%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A weyl.tex GAP documentation Meinolf Geck %% %A @(#)$Id: weyl.tex,v 3.9 1994/06/17 19:05:00 vfelsch Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %H $Log: weyl.tex,v $ %H Revision 3.9 1994/06/17 19:05:00 vfelsch %H examples adjusted to version 3.4 %H %H Revision 3.8 1994/06/10 13:29:00 vfelsch %H updated examples %H %H Revision 3.7 1993/05/28 14:04:37 gap %H fixed a small typo: "A" must be |"A"| in plain text %H %H Revision 3.6 1993/04/28 08:28:00 fceller %H a few minor updates %H %H Revision 3.5 1993/02/19 10:48:42 gap %H adjustments in line length and spelling %H %H Revision 3.4 1993/02/17 13:38:50 felsch %H examples fixed %H %H Revision 3.3 1992/12/02 10:04:30 fceller %H fixed a few bad text alignments %H %H Revision 3.2 92/11/30 13:49:52 fceller %H initial GAP 3.2 revision %% \Chapter{Weyl Groups and Hecke Algebras} In this chapter we describe functions for dealing with finite Weyl groups, associated Hecke algebras over a ring of Laurent polynomials, and their representations. A finite Weyl group is a group with a presentation of the form \begin{center} $\langle s_1, ..., s_n\ \|\ (s_i s_j)^{m(i,j)}=1 $ for all $i, j \rangle$ \end{center} where the $m(i,j)$ are integers contained in $\{ 2, 3, 4, 6 \}$ and where $m(i,i)=1$ for all $i$. These groups play a significant role in various areas of mathematics such as reflection groups, Lie algebras and linear algebraic groups. Each Weyl group is a direct product of indecomposable Weyl groups which are in turn classified by means of the well--known Dynkin diagrams. The Hecke algebra $H$ corresponding to a finite Weyl group as above is defined as follows. Let $A$ be a commutative ring and $q_1, ..., q_n \in A$ such that $q_i=q_j$ whenever $s_i$ and $s_j$ are conjugate in $W$. Then $H$ is the associative $A$--algebra with 1 with generators $a_1, ..., a_n$ and defining relations of the form \begin{center} $a_i^2=q_i 1 + (q_i-1) a_i$ for all $i$ \end{center} and \begin{center} $a_i a_j a_i ... = a_j a_i a_j ...\ \ \ m(i,j)$ factors on each side. \end{center} Usually, $A$ is chosen to be the ring of Laurent polynomials in an indeterminate $v$, and all $q_i$ are chosen to be $v^2$. For each $w \in W$ we define an element $T_w \in H$ as follows. If $w=s_{i_1} \cdots s_{i_k}$ is a reduced expression, for some for some $i_j \in \{1,\ldots,n\}$, then we let $T_w \:= T_{a_{i_1}} \cdots T_{a_{i_k}}$. It can be shown that $T_w$ is indeed independent of the choice of the reduced expression. The elements $T_w$, $w \in W$, form an $A$--basis of the algebra $H$. A suitable reference for the general theory is, for example, the book by J.E.Humpreys on `Reflection Groups and Coxeter Groups\'{} (Cambridge Studies in advanced Mathematics 29). A good reference for applications of Weyl groups and Hecke algebras are Chapters 1,2 in the book by Shi Jian-Yi on `The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups\'{} (Springer Lecture Notes in Mathematics 1179). At first, let us describe in an informal way some examples of how these programs can be used. The starting point for our programs is to specify a Cartan matrix. This is a square matrix corresponding to a (finite) root system $R$ in some Euclidean space $V$ with $(i,j)$-entry given by $2(r_i,r_j)/(r_i,r_i)$ where $r_i$ and $r_j$ are fundamental roots in $R$. The Cartan matrices for the indecomposable root system of type $A_n$, $B_n$, $C_n$, $D_n$, $G_2$, $F_4$, $E_6$, $E_7$, $E_8$ are accessible through the function 'CartanMat'. | gap> RequirePackage( "weyl" ); gap> C := CartanMat( "D", 4 );; gap> PrintArray( last ); [ [ 2, 0, -1, 0 ], [ 0, 2, -1, 0 ], [ -1, -1, 2, -1 ], [ 0, 0, -1, 2 ] ]| The Cartan matrices corresponding to the non-crystallographic groups are now also available under the names $H_3$, $H_4$ and $I_2(m)$. Given two Cartan matrices, their matrix direct sum (corresponding to the orthogonal direct sum of the root systems) is produced by using the function 'DirectSumCartanMat'. This Cartan matrix conversely determines the root system and hence also the corresponding Weyl group $W$, i.e., the finite subgroup of the full orthogonal group of $V$ generated by the reflections along the vectors in $R$. We choose a basis of $V$ consisting of the fundamental root vectors, and express every root vector as a linear combination of and every reflection matrix with respect to this basis. This is performed by the functions 'RootSystem' and 'SimpleReflectionMatrices'. | gap> S:=SimpleReflectionMatrices(C); [ [ [ -1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 1, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 1, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, -1 ] ] ] gap> R := Rootsystem(C); [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 1, 0, 1, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 1 ], [ 1, 1, 1, 0 ], [ 1, 0, 1, 1 ], [ 0, 1, 1, 1 ], [ 1, 1, 1, 1 ], [ 1, 1, 2, 1 ] ]| We notice that the root vectors in $R$ are ordered by increasing height (and that only the positive root vectors are returned; whenever the negative roots are needed, they can be computed by simply typing |-R|). If we label the (positive) roots by their position in the above list, and the negative roots by the next consecutive numbers, then we can also represent each fundamental reflection by the permutation which it induces on the root vectors. | gap> P := PermRepresentationRoots( S, R ); [ ( 1,13)( 3, 5)( 6, 8)( 7, 9)(10,11)(15,17)(18,20)(19,21)(22,23), ( 2,14)( 3, 6)( 5, 8)( 7,10)( 9,11)(15,18)(17,20)(19,22)(21,23), ( 1, 5)( 2, 6)( 3,15)( 4, 7)(11,12)(13,17)(14,18)(16,19)(23,24), ( 3, 7)( 4,16)( 5, 9)( 6,10)( 8,11)(15,19)(17,21)(18,22)(20,23) ]| This is maybe the most fundamental command, and in fact, every function which has to perform computations with group elements in $W$, does so by working with this (faithful) permutation representation of $W$. The procedure up to now can be abbreviated by one command, namely | gap> W := Weyl(C);;| which produces a record with entries\: 'isWeylGroup':\\ is set to 'true'. 'cartan':\\ contains the Cartan matrix 'C'. 'dim':\\ contains the length of 'C'. 'degree':\\ number of positive and negative roots. 'N':\\ number of positive roots. 'roots':\\ contains the root system 'R'. 'matgens':\\ contains the matrix generators 'S'. 'permgens':\\ contains the permutation generators 'P'. This record is all that the following programs and commands need. For a given element $w \in W$, the length $l(w)$ is defined to be smallest integer $k$ such that $w$ can be written as a product of $k$ fundamental reflections. Such an expression of shortest possible length will be called a reduced word for $w$. A user might be interested to think of the elements of $W$ as such words in the generating fundamental reflections. For these programs, we represent a word simply as a list of integers corresponding to the fundamental roots, e.g., |[]| is the identity element, and |[1], [2]|, etc. represent the reflection along the first, the second etc. fundamental root vector. For other purposes, it might be better to see the permutation of an element $w$ on the root vectors. The functions 'WeylWordPerm' and 'PermWeylWord' will do the conversion of one form into the other. | gap> PermWeylWord( W, [ 1, 3, 2, 1, 3 ] ); ( 1,14,13, 2)( 3,17, 8,18)( 4,12)( 5,20, 6,15)( 7,10,11, 9)(16,24) (19,22,23,21) gap> WeylWordPerm( W, last ); [ 1, 3, 1, 2, 3 ]| We notice that the word we started with and the one that we ended up with, are not the same. But of course, they represent the same element of $W$. The reason for this difference is that the function 'WeylWordPerm' always computes a reduced word which is the lexicographically smallest among all possible expressions of an element of $W$ as a word in the fundamental reflections! The function 'ReducedWeylWord' does the same but with an arbitrary word as input (and not with a permutation). In particular, the element used in the above example has length 5. Sometimes, it is not necessary to compute a reduced word for an element $w$ and one is only interested in the length $l(w)$. The crucial point of working with the permutation representation of $W$ on $R$ is that $l(w)$ can be computed very fast and effectively from the permutation, since it also given by the number of positive roots mapped to negative roots by $w$, i.e., by the number of $i \in \{ 1,... ,N \}$ such that $i^w > N$. | gap> LongestWeylWord( W ); & the (unique) longest element in W [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4 ] gap> PermWeylWord( W, last ); ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,19)( 8,20)( 9,21)(10,22) (11,23)(12,24) gap> WeylLengthPerm( W, last ); 12| These are the most basic functions available. We now give an example of how the other commands do work. | gap> WeylReflections( W ); & all reflections in W [ [ 1 ], [ 2 ], [ 3 ], [ 4 ], [ 1, 3, 1 ], [ 2, 3, 2 ], [ 3, 4, 3 ], [ 1, 2, 3, 1, 2 ], [ 1, 3, 4, 3, 1 ], [ 2, 3, 4, 3, 2 ], [ 1, 2, 3, 4, 3, 1, 2 ], [ 3, 1, 2, 3, 4, 3, 1, 2, 3 ] ] gap> WeylElements( W );; &I Order = 192 gap> List( last, Length ); [ 1, 4, 9, 16, 23, 28, 30, 28, 23, 16, 9, 4, 1 ]| The last line tells us that