  12. Hecke elements of the primed C basis
  13. Hecke elements of the D basis
  14. Hecke elements of the primed D basis
  15. Asymptotic algebra
  16. Lusztigaw
  17. LusztigAw

93.1 KazhdanLusztigPolynomial

KazhdanLusztigPolynomial( W, y, w)

returns the coefficients of the Kazhdan-Lusztig polynomial Py,w(q) attached to the elements y and w of the Coxeter group W and to Hecke(W,q). If one prefers to give as input just two Coxeter words, one can define a new function as follows (for example):

    gap> klpol := function( W, x, y)
    >   return KazhdanLusztigPolynomial(W, EltWord(W, x), EltWord(W, y));
    >   end;
    function ( W, x, y ) ... end 

We use this function in the following example where we compute the polynomials P1,w for all elements w in the Coxeter group of type A3.

    gap> q := X( Rationals );; q.name := "q";;
    gap> W := CoxeterGroup( "B", 3 );;
    gap> el := CoxeterWords( W );
    [ [  ], [ 3 ], [ 2 ], [ 1 ], [ 3, 2 ], [ 2, 1 ], [ 2, 3 ], [ 1, 3 ],
      [ 1, 2 ], [ 2, 1, 2 ], [ 3, 2, 1 ], [ 2, 3, 2 ], [ 2, 1, 3 ],
      [ 1, 2, 1 ], [ 1, 3, 2 ], [ 1, 2, 3 ], [ 3, 2, 1, 2 ],
      [ 2, 1, 2, 3 ], [ 2, 3, 2, 1 ], [ 2, 1, 3, 2 ], [ 1, 2, 1, 2 ],
      [ 1, 3, 2, 1 ], [ 1, 2, 1, 3 ], [ 1, 2, 3, 2 ], [ 3, 2, 1, 2, 3 ],
      [ 2, 1, 2, 3, 2 ], [ 2, 3, 2, 1, 2 ], [ 2, 1, 3, 2, 1 ],
      [ 1, 3, 2, 1, 2 ], [ 1, 2, 1, 2, 3 ], [ 1, 2, 1, 3, 2 ],
      [ 1, 2, 3, 2, 1 ], [ 2, 3, 2, 1, 2, 3 ], [ 2, 1, 2, 3, 2, 1 ],
      [ 2, 1, 3, 2, 1, 2 ], [ 1, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2 ],
      [ 1, 2, 1, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 2 ], [ 2, 1, 2, 3, 2, 1, 2 ], 
      [ 2, 1, 3, 2, 1, 2, 3 ], [ 1, 2, 3, 2, 1, 2, 3 ], 
      [ 1, 2, 1, 2, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1, 2 ], 
      [ 2, 1, 2, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2, 1, 2 ], 
      [ 1, 2, 1, 3, 2, 1, 2, 3 ], [ 1, 2, 1, 2, 3, 2, 1, 2, 3 ] ]
    gap> List( el, w -> Polynomial(Rationals,klpol( W, [], w )));
    [ q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0, q^0,
      q^0, q^0, q^0, q^0, q^0, q^0, q + 1, q^0, q^0, q^0, q^0, q + 1,
      q^0, q^0, q + 1, q^0, q^0, q + 1, q + 1, q^0, q + 1, q^0, q + 1,
      q^0, q^2 + 1, q + 1, q^2 + q + 1, q + 1, q + 1, q^0, q^0, q^2 + 1,
      q^0, q + 1, q^0 ] 

Kazhdan--Lusztig polynomials for critical pairs are stored in the record component klpol of W, which allows the function to work much faster after the first time it is called.

93.2 CriticalPair

CriticalPair( W, y, w )

Given elements y and w in the Coxeter group W the function CriticalPair returns the longest element in the double coset W L(w)y W R(w); it is such that the Kazhdan--Lusztig polynomials Pz,w and Py,w are equal.

    gap> W := CoxeterGroup( "F", 4 );
    CoxeterGroup("F",4)
    gap> w := LongestCoxeterElement( W ) * W.generators[1];;
    gap> CoxeterLength( W, w );
    23
    gap> y := EltWord( W, [ 1, 2, 3, 4 ] );;
    gap> cr := CriticalPair( W, y, w );;
    gap> CoxeterWord( W, cr);
    [ 2, 3, 2, 1, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 3 ]
    gap> KazhdanLusztigPolynomial( W, y, w);
    [ 1, 0, 0, 1 ]
    gap> KazhdanLusztigPolynomial( W, cr, w);
    [ 1, 0, 0, 1 ]

