There is a number of programs in CHEVIE for computing Kazhdan-Lusztig polynomials, left cells, and the various Kazhdan-Lusztig bases of Iwahori-Hecke algebras (see KL79). Most of the code deals with the equal parameters case.
From a computational point of view, Kazhdan-Lusztig polynomials are quite a challenge, even in the equal parameters case, and are even more difficult to compute when the algebra has unequal parameters. For the equal parameters case it seems that the best approach still is by using the recursion formula in the original article KL79. 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. (This is different to earlier versions of CHEVIE.)
A good example to see how long the programs will take for computations in big Coxeter groups is the following:
LeftCells( CoxeterGroup( "F", 4 ) );
which takes 5 minutes cpu time on a PIII/700 computer. The computation of all Kazhdan-Lusztig polynomials for type F4 takes a bit more than 6 minutes.
However, it still seems to be out of range to re-do Alvis' computation of the Kazhdan--Lusztig polynomials of the Coxeter group of type H4 in a computer algebra system like GAP. 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 few programs which exist for the unequal
parameters case have been written by F.Digne and J.Michel.
We recall now some theory to explain the computation done. The most general case when Kazhdan-Lusztig bases and polynomials can be defined is when the parameters of the Hecke algebra belong to a totally ordered abelian group Γ (a group for the multiplication of parameters). Thus coefficients of elements of the Hecke algebra that we will construct will be elements of the group algebra Z[Γ]. We assume there is for each Ts an element qs∈ Γ+ such that (Ts-qs)(Ts+1)=0 and which has a square root qs1/2 in Γ. We extend this notation to define an element qw∈Γ+ by setting qw=qs1... qsn if w=s1... sn is a reduced expression for some element of the Coxeter group. We have two operations on Z[Γ]: the bar involution \overline∑γ∈Γaγγ= ∑γ∈Γaγγ-1 and truncation:\ τ ≤ν∑γ∈Γaγγ= ∑γ ≤νaγγ. We then define elements Rx,y of Z[Γ] by Ty-1=∑x \overlineRx,y-1qx-1Tx. We then define inductively the Kazhdan-Lusztig polynomials (in this general context we should say the Kazhdan-Lusztig elements of Z[Γ]) by
Px,w=τ ≤(qw/qx)1/2 (∑x<y ≤ wRx,yPy,w) |
The C′ basis is then defined by C′w=∑y qw-1/2 Py,wTy. The other Kazhdan-Lusztig bases are computed in CHEVIE in terms of the C′ basis.
CHEVIE is able to define automatically the two operations above on
Z(Γ) when all parameters are powers of the same indeterminate
q. Then the total order on Γ is by the power of q, the
bar involution is evaluating a Laurent polynomial at q-1, 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, say 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∈Γaγγ→
∑γ∈Γaγγ-1,
∑γ∈Γaγγ→
∑γ ≥ 1aγγ and
∑γ∈Γaγγ→∑γ ≤ 1aγγ
on elements p
of Z[Γ]. Then define some Kahzdan-Lusztig
basis. The operations above will be used internally by CHEVIE to
compute them.
KazhdanLusztigPolynomial( W, y, w [, ly, lw ] )
returns the Kazhdan-Lusztig polynomial corresponding to the elements y
and w of the Coxeter group W when all the parameters of the Hecke
algebra are equal to the indeterminate X(Rationals)
. The optional
variables ly and lw contain the length of y and w, respectively.
The above form for the input has been chosen for efficiency reasons (the
program calls recursively itself and the length computation may be a
substantial overhead). 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 ), Length( x ), Length( 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 -> 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 ]
The set of Kazhdan--Lusztig polynomials that were found during this
computation is contained in the record component klpol
(as lists of
coefficients):
gap> W.klpol; [ [ 1, 1 ], [ 1 ], [ 1, 0, 1 ], [ 1, 1, 1 ] ]
Thus, we have found four different Kazhdan-Lusztig polynomials, namely 1+q, 1, 1+q2 and 1+q+q2.
This function requires the package "chevie" (see RequirePackage).
CriticalPair( W, y, w )
Given an element y of length in the Coxeter group W and an element
w the function CriticalPair
returns a triple [z,w,lz-ly]
(the
third element is the difference between the lengths of z and y)
where y ≤ z ≤ w and where (z,w) is a critical pair (i.e., we have
L(w) ⊆ L(z) and R(w)⊆
R(z)). The actual critical pair chosen is 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[1] ); [ 2, 3, 2, 1, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 3 ] gap> cr[3]; 12 gap> KazhdanLusztigPolynomial( W, y, w, 4, 23 ); q^3 + 1 gap> KazhdanLusztigPolynomial( W, cr[1], cr[2], 16, 23 ); q^3 + 1
This function requires the package "chevie" (see RequirePackage).
