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 w_{1}, w_{2} ∈ W such that w_{1} ≤ w_{2} is critical if L(w_{2}) ⊆ L(w_{1}) and R(w_{2}) ⊆ R(w_{1}), 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 P_{y,w} and P_{z,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:
gap> W:=CoxeterGroup("F",4);; gap> LeftCells(W);;
which takes 30 seconds cpu time on 2Ghz computer. The computation of all Kazhdan-Lusztig polynomials for type F_{4} 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, it still seems to be out of range to re-do Alvis' computation of the Kazhdan--Lusztig polynomials of the Coxeter group of type H_{4} 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 extension to the unequal parameters case has
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 we see the Hecke algebra as being defined over the ring Z[Γ], the group algebra of Γ over Z. We assume there is for each T_{s} an element q_{s}∈ Γ_{+} such that (T_{s}-q_{s})(T_{s}+1)=0 and which has a square root q_{s}^{1/2} in Γ. We extend this notation to define an element q_{w}∈Γ_{+} by setting q_{w}=q_{s1}... q_{sn} if w=s_{1}... s_{n} is a reduced expression for some element of the Coxeter group. We have two operations on Z[Γ]: the bar involution ∑_{γ∈Γ}a_{γ}γ= ∑_{γ∈Γ}a_{γ}γ^{-1} and truncation:\ τ_{ ≤ν}∑_{γ∈Γ}a_{γ}γ= ∑_{γ ≤ν}a_{γ}γ. We then define elements R_{x,y} of Z[Γ] by T_{y}^{-1}=∑_{x} R_{x,y-1}q_{x}^{-1}T_{x}. We then define inductively the Kazhdan-Lusztig polynomials (in this general context we should say the Kazhdan-Lusztig elements of Z[Γ]) by
P_{x,w}=τ_{ ≤(qw/qx)1/2} (∑_{x<y ≤ w}R_{x,y}P_{y,w}) |
The C' basis is then defined by C'_{w}=∑_{y} q_{w}^{-1/2} P_{y,w}T_{y}. If we extend the bar involution to the whole Hecke algebra by T_{s}=T_{s}^{-1} the basis C'_{w} is characterized as the only basis stable by the bar involution and congruent to q_{w}^{-1/2}T_{w} modulo Γ_{-} in the basis q_{w}^{-1/2}T_{w}.
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,
with total order on Γ by the power of q, or when the parameters
are powers of some Mvp
s, with the lexicographic order on the variables.
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, 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∈Γ} a_{γ}γ→
∑_{γ∈Γ} a_{γ}γ^{-1}, ∑_{γ∈Γ}
a_{γ}γ→ ∑_{γ ≥ 1} a_{γ}γ and
∑_{γ∈Γ} a_{γ}γ→ ∑_{γ ≤ 1}
a_{γ}γ on elements p
of Z[Γ]. It is then possible to
define some 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 corresponding to the elements y and w of the Coxeter group W when all the parameters of the Hecke algebra are equal. 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 P_{1,w} for all elements w in the Coxeter group of type A_{3}.
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 ]
The set of Kazhdan--Lusztig polynomials for critical pairs is stored record
component klpol
of W. Thus the function is much faster the second time.
This function requires the package "chevie" (see RequirePackage).
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
P_{z,w} and P_{y,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 ]
This function requires the package "chevie" (see RequirePackage).
KazhdanLusztigCoefficient( W, y, w, k )
returns the coefficient of q^{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 the indeterminate 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+q^{2}.
This function requires the package "chevie" (see RequirePackage).
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 corresponding to y and 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.
This function requires the package "chevie" (see RequirePackage).
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 μ_{y,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( Group( 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).
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
Mvp
s (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 P_{x,y} be the corresponding Kazhdan--Lusztig polynomial. If
{T_{w} | w∈ W} denotes the usual T-basis, then
C_{x}:=∑_{y
≤ x} (-1)^{l(x)-l(y)}P_{x,y}(q^{-1})q_{x}^{1/2}q_{y}^{-1} T_{y} for every x ∈ W. |
C_{s} . C_{x} ={ |
| . |
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).
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
C_{x}' := ∑_{y ≤ x} P_{x,y}q_{x}^{-1/2} T_{y} 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 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).
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
D_{x} := v^{-N}C_{xw0}' T_{w0} 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).
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
D_{x}' := v^{-N}C_{xw0} T_{w0} 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).
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GAP 3.4.4