This webpage contains GAP3 programs that allow to write any word in the generators of the generic Hecke algebra associated with the group Gn,
for n=4,5,6,7,8,13, as a linear combination of the elements of the basis Bn, used in the
papers BCCK and BCC
for the proof of the BMM symmetrising trace conjecture.
The algorithm is explained in detail in Section 3.3 of the associated paper.
INSTRUCTIONS
The user should run the file corresponding to the group they are interested in G4, G5, G6, G7, G8, G13 and then:
If they want to express a word w as a linear combination of elements of Bn:
Step 1. Write w as a list l of integers 1,2,3,-1,-2,-3 for s1, s2, s3, s1-1, s2-1, s3-1 respectively. For example, for w=s12s2s1-1s2-2, take l=[1,1,2,-1,-2,-2].
Step 2. Calculate F(l); The output is a list where each element is the coefficient of the corresponding element in Bn .
If they want to check whether a family of |Gn| words is a basis of the generic Hecke algebra:
Step 1. Create a list B where each word is presented as a list of integers as in Step 1 above.
Step 2. Calculate M:=FindMat(B); This is the change of basis matrix.
Step 3. Calculate DeterminantMat(r*M); If this is a unit, then the initial family is a basis. The factor r is used, because GAP3 (and SAGE as
well) does not calculate determinants with indterminates to negative powers.
INSTRUCTIONS
The user should run the file corresponding to the group they are interested in G4, G5, G6, G7, G8, G13 and then:
Step 1. Write w as a list l of integers 1,2,3,-1,-2,-3 for s1, s2, s3, s1-1, s2-1, s3-1 respectively. For example, for w=s12s2s1-1s2-2, take l=[1,1,2,-1,-2,-2].
Step 2. Calculate F(l); The output is a list where each element is the coefficient of the corresponding element in Bn .
Step 1. Create a list B where each word is presented as a list of integers as in Step 1 above.
Step 2. Calculate M:=FindMat(B); This is the change of basis matrix.
Step 3. Calculate DeterminantMat(r*M); If this is a unit, then the initial family is a basis. The factor r is used, because GAP3 (and SAGE as
well) does not calculate determinants with indterminates to negative powers.
PARABOLIC BASES
In the following files we give parabolic bases for the generic Hecke algebras
for the groups discussed: G4,
G5, G6,
G7, G8, G13.