85.3 KazhdanLusztigCoefficient
KazhdanLusztigCoefficient( W, y, w, [ ly, lw,] k )
returns the coefficient of X(Rationals)^k
in the Kazhdan-Lusztig
polynomial corresponding to the elements y and w of the Coxeter
group W when all the parameters of the Hecke algebra are equal to
indeterminate X(Rationals)
. Again, the optional variables ly and
lw contain the length of y and w, respectively.
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, 1, 7, i ) ); [ 1, 2, 1, 0 ]
So the Kazhdan-Lusztig polynomial corresponding to x and y is 1+2u+u2.
This function requires the package "chevie" (see RequirePackage).
KazhdanLusztigMue( W, y, w [, ly, lw ] )
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 corresponding to y and w. The optional variables ly and lw contain the length of y and w, respectively.
Of course, the result of this function could also be obtained by
KazhdanLusztigCoefficient( W, y, w, ly, lw, (lw - ly -1)/2)
but there are some speed-ups compared to this general function.
This function requires the package "chevie" (see RequirePackage).
85.5 LeftCells
LeftCells( W )
returns a list of pairs. The first component of each pair consists of the reduced words in the Coxeter group W which lie in one left cell C, the second component consists of the corresponding matrix of highest coefficients μy,w, where y,w are in C.
gap> W := CoxeterGroup( "G", 2 );; gap> LeftCells(W); [ [ [ [ ] ], [ [ 0 ] ] ], [ [ [ 1 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ] ], [ [ 0, 1, 0, 0, 0 ], [ 1, 0, 1, 0, 0 ], [ 0, 1, 0, 1, 0 ], [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ] ] ], [ [ [ 1, 2, 1, 2, 1, 2 ] ], [ [ 0 ] ] ], [ [ [ 2 ], [ 1, 2 ], [ 2, 1, 2 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1, 2 ] ], [ [ 0, 1, 0, 0, 0 ], [ 1, 0, 1, 0, 0 ], [ 0, 1, 0, 1, 0 ], [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ] ] ] ]
This function requires the package "chevie" (see RequirePackage).
LeftCellRepresentation( W , cell )
returns a list of matrices giving the left cell representation of the
Iwahori-Hecke algebra W. The argument cell is a pair with first
component a list of reduced words which form a left cell, and second
component the corresponding matrix of highest coefficients of the
corresponding Kazhdan-Lusztig polynomials. Typically, cell is an entry
from the result of the function LeftCells
.
gap> v := X( Cyclotomics ) ;; v.name := "v";; gap> H := Hecke(CoxeterGroup( "H", 3), v^2, v ); Hecke(CoxeterGroup("H",3),v^2,v) gap> c := LeftCells( CoxeterGroup( H ) );; gap> List( c, i -> Length( i[ 1 ] ) ); [ 1, 6, 5, 8, 5, 6, 1, 5, 8, 5, 5, 6, 6, 5, 8, 5, 5, 8, 5, 6, 6, 5 ] gap> LeftCellRepresentation(H,c[3]); [ [ [ -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, v, -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 ] ], [ [ v^2, 0*v^0, 0*v^0, 0*v^0, 0*v^0 ], [ v, -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, v, -v^0, v ], [ 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, 0*v^0, v^2, 0*v^0 ], [ 0*v^0, 0*v^0, 0*v^0, v, -v^0 ] ] ]
This function requires the package "chevie" (see RequirePackage).
85.7 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 (see the introduction). This 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)Px,y(q-1)qx1/2qy-1 Ty for every x ∈ W. |
Cs . Cx ={ |
| . |
gap> W := CoxeterGroup( "B", 3 );; gap> v := X( Rationals );; v.name := "v";; gap> H := Hecke( W, v^2, v ); Hecke(CoxeterGroup("B",3),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, 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 ]
This function requires the package "chevie" (see RequirePackage).
85.8 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 (see the introduction). This is defined by
Cx′ := ∑y ≤ x Px,yqx-1/2 Ty for every 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 #warning: C' basis: v chosen as 2nd root of v\^2 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)
This function requires the package "chevie" (see RequirePackage).
85.9 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(CoxeterGroup("B",2),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)
This function requires the package "chevie" (see RequirePackage).
85.10 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(CoxeterGroup("B",2),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)
This function requires the package "chevie" (see RequirePackage).
GAP 3.4.4