# 71.15 InducedDecompositionMatrix

`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
April 1997