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,0us,1 has a square root, we can scale the basis Ts to -Ts/√-us,0us,1 to get a new basis ts with quadratic relations (ts-vs) (ts+vs-1)=0 where vs=√-us,0/us,1. 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 ℤ, 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=τ ≤(qw/qx)1/2 (∑x<y ≤ wRx,yPy,w) where τ is the truncation : τ ≤ν ∑γ∈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.
There is a separate function to compute one-parameter kazhdan-Lusztig
polynomials. From a computational point of view, even this case is quite a
challenge. 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 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 then there always exists a
critical pair (z,w) with y ≤ z ≤ w and then Py,w=Pz,w. Given
two elements y and w, such a critical pair is found by the function
CriticalPair
. Whenever the polynomial corresponding to a critical pair is
computed then this pair and the polynomial are stored in the field .klpol
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("B",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 around 1994, 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 around 1999.
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 Mvp
s, 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.
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.
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.
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.
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=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}>, LeftCell<G2: duflo=1,2 character=phi{1,6}> ]
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}> ]
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}>
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.
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 ] ] ]
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)
.
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)={ |
| . |
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 ] ] ]
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.
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
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
Mvp
s (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 ={ |
| . |
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, ChevieClassInfo(W).classtext); [ 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. |
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, |
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, |
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)
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,z=γx,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(12), t(212), t(1212), t(21212), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, t(12), t(2)+t(212), t(12)+t(1212), t(212)+t(21212), t(1212) ], [ t(212), t(12)+t(1212), t(2)+t(212)+t(21212), t(12)+t(1212), t(212), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, t(1212), t(212)+t(21212), t(12)+t(1212), t(2)+t(212), t(12) ], [ t(21212), t(1212), t(212), t(12), t(2), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, t(1), t(21), t(121), t(2121), t(12121) ], [ t(21), t(1)+t(121), t(21)+t(2121), t(121)+t(12121), t(2121), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, t(121), t(21)+t(2121), t(1)+t(121)+t(12121), t(21)+t(2121), t(121) ], [ t(2121), t(121)+t(12121), t(21)+t(2121), t(1)+t(121), t(21), 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, t(12121), t(2121), t(121), t(21), t(1) ] ]
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}>
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}>Previous Up Next
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