there is 1 element of length 0, there are 4 elements of length 4, etc. | gap> WeylConjugacyClasses( W ); &I Number of cosets = 8 &I Number of cosets = 6 &I Number of cosets = 2 &I 30 elements to consider &I Still 18 elements to consider &I 13 conjugacy classes found [ [ ], [ 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ], [ 2, 4 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ], [ 1, 2, 3, 4 ], [ 1, 3, 1, 2, 3, 4 ], [ 1, 2, 3, 1, 2, 3, 4, 3, 1, 2, 3, 4 ] ]| This are the representatives of the conjugacy classes of minimal length. | gap> LeftCells( W );; &I Order = 192 &I R(w) = [ ] : Size = 1, 1 new cell &I R(w) = [ 1 ] : Size = 7, 2 new cells &I R(w) = [ 1, 2 ] : Size = 17, 2 new cells &I R(w) = [ 1, 2, 3 ] : Size = 7, 2 new cells &I R(w) = [ 1, 2, 3, 4 ] : Size = 1, 1 new cell &I R(w) = [ 1, 2, 4 ] : Size = 23, 5 new cells &I R(w) = [ 1, 3 ] : Size = 17, 2 new cells &I R(w) = [ 1, 3, 4 ] : Size = 7, 2 new cells &I R(w) = [ 1, 4 ] : Size = 17, 2 new cells &I R(w) = [ 2 ] : Size = 7, 2 new cells &I R(w) = [ 2, 3 ] : Size = 17, 2 new cells &I R(w) = [ 2, 3, 4 ] : Size = 7, 2 new cells &I R(w) = [ 2, 4 ] : Size = 17, 2 new cells &I R(w) = [ 3 ] : Size = 23, 5 new cells &I R(w) = [ 3, 4 ] : Size = 17, 2 new cells &I R(w) = [ 4 ] : Size = 7, 2 new cells gap> last[3]; [ [ [ 2, 4, 3, 1 ], [ 3, 2, 4, 3, 1 ], [ 1, 3, 2, 4, 3, 1 ] ], [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ] ] ]| This is a left cell consisting of three elements with the corresponding matrix of $\mu_{x,y}$, the coefficient of $v^{(l(y)-l(x)-1)}$ in the Kazhdan-Lusztig polynomial $P_{x,y}(v^2)$. | gap> LeftCellRepresentation( W, last, E(4) );; gap> PrintArray( last ); [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, E(4), -1 ] ], [ [ -1, E(4), 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ], [ [ -1, 0, 0 ], [ E(4), -1, E(4) ], [ 0, 0, -1 ] ], [ [ -1, E(4), 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ]| The corresponding left cell representation with the indeterminate $v$ replaced by 'E(4)'. Of course, the user should observe limitations on storage for working with these programs, e.g., the command 'WeylElements' applied to a Weyl group of type $E_8$ will try to compute all elements as words in the fundamental reflections. Every computer will run out of memory! Some good examples to see how long the programs will take for computations in big Weyl groups are the following. | LeftCells( Weyl( CartanMat( "F", 4 ) ) );| (which takes roughly 5 hours cpu time on a HP computer) or | WeylConjugacyClasses( Weyl( CartanMat( "E", 8 ) ) ) ;| (which takes a few days...) Finally, we give an example of some computations with Hecke algebras. We start with the Weyl group of type $A_4$. | gap> W := Weyl( CartanMat("A",4) );;| We wish to work over the ring of Laurent polynomials in an indeterminate $v$, and we want to have $u=v^2$ as a parameter for the generic Hecke algebra $H$ of $W$. | gap> v := Indeterminate( Rationals );; v.name := "v";; gap> Hecke( W, v^2 );| From $v$ we can build arbitrary Laurent polynomials by using the operations |+|, |-|, |*|, and |^|. | gap> 2*v+3*v^-2-7*(v+v^-1)*(3*v^2-1); -21*v^3 - 12*v + 7*v^(-1) + 3*v^(-2)| The command 'Hecke' produces an additional component 'T' in the record of $W$ which gives, for an element $w \in W$, the corresponding basis element $T_w$ in $H$. Computations like the following are possible. | gap> W.T([1]); (v^0)*T([ 1 ]) gap> last^2; & the quadratic relation for a standard base element (v^2)*T([ ])+(v^2 - 1)*T([ 1 ]) gap> h := W.T([1,2,3,4]); (v^0)*T([ 1, 2, 3, 4 ]) gap> inv := h^-1; (1 - 4*v^(-2) + 6*v^(-4) - 4*v^(-6) + v^(-8))*T([ ])+(-v^(-2) + 3*v^( -4) - 3*v^(-6) + v^(-8))*T([ 1 ])+(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^( -8))*T([ 2 ])+(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 3 ])+(-v^( -2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 4 ])+(v^(-4) - 2*v^(-6) + v^( -8))*T([ 1, 3 ])+(v^(-4) - 2*v^(-6) + v^(-8))*T([ 1, 4 ])+(v^(-4) - 2*v^(-6) + v^(-8))*T([ 2, 1 ])+(v^(-4) - 2*v^(-6) + v^(-8))*T( [ 2, 4 ])+(v^(-4) - 2*v^(-6) + v^(-8))*T([ 3, 2 ])+(v^(-4) - 2*v^( -6) + v^(-8))*T([ 4, 3 ])+(-v^(-6) + v^(-8))*T([ 1, 4, 3 ])+(-v^( -6) + v^(-8))*T([ 2, 1, 4 ])+(-v^(-6) + v^(-8))*T([ 3, 2, 1 ])+(-v^( -6) + v^(-8))*T([ 4, 3, 2 ])+(v^(-8))*T([ 4, 3, 2, 1 ]) gap> h*inv; (v^0)*T([ ])| More clearly arranged, is | (1 - 4*v^(-2) + 6*v^(-4) - 4*v^(-6) + v^(-8))*T([ ]) +(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 1 ]) +(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 2 ]) +(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 3 ]) +(-v^(-2) + 