CalculateDecompositionMatrix(H,n)
CalculateDecompositionMatrix(H,n) is similar to the function
DecompositionMatrix DecompositionMatrix in that both functions try
to return the decomposition matrix d of 'H'(Sym_n); the
difference is that this function tries to calculate this matrix
whereas the later reads the matrix from the library files (in
characteristic zero both functions apply the algorithm of [LLT] to
compute~d). In effect this function is only needed when working with
Hecke algebras defined over fields of positive characteristic (or when
you wish to avoid the libraries).
For example, if you want to do calculations with the decomposition
matrix of the symmetrix group Sym_{15} over a field of
characteristic two, DecompositionMatrix returns false whereas
CalculateDecompositionMatrix; returns a part of the decomposition
matrix.
gap> H:=Specht(2,2);
Specht(e=2, p=2, S(), P(), D())
gap> d:=DecompositionMatrix(H,15);
# This decomposition matrix is not known; use CalculateDecompositionMatrix()
# or InducedDecompositionMatrix() to calculate with this matrix.
false
gap> d:=CalculateDecompositionMatrix(H,15);;
# Projective indecomposable P(6,4,3,2) not known.
# Projective indecomposable P(6,5,3,1) not known.
...
gap> MissingIndecomposables(d);
The following projectives are missing from <d>:
[ 15 ] [ 14, 1 ] [ 13, 2 ] [ 12, 3 ] [ 12, 2, 1 ] [ 11, 4 ]
[ 11, 3, 1 ] [ 10, 5 ] [ 10, 4, 1 ] [ 10, 3, 2 ] [ 9, 6 ] [ 9, 5, 1 ]
[ 9, 4, 2 ] [ 9, 3, 2, 1 ] [ 8, 7 ] [ 8, 6, 1 ] [ 8, 5, 2 ] [ 8, 4, 3]
[ 8, 4, 2, 1 ] [ 7, 6, 2 ] [ 7, 5, 3 ] [ 7, 5, 2, 1 ] [ 7, 4, 3, 1 ]
[ 6, 5, 4 ] [ 6, 5, 3, 1 ] [ 6, 4, 3, 2 ]
Actually, you are much better starting with the decompositon matrix of
Sym_{14} and then applying InducedDecompositionMatrix to this
matrix.
See also DecompositionMatrix DecompositionMatrix. This function
requires the package ``specht'' (see RequirePackage).
GAP 3.4.4