93.3 KazhdanLusztigCoefficient

KazhdanLusztigCoefficient( W, y, w, k )

returns the coefficient of qk in the Kazhdan-Lusztig polynomial Py,w attached to the elements y and w of the Coxeter group W and to Hecke(W,q).

    gap> W := CoxeterGroup( "B", 4 );;
    gap> y := [ 1, 2, 3, 4, 3, 2, 1 ];;
    gap> py := EltWord( W, y );
    ( 1,28)( 2,15)( 4,27)( 6,16)( 7,24)( 8,23)(11,20)(12,17)(14,30)(18,31)
    (22,32)
    gap> x := [ 1 ];;
    gap> px := EltWord( W, x );
    ( 1,17)( 2, 8)( 6,11)(10,14)(18,24)(22,27)(26,30)
    gap> Bruhat( W, px, py );
    true
    gap> List([0..3],i->KazhdanLusztigCoefficient( W, px, py, i ) );
    [ 1, 2, 1, 0 ] 

So the Kazhdan-Lusztig polynomial corresponding to x and y is 1+2q+q2.

93.4 KazhdanLusztigMue

KazhdanLusztigMue( W, y, w )

given elements y and w in the Coxeter group W, this function returns the coefficient of degree (l(w)-l(y)-1)/2 of the Kazhdan-Lusztig polynomial Py,w.

Of course, the result of this function could also be obtained by

KazhdanLusztigCoefficient(W,y,w,(CoxeterLength(W,w)-CoxeterLength(W,y)-1)/2)

but there are some speed-ups compared to this general function.

93.5 LeftCells

LeftCells( W [, i])

returns a list of records describing left cells of W for Hecke(W,q). The program uses precomputed data(see GH14) for exceptional types and for type A, so is quite fast for these types (it takes 32 seconds to compute the 101796 left cells for type E8). For other types, left cells are computed from first principles, thus computing many Kazhdan-Lusztig polynomials. It takes 10 seconds to compute the left cells of D5, for example.

    gap> W := CoxeterGroup( "G", 2 );;
    gap> LeftCells(W);
    [ LeftCell<G2: duflo= character=phi{1,0}>, 
      LeftCell<G2: duflo=1,2 character=phi{1,6}>,
      LeftCell<G2: duflo=2 character=phi{2,1}+phi{1,3}'+phi{2,2}>, 
      LeftCell<G2: duflo=1 character=phi{2,1}+phi{1,3}''+phi{2,2}> ]

Printing such a record displays the character afforded by the left cell and its Duflo involution; the Duflo involution r is printed as a subset I of [1..W.N] such that r=LongestCoxeterElement(ReflectionSubgroup(W,I)), see DescribeInvolution.

If a second argument i is given, the program returns only the left cells which are in the i-th two-sided cell, that is whose character is in the i-th family of W (see Families of unipotent characters).

    gap> LeftCells(W,1);
    [ LeftCell<G2: duflo=2 character=phi{2,1}+phi{1,3}'+phi{2,2}>, 
      LeftCell<G2: duflo=1 character=phi{2,1}+phi{1,3}''+phi{2,2}> ]

93.6 LeftCell

LeftCell( W, x)

returns a record describing the left cell of W for Hecke(W,q) containing element x.

    gap> W := CoxeterGroup( "E", 8 );;
    gap> LeftCell(W, Random(W));
    LeftCell<E8: duflo=6,11,82,120 character=phi{2268,30}+phi{1296,33}>

93.7 Functions for LeftCells

Size( cell)

Returns the number of elements of the cell.

    gap> W:=CoxeterGroup( "H", 3);;
    gap> c := LeftCells( W );;
    gap> List( c, Size);
    [ 1, 6, 5, 8, 5, 6, 1, 5, 8, 5, 5, 6, 6, 5, 8, 5, 5, 8, 5, 6, 6, 5 ]

Elements( cell)

Returns the list of elements of the cell.

The operations in and = are defined for left cells.

