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