# 71.30 Schaper

`Schaper(H, mu)`

Given a partition mu, and a Hecke algebra H, `Schaper` returns a linear combination of Specht modules which have the same composition factors as the sum of the modules in the ``Jantzen filtration'' of `S`(mu); see [JM2]. In particular, if nu strictly dominates mu then `D`(nu) is a composition factor of `S`(mu) if and only if it is a composition factor of `Schaper(mu)`.

`Schaper` uses the valuation map `H.valuation` attached to H (see Specht and [JM2]).

One way in which the q--Schaper theorem can be applied is as follows. Suppose that we have a projective module x, written as a linear combination of Specht modules, and suppose that we are trying to decide whether the projective indecomposable `P`(mu) is a direct summand of x. Then, providing that we know that `P`(nu) is not a summand of x for all (e--regular) partitions nu which strictly dominate mu (see Dominates), `P`(mu) is a summand of x if and only if `InnerProduct(Schaper(H,mu),x)` is non--zero (note, in particular, that we don't need to know the indecomposable `P`(mu) in order to perform this calculation).

The q--Schaper theorem can also be used to check for irreduciblity; in fact, this is the basis for the criterion employed by `IsSimpleModule`.

```gap> H:=Specht(2);;
gap> Schaper(H,9,5,3,2,1);
S(17,2,1)-S(15,2,1,1,1)+S(13,2,2,2,1)-S(11,3,3,2,1)+S(10,4,3,2,1)-S(9,8,3)
-S(9,8,1,1,1)+S(9,6,3,2)+S(9,6,3,1,1)+S(9,6,2,2,1)
gap> Schaper(H,9,6,5,2);
0*S(0)```

The last calculation shows that `S`(9,6,5,2) is irreducible when R is a field of characteristic 0 and `e=2` (cf. `IsSimpleModule(H,9,6,5,2)`).

This function requires the package ``specht'' (see RequirePackage).

GAP 3.4.4
April 1997