Representation( cell, H )

returns a list of matrices giving the representation of Hecke(W,v^2,v) on the left cell c.

    gap> v := X( Cyclotomics ) ;; v.name := "v";;
    gap> H := Hecke(W, v^2, v );
    Hecke(H3,v^2,v)
    gap> Representation(c[3],H);
    [ [ [ -v^0, 0*v^0, 0*v^0, 0*v^0, 0*v^0 ], 
        [ 0*v^0, -v^0, 0*v^0, 0*v^0, v ], 
        [ 0*v^0, 0*v^0, -v^0, v, v ], 
        [ 0*v^0, 0*v^0, 0*v^0, v^2, 0*v^0 ], 
        [ 0*v^0, 0*v^0, 0*v^0, 0*v^0, v^2 ] ], 
      [ [ -v^0, v, 0*v^0, 0*v^0, 0*v^0 ], 
        [ 0*v^0, v^2, 0*v^0, 0*v^0, 0*v^0 ], 
        [ 0*v^0, 0*v^0, v^2, 0*v^0, 0*v^0 ], 
        [ 0*v^0, 0*v^0, v, -v^0, 0*v^0 ], 
        [ 0*v^0, v, v, 0*v^0, -v^0 ] ], 
      [ [ v^2, 0*v^0, 0*v^0, 0*v^0, 0*v^0 ], 
        [ v, -v^0, 0*v^0, 0*v^0, 0*v^0 ], 
        [ 0*v^0, 0*v^0, -v^0, v, 0*v^0 ], 
        [ 0*v^0, 0*v^0, 0*v^0, v^2, 0*v^0 ], 
        [ 0*v^0, 0*v^0, 0*v^0, 0*v^0, -v^0 ] ] ]

Character(c)

Returns a list l such that the character of W afforded by the left cell c is Sum(CharTable(W).irreducibles{l}).

    gap> Character(c[13]); 
    [ 6, 5 ]

See also WGraph below. When Character(c) has been computed, then c.a also has been bound which holds the common value of Lusztig's a-function (see LowestPowerGenericDegrees) for

• The elements of c.

• The irreducible constituents of Character(c).

93.8 W-Graphs

Let H be the 1-parameter Hecke algebra with parameter q associated to the Coxeter system (W,S). A W-graph encodes a representation of H of the kind which is constructed by Kazhdan-Lusztig theory. It consists of a basis V (the vertices of the graph) of a real vector space with a function x→ I(x) from V to the subsets of S and a function μ: V2→ℝ (the nonzero values are thus labels for the edges of the graph). This defines the representation of H given in the basis V by the formulae

Ts(x)={
-x if s∈ I(x)
q x+∑{y∈ V| s∈ I(y)} √ qμ(y,x)y otherwise.
.

There are two points to W-graphs. First, they describe nice, sparse, integral representations of H (and thus of W also). Second, they can be stored very compactly; for example, for the representation of dimension 7168 of the Hecke algebra of type E8, a naive implementation would take more than a gigabyte. The corresponding W-graph takes 500KB.

W-graphs are represented in CHEVIE as a pair.

• The first element is a list describing C; its elements are either a set I(x), or an integer n specifying to repeat the previous element n more times.

• The second element is a list which specifies μ. We first describe the μ-list for symmetric W-graphs (when μ(x,y)=μ(y,x)). There is one element of the μ-list for each non-zero value m taken by μ, which consists of a pair whose first element is m and whose second element is a list of lists; if l is one of these lists each pair [l[1],l[i]] represents an edge x=l[1],y=l[i] such that μ(x,y)=μ(y,x)=m. For non-symmetric W-graphs, the first element of each pair in the μ-list is a pair [m,n] and each edge [x,y] obtained from the lists in the second element has to be interpreted as μ(x,y)=m and μ(y,x)=n.

Here is an example of graph for a Coxeter group, and the corresponding representation. Here v is a variable representing the square root of q.

    gap> W:=CoxeterGroup("H",3);;
    gap> WGraph(W,3);                     
    [ [ [ 2 ], [ 1, 2 ], [ 1, 3 ], [ 1, 3 ], [ 2, 3 ] ], 
      [ [ -1, [ [ 1, 3 ], [ 2, 4 ], [ 3, 5 ], [ 4, 5 ] ] ] ] ]
    gap> WGraphToRepresentation(3,last,Mvp("v"));
    [ [ [ v^2, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0 ], [ -v, 0, -1, 0, -v ], 
        [ 0, 0, 0, -1, -v ], [ 0, 0, 0, 0, v^2 ] ], 
      [ [ -1, 0, -v, 0, 0 ], [ 0, -1, 0, -v, 0 ], [ 0, 0, v^2, 0, 0 ], 
        [ 0, 0, 0, v^2, 0 ], [ 0, 0, -v, -v, -1 ] ], 
      [ [ v^2, 0, 0, 0, 0 ], [ 0, v^2, 0, 0, 0 ], [ -v, 0, -1, 0, 0 ], 
        [ 0, -v, 0, -1, 0 ], [ 0, 0, 0, 0, -1 ] ] ]

93.9 WGraph

WGraph( W, i )

W should be a finite Coxeter group. Returns the W-graph for the i-th representation of the one-parameter Hecke algebra of W (or the i-th representation of W). For the moment this is only implemented for irreducible groups of exceptional type E, F, G, H.