3*v^(-4) - 3*v^(-6) + v^(-8))*T([ 4 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 1, 3 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 1, 4 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 2, 1 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 2, 4 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 3, 2 ]) +(v^(-4) - 2*v^(-6) + v^(-8))*T([ 4, 3 ]) +(-v^(-6) + v^(-8))*T([ 1, 4, 3 ]) +(-v^(-6) + v^(-8))*T([ 2, 1, 4 ]) +(-v^(-6) + v^(-8))*T([ 3, 2, 1 ]) +(-v^(-6) + v^(-8))*T([ 4, 3, 2 ]) +(v^(-8))*T([ 4, 3, 2, 1 ])| Now we turn to some aspects about the representation theory of Hecke algebras. At first we compute the left cells of $W$. | gap> ce := LeftCells(W);; &I Order = 120 &I R(w) = [ ] : Size = 1, 1 new cell &I R(w) = [ 1 ] : Size = 4, 1 new cell &I R(w) = [ 1, 2 ] : Size = 6, 1 new cell &I R(w) = [ 1, 2, 3 ] : Size = 4, 1 new cell &I R(w) = [ 1, 2, 3, 4 ] : Size = 1, 1 new cell &I R(w) = [ 1, 2, 4 ] : Size = 9, 2 new cells &I R(w) = [ 1, 3 ] : Size = 16, 3 new cells &I R(w) = [ 1, 3, 4 ] : Size = 9, 2 new cells &I R(w) = [ 1, 4 ] : Size = 11, 2 new cells &I R(w) = [ 2 ] : Size = 9, 2 new cells &I R(w) = [ 2, 3 ] : Size = 11, 2 new cells &I R(w) = [ 2, 3, 4 ] : Size = 4, 1 new cell &I R(w) = [ 2, 4 ] : Size = 16, 3 new cells &I R(w) = [ 3 ] : Size = 9, 2 new cells &I R(w) = [ 3, 4 ] : Size = 6, 1 new cell &I R(w) = [ 4 ] : Size = 4, 1 new cell | We use 'LeftCellRepresentation' to get the corresponding left cell representations. | gap> l := List( last, i -> LeftCellRepresentation(W,i,v) );; gap> c := WeylConjugacyClasses(W); [ [ ], [ 1 ], [ 1, 3 ], [ 1, 2 ], [ 1, 2, 4 ], [ 1, 2, 3 ], [ 1, 2, 3, 4 ] ] gap> ch := List( l, i -> CharHeckeRepresentation(i,c) );;| The last command computes the character values on representatives of minimal length in the conjugacy classes. | gap> Set(last); [ [ v^0, -v^0, v^0, v^0, -v^0, -v^0, v^0 ], [ v^0, v^2, v^4, v^4, v^6, v^6, v^8 ], [ 4*v^0, v^2 - 3, -2*v^2 + 2, -v^2 + 2, 2*v^2 - 1, v^2 - 1, -v^2 ], [ 4*v^0, 3*v^2 - 1, 2*v^4 - 2*v^2, 2*v^4 - v^2, v^6 - 2*v^4, v^6 - v^4, -v^6 ], [ 5*v^0, 2*v^2 - 3, v^4 - 2*v^2 + 2, -2*v^2 + 1, -v^4 + v^2 - 1, v^2, 0*v^0 ], [ 5*v^0, 3*v^2 - 2, 2*v^4 - 2*v^2 + 1, v^4 - 2*v^2, v^6 - v^4 + v^2, -v^4, 0*v^0 ], [ 6*v^0, 3*v^2 - 3, v^4 - 4*v^2 + 1, v^4 - 2*v^2 + 1, -2*v^4 + 2*v^2, -v^4 + v^2, v^4 ] ]| 'HeckeCharTable' constructs a record of a character table, which can be displayed using 'DisplayCharTable'. | gap> t := HeckeCharTable( W, c, last );; gap> DisplayCharTable(t); H() 2 3 2 3 1 1 2 . 3 1 1 . 1 1 . . 5 1 . . . . . 1 1a 2a 2b 3a 6a 4a 5a X.1 1 -1 1 1 -1 -1 1 X.2 1 v^2 v^4 v^4 v^6 v^6 v^8 X.3 4 v^2-3 -2v^2+2 -v^2+2 2v^2-1 v^2-1 -v^2 X.4 4 3v^2-1 2v^4-2v^2 2v^4-v^2 v^6-2v^4 v^6-v^4 -v^6 X.5 5 2v^2-3 v^4-2v^2+2 -2v^2+1 -v^4+v^2-1 v^2 0 X.6 5 3v^2-2 2v^4-2v^2+1 v^4-2v^2 v^6-v^4+v^2 -v^4 0 X.7 6 3v^2-3 v^4-4v^2+1 v^4-2v^2+1 -2v^4+2v^2 -v^4+v^2 v^4| To compute character values on other elements, we proceed as follows. | gap> w0 := LongestWeylWord(W); [ 1, 2, 1, 3, 2, 1, 4, 3, 2, 1 ] gap> wcl := WeylClassPolynomials( W, c, last ); [ 0, 0, v^8, v^10 - 2*v^8 + v^6, v^10 - v^8 + v^6 - v^4, v^12 - v^10 + v^4 - v^2, v^12 - v^10 - v^2 + 1 ]| In order to get the values of the irreducible characters on $T_{w_0}$, we have to postmultiply the character table of $H$ by 'wcl' regarded as a column vector: | gap> tr := TransposedMat([wcl]) * v^0;; gap> t.irreducibles * tr; [ [ v^0 ], [ v^20 ], [ 0*v^0 ], [ 0*v^0 ], [ v^8 ], [ v^12 ], [ -2*v^10 ] ]| For any comments, suggestions of improvements I would be very grateful, \hfill Aachen, November 1992, Meinolf Geck \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CartanMat}% \index{Cartan matrices} 'CartanMat( , )' returns the Cartan matrix of Dynkin type and rank . Admissable types are the strings |"A"|, |"B"|, |"C"|, |"D"|, |"E"|, |"F"|, |"G"|, |"H"|, |"I2"|. | gap> CartanMat( "F", 4 );; gap> PrintArray( last ); [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -2, 2, -1 ], [ 0, 0, -1, 2 ] ]| | gap> CartanMat( "I2", 5 ); [ [ 2, E(5)^2+E(5)^3 ], [ E(5)^2+E(5)^3, 2 ] ]| (this gives the dihedral group of order 10, so the integer 5 is not the rank in this case.) This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DirectSumCartanMat}% \index{Cartan matrices!direct sum of} 'DirectSumCartanMat( , )' returns the block diagonal direct sum of the Cartan matrices and . | gap> C1 := CartanMat( "A", 4 );; gap> PrintArray( last ); [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -1 ], [ 0, 0, -1, 2 ] ] gap> C2 := CartanMat( "D", 4 );; gap> PrintArray( last ); [ [ 2, 0, -1, 0 ], [ 0, 