InducedDecompositionMatrix(d)
If d is the decomposition matrix of 'H'(Sym_n), then
InducedDecompositionMatrix(d) attempts to calculate the
decomposition matrix of 'H'(Sym_{n+1}). It does this by extracting
each projective indecomposable from d and inducing these modules to
obtain projective modules for 'H'(Sym_{n+1}).
InducedDecompositionMatrix then tries to decompose these projectives
using the function IsNewIndecomposable (see
IsNewIndecomposable). In general there will be columns of the
decomposition matrix which InducedDecompositionMatrix is unable to
decompose and these will have to be calculated ``by hand''.
InducedDecompositionMatrix prints a list of those columns of the
decomposition matrix which it is unable to calculate (this list is
also printed by the function MissingIndecomposables(d)).
gap> gap> d:=DecompositionMatrix(Specht(3,3),14);;
gap> InducedDecompositionMatrix(d);;
# Inducing....
The following projectives are missing from <d>:
[ 15 ] [ 8, 7 ]
Note that the missing indecomposables come in ``pairs'' which map
to each other under the Mullineux map (see Mullineux Mullineux).
Almost all of the decomposition matrices included in Specht were
calculated directly by InducedDecompositionMatrix. When n is
``small'' InducedDecompositionMatrix is usually able to return
the full decomposition matrix for 'H'(Sym_{n+1}).
Finally, although the InducedDecompositionMatrix can also be applied
to the decomposition matrices of the q--Schur algebras (see Schur
Schur), InducedDecompositionMatrix is much less successful in
inducing these decomposition matrices because it contains no special
routines for dealing with the indecomposable modules of the q--Schur
algebra which are indexed by e--singular partitions. Note also that
we use a non--standard labeling of the decomposition matrices of
q--Schur algebras; see Schur.
GAP 3.4.4