WGraph( c )

c should be a left cell for the one-parameter Hecke algebra of a finite Coxeter group W. Returns the corresponding W-graph.

93.10 WGraphToRepresentation

WGraphToRepresentation (r, graph, v)

graph should be a W-graph for some finite Coxeter group W of semisimple rank r. The function returns the r matrices defining the representation defined by graph of the Hecke algebra for W with parameters -1 and v2.

    gap> W:=CoxeterGroup("H",3);;
    gap> g:=WGraph(W,3);                     
    [ [ [ 2 ], [ 1, 2 ], [ 1, 3 ], [ 1, 3 ], [ 2, 3 ] ], 
      [ [ -1, [ [ 1, 3 ], [ 2, 4 ], [ 3, 5 ], [ 4, 5 ] ] ] ] ]
    gap> WGraphToRepresentation(3,g,Mvp("v"));
    [ [ [ v^2, 0, 0, 0, 0 ], [ 0, -1, 0, 0, 0 ], [ -v, 0, -1, 0, -v ], 
        [ 0, 0, 0, -1, -v ], [ 0, 0, 0, 0, v^2 ] ], 
      [ [ -1, 0, -v, 0, 0 ], [ 0, -1, 0, -v, 0 ], [ 0, 0, v^2, 0, 0 ], 
        [ 0, 0, 0, v^2, 0 ], [ 0, 0, -v, -v, -1 ] ], 
      [ [ v^2, 0, 0, 0, 0 ], [ 0, v^2, 0, 0, 0 ], [ -v, 0, -1, 0, 0 ], 
        [ 0, -v, 0, -1, 0 ], [ 0, 0, 0, 0, -1 ] ] ]

WGraphToRepresentation(H, graph)

H should be a one-parameter Hecke algebra for a finite Coxeter group. The function returns the matrices of the representation defined by graph of H.

    gap> H:=Hecke(W,[[Mvp("v"),-Mvp("v")^-1]]);
    Hecke(H3,[[v,-v^-1]])
    gap> WGraphToRepresentation(H,g);
    [ [ [ v, 0, 0, 0, 0 ], [ 0, -v^-1, 0, 0, 0 ], [ -1, 0, -v^-1, 0, -1 ], 
      [ 0, 0, 0, -v^-1, -1 ], [ 0, 0, 0, 0, v ] ], 
    [ [ -v^-1, 0, -1, 0, 0 ], [ 0, -v^-1, 0, -1, 0 ], [ 0, 0, v, 0, 0 ], 
        [ 0, 0, 0, v, 0 ], [ 0, 0, -1, -1, -v^-1 ] ], 
    [ [ v, 0, 0, 0, 0 ], [ 0, v, 0, 0, 0 ], [ -1, 0, -v^-1, 0, 0 ], 
        [ 0, -1, 0, -v^-1, 0 ], [ 0, 0, 0, 0, -v^-1 ] ] ]

93.11 Hecke elements of the C basis

Basis( H, "C" )

returns a function which gives the C-basis of the Iwahori-Hecke algebra H. The parameters of H should be powers of a single indeterminate or Mvps (see the introduction). This basis is defined as follows (see e.g. Lus85, (5.1)). Let W be the underlying Coxeter group. For x,y ∈ W let Px,y be the corresponding Kazhdan--Lusztig polynomial. If {Tw | w∈ W} denotes the usual T-basis, then Cx:=∑y ≤ x (-1)l(x)-l(y)Py,x(q-1)qx1/2qy-1 Ty for x ∈ W. For example, we have Cs=qs-1/2Ts-qs1/2T1 for s ∈ S. Thus, the transformation matrix between the T-basis and the C-basis is lower unitriangular, with powers of v along the diagonal. In the one-parameter case (all qs are equal to v2) the multiplication rules for the C basis are given by:

Cs . Cx ={
-(v+v-1)Cx , if sx<x
Csx+∑y μ(y,x)Cy , if sx>x
.
where the sum is over all y such that y<x, l(y) ≢ l(x) mod 2 and sy<y. The coefficient μ(y,x) is the coefficient of degree (l(x)-l(y)-1)/2 in the Kazhdan--Lusztig polynomial Px,y.