2, -1, 0 ], [ -1, -1, 2, -1 ], [ 0, 0, -1, 2 ] ] gap> DirectSumCartanMat( C1, C2 );; gap> PrintArray( last ); [ [ 2, -1, 0, 0, 0, 0, 0, 0 ], [ -1, 2, -1, 0, 0, 0, 0, 0 ], [ 0, -1, 2, -1, 0, 0, 0, 0 ], [ 0, 0, -1, 2, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 2, 0, -1, 0 ], [ 0, 0, 0, 0, 0, 2, -1, 0 ], [ 0, 0, 0, 0, -1, -1, 2, -1 ], [ 0, 0, 0, 0, 0, 0, -1, 2 ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{SimpleReflectionMatrices} 'SimpleReflectionMatrices( )' returns the matrices (in row convention) of the simple reflections of the Weyl group determined by the Cartan matrix with respect to the basis consisting of the fundamental root vectors corresponding to the rows of . (This function is only used in the function 'Weyl'.) | gap> C := CartanMat( "A", 4 );; gap> PrintArray( last ); [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -1 ], [ 0, 0, -1, 2 ] ] gap> SimpleReflectionMatrices( C ); [ [ [ -1, 0, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 1, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 1, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 0, -1 ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Rootsystem} 'Rootsystem( )' returns the positive roots of the root system determined by the Cartan matrix , given by their coefficients when expressed as linear combinations of fundamental roots. Thus the fundamental roots are the standard basis vectors $e_i$, corresponding to the rows $i=1,... ,n$ of . The roots are ordered by increasing height. Note that we use the convention that '[i,j]' equals $2(e_i,e_j)/(e_i,e_i)$. (This function is only used in the function 'Weyl'.) | gap> C := CartanMat( "A", 4 );; gap> PrintArray( last ); [ [ 2, -1, 0, 0 ], [ -1, 2, -1, 0 ], [ 0, -1, 2, -1 ], [ 0, 0, -1, 2 ] ] gap> Rootsystem( C ); [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ], [ 1, 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 0, 1, 1 ], [ 1, 1, 1, 0 ], [ 0, 1, 1, 1 ], [ 1, 1, 1, 1 ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PermRepresentationRoots} 'PermRepresentationRoots( , )' returns the permutations induced by the fundamental reflections of a given Weyl group on the corresponding root system . If there are $2N$ roots in total, then $1,... ,N$ correspond to the positive roots, and $N+1,... ,2N$ to the negative roots. (This function is only used in the function 'Weyl'.) | gap> C := CartanMat( "A", 4 );; gap> S := SimpleReflectionMatrices( C );; gap> R := Rootsystem( C );; gap> PermRepresentationRoots( S, R ); [ ( 1,11)( 2, 5)( 6, 8)( 9,10)(12,15)(16,18)(19,20), ( 1, 5)( 2,12)( 3, 6)( 7, 9)(11,15)(13,16)(17,19), ( 2, 6)( 3,13)( 4, 7)( 5, 8)(12,16)(14,17)(15,18), ( 3, 7)( 4,14)( 6, 9)( 8,10)(13,17)(16,19)(18,20) ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Weyl} 'Weyl( )' returns a record containing the basic information about the Weyl group determined by the Cartan matrix . This record has the following entries. 'cartan':\\ the Cartan matrix 'dim':\\ the size of 'degree':\\ the number of positive roots 'N':\\ the total number of roots 'roots':\\ the root vectors 'matgens':\\ the matrices of the simple reflections 'permgens':\\ the permutations on the root vectors | gap> W := Weyl( CartanMat("G",2) ); Weyl( [ [ 2, -1 ], [ -3, 2 ] ] ) gap> PrintRec( W ); Print( "\n" ); rec( isDomain := true, isWeylGroup := true, cartan := [ [ 2, -1 ], [ -3, 2 ] ], dim := 2, degree := 12, N := 6, roots := [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 3 ] ], matgens := [ [ [ -1, 0 ], [ 1, 1 ] ], [ [ 1, 3 ], [ 0, -1 ] ] ], permgens := [ ( 1, 7)( 2, 3)( 5, 6)( 8, 9)(11,12), ( 1, 5)( 2, 8)( 3, 4)( 7,11)( 9,10) ], parameter := [ 1, 1 ], operations := ... )| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PermWeylWord} 'PermWeylWord( , )' returns the permutation on the root vectors determined by an element which is given as a list of integers representing the standard generators of the Weyl group . | gap> W := Weyl( CartanMat("G",2) );; gap> PermWeylWord( W, [1,2,1] ); ( 1,12)( 2, 4)( 3, 9)( 6, 7)( 8,10)| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylWordPerm} 'WeylWordPerm( , )' returns a reduced word in the standard generators of the Weyl group determined by the permutation on the root vectors. | gap> W := Weyl( CartanMat("G",2) );; gap> WeylWordPerm( W, ( 1,12)( 2, 4)( 3, 9)( 6, 7)( 8,10) ); [ 1, 2, 1 ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylLengthPerm} 'WeylLengthPerm( , )' returns the length of the permutation , which corresponds to an element in the Weyl group , as a reduced expression in the standard generators. | gap> W := Weyl( CartanMat("F",4) );; gap> p := PermWeylWord( W, [1,2,3,4,2] ); ( 1,44,38,25,20,14)( 2, 5,40,47,48,35)( 3, 7,13,21,19,15) ( 4, 6,12,28,30,36)( 8,34,41,32,10,17)( 9,18)(11,26,29,16,23,24) (27,31,37,45,43,39)(33,42) gap> WeylLengthPerm( W, p ); 5 gap> WeylWordPerm( W, p ); [ 1, 2, 3, 2, 4 ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ReducedWeylWord} 'ReducedWeylWord( , )' returns a reduced expression for an element of the Weyl group , which is given as a list of integers where each entry 'i' in this list represents the 'i'-th standard generator of . | gap> W := Weyl( CartanMat("E",6) );; gap> ReducedWeylWord( W, [1,1,1,1,1,2,2,2,3] ); [ 1, 2, 3 ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{LongestWeylWord} 'LongestWeylWord( )' returns a reduced expression in the standard generators of the unique element of maximal length of the Weyl group . | gap> W := Weyl( CartanMat("E",6) );; gap> w0 := LongestWeylWord(W); [ 1, 2, 3, 1, 4, 2, 3, 1, 4, 3, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1 ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylReflections} 'WeylReflections( )' returns a list of reduced expressions in the standard generators for the set of reflections in the Weyl group . The 'i'-th entry in this list is the reflection along the 'i'-th root in '.roots'. | gap> W := Weyl( CartanMat("B",2) );; gap> W.roots; [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ], [ 2, 1 ] ] gap> WeylReflections(W); [ [ 1 ], [ 2 ], [ 2, 1, 2 ], [ 1, 2, 1 ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylRightCosetRepresentatives} 'WeylRightCosetRepresentatives( , , )' returns a list of reduced words in the Weyl group which are distinguished representatives for the right cosets of $W(J)$ in $W(I)$ where, for each sublist 'K' of '[1..W.dim]', $W(K)$ is the parabolic subgroup generated by the simple reflections corresponding to elements in 'K'. | gap> W := Weyl( CartanMat("F",4) );; gap> WeylRightCosetRepresentatives( W, [1,2,3], [1,2] ); &I Number of cosets = 8 [ [ ], [ 3 ], [ 3, 2 ], [ 3, 2, 1 ], [ 3, 2, 3 ], [ 3, 2, 1, 3 ], [ 3, 2, 1, 3, 2 ], [ 3, 2, 1, 3, 2, 3 ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylCosetPermRepresentation} 'WeylCosetPermRepresentation( , )' returns the list of permutations induced by the standard generators of the Weyl group on the cosets of the parabolic subgroup generated by the elements in the set . The cosets are in the order determined by the result of the function 'WeylRightCosetRepresentatives( , [1...dim], )' | gap> W := Weyl( CartanMat("F",4) );; gap> WeylCosetPermRepresentation( W, [1,2,3] ); &I Number of cosets = 24 [ ( 4, 5)( 6, 7)( 8,10)(16,18)(17,19)(20,21), ( 3, 4)( 7, 9)(10,12)(14,16)(15,17)(21,22), ( 2, 3)( 4, 6)( 5, 7)( 9,11)(12,14)(13,15)(17,20)(19,21)(22,23), ( 1, 2)( 6, 8)( 7,10)( 9,12)(11,13)(14,15)(16,17)(18,19)(23,24) ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylElements} 'WeylElements( )' returns a list of which the 'i' - th entry is the list of reduced words of length 'i-1' in the Weyl group . | gap> W := Weyl( CartanMat("G",2) );; gap> WeylElements(W); &I Order = 12 [ [ [ ] ], [ [ 1 ], [ 2 ] ], [ [ 1, 2 ], [ 2, 1 ] ], [ [ 1, 2, 1 ], [ 2, 1, 2 ] ], [ [ 1, 2, 1, 2 ], [ 2, 1, 2, 1 ] ], [ [ 1, 2, 1, 2, 1 ], [ 2, 1, 2, 1, 2 ] ], [ [ 1, 2, 1, 2, 1, 2 ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylConjugacyClasses} 'WeylConjugacyClasses( )' returns a list of representatives of the conjugacy classes of the Weyl group . Each element in this list is given as a word in the standard generators, and it has the property that it is of minimal length in its conjugacy class. | gap> W := Weyl( CartanMat("F",4) );; gap> WeylConjugacyClasses(W); &I Number of cosets = 24 &I Number of cosets = 8 &I Number of cosets = 3 &I 108 elements to consider &I Still 34 elements to consider &I 25 conjugacy classes found [ [ ], [ 1 ], [ 3 ], [ 1, 2 ], [ 1, 3 ], [ 2, 3 ], [ 3, 4 ], [ 1, 2, 3 ], [ 1, 2, 4 ], [ 1, 3, 4 ], [ 2, 3, 4 ], [ 1, 2, 3, 4 ], [ 2, 3, 2, 3 ], [ 1, 2, 3, 2, 3 ], [ 2, 3, 2, 3, 4 ], [ 1, 2, 3, 2, 3, 4 ], [ 1, 3, 2, 1, 3, 2, 3, 4 ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3 ], [ 2, 3, 2, 3, 4, 3, 2, 3, 4 ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3, 4 ], [ 1, 2, 3, 2, 3, 4, 3, 2, 3, 4 ], [ 1, 2, 1, 3, 2, 1, 3, 4, 3, 2, 3, 4 ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3, 4, 3, 2, 3, 4 ], [ 1, 2, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4 ], [ 1, 2, 1, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4, 3, 2, 1, 3, 2, 3, 4 ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{ParametersCentralizers} 