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).

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GAP 3.4.4
April 1997