    gap> W := CoxeterGroup( "B", 3 );;
    gap> v := X( Rationals );; v.name := "v";;
    gap> H := Hecke( W, v^2, v );
    Hecke(B3,v^2,v)
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> C := Basis( H, "C" );
    function ( arg ) ... end
    gap> T( C( 1 ) );
    -vT()+v^-1T(1)
    gap> C( T( 1 ) );
    v^2C()+vC(1) 

We can also compute character values on elements in the C-basis as follows:

    gap> ref := HeckeReflectionRepresentation( H );;
    gap> c := CharRepresentationWords( ref, WordsClassRepresentatives( W ) );
    [ 3*v^0, 2*v^2 - 1, v^8 - 2*v^4, -3*v^12, 2*v^2 - 1, v^4, 
     v^4 - 2*v^2, -v^6, v^4 - v^2, 0*v^0 ]
    gap> List( ChevieClassInfo( W ).classtext, i ->
    >                             HeckeCharValues( C( i ), c ) );
    [ 3*v^0, -v - v^(-1), 0*v^0, 0*v^0, -v - v^(-1), 2*v^0, 0*v^0, 0*v^0, 
      v^0, 0*v^0 ]

93.12 Hecke elements of the primed C basis

Basis( H, "C'" )

returns a function which gives the C'-basis of the Iwahori-Hecke algebra H (see Lus85, (5.1)) The parameters of H should be powers of a single indeterminate or monomials in Mvp's (see the introduction). This basis is defined by

Cx' := ∑y ≤ xPy,xqx-1/2 Ty for x ∈ W.
We have Cx'=(-1)l(x)Alt(Cx) for all x ∈ W (see AltInvolution in section "Operations for Hecke elements of the T basis").

    gap>  v := X( Rationals );; v.name := "v";;
    gap>  H := Hecke( CoxeterGroup( "B", 2 ), [v ^4, v^2] );;
    gap>  h := Basis( H, "C'" )( 1 );
    #warning: C' basis: v\^2 chosen as 2nd root of v\^4
    C'(1)
    gap>  h2 := h * h;
    (v^2+v^-2)C'(1)
    gap>  Basis( H, "T" )( h2 );
    (1+v^-4)T()+(1+v^-4)T(1)
    gap> Basis(H,"C'")(last);
    (v^2+v^-2)C'(1)

93.13 Hecke elements of the D basis

Basis( H, "D" )

returns a function which gives the D-basis of the (one parameter generic) Iwahori-Hecke algebra H (see Lus85, (5.1)) of the finite Coxeter group W. This can be defined by

Dx := v-NCxw0' Tw0 for every x ∈ W,
where N denotes the number of positive roots in the root system of W and w0 is the longest element of W. The D-basis is dual to the C-basis with respect to the non-degenerate form H × H → ℤ[v,v-1], (h1,h2) → τ(h1 . h2) where τ : H → ℤ[v,v-1] is the linear form such that τ(T1)=1 and τ(Tx)=0 for x ≠ 1. We have Dx=β(Cw0x') for all x ∈ W (see BetaInvolution in section "Operations for Hecke elements of the T basis").

    gap> W := CoxeterGroup( "B", 2 );;
    gap> v := X( Rationals );; v.name := "v";;
    gap> H := Hecke( W, v^2, v );
    Hecke(B2,v^2,v)
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> D := Basis( H, "D" );
    function ( arg ) ... end
    gap> D( T( 1 ) );
    vD(1)-v^2D(1,2)-v^2D(2,1)+v^3D(1,2,1)+v^3D(2,1,2)-v^4D(1,2,1,2)
    gap> BetaInvolution( D( 1 ) );
    C'(2,1,2) 

93.14 Hecke elements of the primed D basis

Basis( H, "D'" )

returns a function which gives the D'-basis of the (one parameter generic) Iwahori-Hecke algebra H of the finite Coxeter group W (see Lus85, (5.1)). This can be defined by

Dx' := v-NCxw0 Tw0 for every x ∈ W,
where N denotes the number of positive roots in the root system of W and w0 is the longest element of W. The D'-basis is basis dual to the C'-basis with respect to the non-degenerate form H × H → ℤ[v,v-1], (h1,h2) → τ(h1 . h2) where τ : H → ℤ[v,v-1] is the linear form such that τ(T1)=1 and τ(Tx)=0 for x ≠ 1. We have Dx'=Alt(Dx) for all x ∈ W (see AltInvolution in section "Operations for Hecke elements of the T basis").