'ParametersCentralizers( )' returns a list of pairs which parametrizes the conjugacy classes of centralizers of semisimple elements in a Chevalley group with Weyl group . Each pair consists of a subset which contains fundamental roots or the highest root in the root system of , and an element in the normalizer in of the subgroup generated by the reflections along the roots in the given subset. | gap> W := Weyl( CartanMat( "A", 4 ) );; gap> ParametersCentralizers( W ); &I Number of Psi's = 7 &I 7 conjugacy classes &I 3 conjugacy classes &I 2 conjugacy classes &I 2 conjugacy classes &I 1 conjugacy classes &I 1 conjugacy classes &I 1 conjugacy classes &I Number of Pairs = 17 [ [ [ ], [ ] ], [ [ ], [ 4 ] ], [ [ ], [ 4, 3 ] ], [ [ ], [ 4, 3, 2, 1 ] ], [ [ ], [ 3, 2, 1 ] ], [ [ ], [ 2, 1, 4 ] ], [ [ ], [ 2, 1, 3, 2 ] ], [ [ 1 ], [ ] ], [ [ 1 ], [ 4 ] ], [ [ 1 ], [ 4, 3 ] ], [ [ 1, 2 ], [ ] ], [ [ 1, 2 ], [ 4 ] ], [ [ 1, 3 ], [ ] ], [ [ 1, 3 ], [ 2, 1, 3, 2 ] ], [ [ 1, 2, 3 ], [ ] ], [ [ 1, 2, 4 ], [ ] ], [ [ 1, 2, 3, 4 ], [ ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Bruhat} 'Bruhat( , , )' returns true, if the element is less than or equal to the element of the Weyl group , and false otherwise. Both and must be given as permutations on the root vectors of . This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{KazhdanLusztigPolynomial} 'KazhdanLusztigPolynomial( , , , )' returns the Kazhdan - Lusztig polynomial in the indeterminate corresponding to the elments and (given as reduced expressions in the standard generators) of the Weyl group . | gap> W := Weyl( CartanMat("B",4) );; gap> y := [1,2,3,4,3,2,1];; gap> x := [1];; gap> u := Indeterminate(Rationals);; u.name := "u";; gap> KazhdanLusztigPolynomial( W, x, y, u ); &I Order = 384 u^2 + 2*u + 1| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{KLCoefficient} 'KLCoefficient( , , , )' returns the -th coefficient of the Kazhdan - Lusztig polynomial corresponding to the elments and , which must be given as permutations on the root vectors, of the Weyl group . (This function does not use the polynomial arithmetic and is slightly more efficient than the function 'KazhdanLusztigPolynomial'.) | gap> W := Weyl( CartanMat("B",4) );; gap> y := [1,2,3,4,3,2,1];; gap> py := PermWeylWord( 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 := PermWeylWord( 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 -> KLCoefficient( W, px, py, i ) ); &I Order = 384 [ 1, 2, 1, 0 ]| So the Kazhdan-Lusztig polynomial corresponding to $x$ and $y$ is $1+2u+u^2$. This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylMueMat} 'WeylMueMat( , )' returns the list of highest coefficients $\mu_{y,w}$ of the Kazhdan- Lusztig polynomials corresponding to pairs of elements of the Weyl group from a list . The elements in must be reduced expressions in the standard generators, ordered by increasing length. E.g., is the result of 'Iterated(WeylElements(),Concatenation)'. (See the next section for an example.) This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{DecomposedLeftCells} 'DecomposedLeftCells( , , )' given a list of reduced words in the Weyl group with corresponding matrix of highest coefficients of the corresponding Kazhdan-Lusztig polynomials, it returns a list of pairs. The first component of each pair consists of the reduced words in the list which lie in one left cell $C$, the second component consists of the corresponding matrix of highest coefficients $\mu_{y,w}$, where $y,w$ are in $C$. | gap> W := Weyl( CartanMat("G",2) );; gap> WeylElements(W);; &I Order = 12 gap> c := Iterated( last, Concatenation ); [ [ ], [ 1 ], [ 2 ], [ 1, 2 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ], [ 2, 1, 2, 1, 2 ], [ 1, 2, 1, 2, 1, 2 ] ] gap> mue := WeylMueMat( W, c ); &I Order = 12 [ [ 0 ], [ 1, 0 ], [ 1, 0, 0 ], [ 0, 1, 1, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0 ] ] gap> DecomposedLeftCells( W, c, mue ); [ [ [ [ ] ], [ [ 0 ] ] ], [ [ [ 1 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ] ], [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0 ] ] ], [ [ [ 2 ], [ 1, 2 ], [ 2, 1, 2 ], [ 1, 2, 1, 2 ], [ 2, 1, 2, 1, 2 ] ], [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0 ] ] ], [ [ [ 1, 2, 1, 2, 1, 2 ] ], [ [ 0 ] ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{LeftCells} 'LeftCells( )' returns a list of pairs. The first component of each pair consists of the reduced words in the Weyl group which lie in one left cell $C$, the second component consists of the corresponding matrix of highest coefficients $\mu_{y,w}$, where $y,w$ are in $C$. | gap> W := Weyl( CartanMat("G",2) );; gap> LeftCells(W); &I Order = 12 &I R(w) = [ ] : Size = 1, 1 new cell &I R(w) = [ 1 ] : Size = 5, 1 new cell &I R(w) = [ 1, 2 ] : Size = 1, 1 new cell &I R(w) = [ 2 ] : Size = 5, 1 new cell [ [ [ [ ] ], [ [ 0 ] ] ], [ [ [ 1 ], [ 2, 1 ], [ 1, 2, 1 ], [ 2, 1, 2, 1 ], [ 1, 2, 1, 2, 1 ] ], [ [ 0 ], [ 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1, 0 ], [ 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, 1, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1, 0 ] ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{LeftCellRepresentation} 'LeftCellRepresentation( , , )' returns a list of matrices giving the left cell representation of the Hecke algebra with parameter $^2$ associated with the Weyl group . The argument 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, is the result of the function 'WeylMueMat'. This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Hecke} 'Hecke( , )' adds to the record of the Weyl group a component 'T' which is a function that produces, for each element $w \in W$, the corresponding basis element $T_w$ in the generic Hecke algebra. specifies the index parameters corresponding to the standard generators. If is a single field element or a polynomial, then all parameters are assumed to be equal to this element. Alternatively, may be a list of such numbers or polynomials. Accordingly, a component 'parameter' is added to the record of . | gap> W := Weyl( CartanMat("B",3) );; gap> q := Indeterminate(Rationals);; q.name := "q";; gap> Hecke( W, 1 );| or | gap> Hecke( W, [q,q^2,q^2] );| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{WeylClassPolynomials} 'WeylClassPolynomials( , , )' returns the class polynomials of the element with respect to representatives of minimal lengths in the conjugacy classes of the Weyl group . Typically, is the result of the 'WeylConjugacyClasses' and is a reduced expression in the standard generators of . (For the definition of these polynomials, see the article by G.Pfeiffer and M.Geck in Advances in Mathematics (to appear).) This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{HeckeReflectionRepresentation} 'HeckeReflectionRepresentation( )' returns a list of matrices which give the reflection representation of the Hecke algebra corresponding to the Weyl group . The function 'Hecke' must have been applied to the record . | gap> W := Weyl( CartanMat("F",4) );; gap> Hecke(W,1); & the parameters are set to 1, so that the gap> & Hecke algebra is isomorphic to the gap> & group algebra of the Weyl group. gap> HeckeReflectionRepresentation(W); [ [ [ -1, 0, 0, 0 ], [ -1, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, -1, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, -1, 1, 0 ], [ 0, 0, 0, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, -2, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, -1, 1 ] ], [ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, -1 ], [ 0, 0, 0, -1 ] ] ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CheckHeckeDefiningRelations} 'CheckHeckeDefiningRelations( , )' returns true or false, according to whether a given set of matrices corresponding to the standard generators of the Weyl group defines a representation of the associated Hecke algebra or not. | gap> W := Weyl( CartanMat("F",4) );; gap> Hecke( W, 1 ); gap> HeckeReflectionRepresentation(W);; gap> CheckHeckeDefiningRelations( W, last ); true| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CharHeckeRepresentation} 'CharHeckeRepresentation( , )' given a list of matrices corresponding to the standard generators of a finite Weyl group and a list of words in the standard generators it returns the list of traces of the corresponding representation on the elements in . | gap> W := Weyl( CartanMat("F",4) );; gap> Hecke( W, 1 ); gap> r := WeylConjugacyClasses(W);; &I Number of cosets = 24 &I Number of cosets = 8 &I Number of cosets = 3 &I 108 elements to consider &I Still 34 elements to consider &I 25 conjugacy classes found gap> t := HeckeReflectionRepresentation(W);; gap> CharHeckeRepresentation( t, r ); [ 4, 2, 2, 1, 0, 2, 1, 1, -1, -1, 1, 0, 0, 0, 0, 0, 2, -2, -2, -1, -1, 0, -2, -2, -4 ]| This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{HeckeCharTable} 'HeckeCharTable( , , )' given a Weyl group , a list of characters and a list of representatives of minimal lengths in the conjugacy classes it returns a character table record. This function requires the package \"weyl\" (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section\\|%E" %% fill-column: 73 %% eval: (hide-body) %% End: %%