    gap> W := CoxeterGroup( "B", 2 );;
    gap> v := X( Rationals );; v.name := "v";;
    gap> H := Hecke( W, v^2, v );
    Hecke(B2,v^2,v)
    gap> T := Basis( H, "T" );
    function ( arg ) ... end
    gap> Dp := Basis( H, "D'" );
    function ( arg ) ... end
    gap> AltInvolution( Dp( 1 ) );
    D(1)
    gap> Dp( 1 )^3;
    (v+2v^-1-5v^-5-9v^-7-8v^-9-4v^-11-v^-13)D'()+(v^2+2+v^-2)D'(1) 

93.15 Asymptotic algebra

AsymptoticAlgebra( W, i)

The asymptotic algebra A associated to the algebra H=Hecke(W,q) is an algebra with basis {tx}x∈ W and structure constants txty=∑zγx,y,z tz given by: let hx,y,z be the coefficient of Cx Cy on Cz. Then hx,y,zx,y,z-1 qa(z)/2+lower terms, where qa(z)/2 is the maximum over x,y of the degree of hx,y,z.

The algebra A is the direct product of the subalgebras AC generated by the elements {tx}x∈C, where C runs over the two-sided cells of W. If C is the i-th two-sided cell of W, the command AsymptoticAlgebra(W,i) returns the algebra AC. Note that the function a(z) is constant over a two-sided cell, equal to the common value of the a-function attached to the characters of the two-sided cell (see Character for left cells).

    gap> W:=CoxeterGroup("G",2);;
    gap> A:=AsymptoticAlgebra(W,1);
    Asymptotic algebra dim.10
    gap> b:=A.basis;
    [ t(2), t(12), t(212), t(1212), t(21212), t(1), t(21), t(121), 
      t(2121), t(12121) ]
    gap> List(b,x->b*x);
    [ [ t(2), t(21), t(212), t(2121), t(21212), 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, t(21), t(2)+t(212), t(21)+t(2121), t(212)+t(21212),
          t(2121) ], 
      [ t(212), t(21)+t(2121), t(2)+t(212)+t(21212), t(21)+t(2121), 
          t(212), 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, t(2121), t(212)+t(21212), t(21)+t(2121), 
          t(2)+t(212), t(21) ], 
      [ t(21212), t(2121), t(212), t(21), t(2), 0, 0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, t(1), t(12), t(121), t(1212), t(12121) ], 
      [ t(12), t(1)+t(121), t(12)+t(1212), t(121)+t(12121), t(1212), 0, 
          0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, t(121), t(12)+t(1212), t(1)+t(121)+t(12121), 
          t(12)+t(1212), t(121) ], 
      [ t(1212), t(121)+t(12121), t(12)+t(1212), t(1)+t(121), t(12), 0, 
          0, 0, 0, 0 ], 
      [ 0, 0, 0, 0, 0, t(12121), t(1212), t(121), t(12), t(1) ] ]

93.16 Lusztigaw

Lusztigaw( W, w)

For w an element of the Coxeter groups W, this function returns the coefficients on the irreducible characters of the virtual Character aw defined in Lus85, 5.10.2. This character has the property that the corresponding almost character is integral and positive.

    gap> W:=CoxeterGroup("G",2);
    CoxeterGroup("G",2)
    gap> Lusztigaw(W,Reflection(W,1));
    [ 0, 0, 1, 0, 1, 1 ]
    gap> last*List([1..NrConjugacyClasses(W)],i->AlmostCharacter(W,i));
    [G2]=<phi{1,3}'>+<phi{2,1}>+<phi{2,2}>

93.17 LusztigAw

LusztigAw( W, w)

For w an element of the Coxeter groups W, this function returns the coefficients on the irreducible characters of the virtual Character Aw defined in Lus85, 5.10.2. This character has the property that the corresponding almost character is integral and positive.

    gap> W:=CoxeterGroup("G",2);      
    CoxeterGroup("G",2)
    gap> LusztigAw(W,Reflection(W,1));
    [ 0, 0, 0, 1, 1, 1 ]
    gap> last*List([1..NrConjugacyClasses(W)],i->AlmostCharacter(W,i));
    [G2]=<phi{1,3}''>+<phi{2,1}>+<phi{2,2}>
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gap3-jm
19 Feb 2018