71.59 ERegulars

ERegulars(x)
ERegulars(d)
ListERegulars(x)

ERegulars(x) prints a list of the e--regular partitions, together with multiplicities, which occur in the module x. ListERegulars(x) returns an actual list of these partitions rather than printing them.

gap> H:=Specht(8);;
gap> x:=H.S(InducedModule(H.P(8,5,3)) );  
S(9,5,3)+S(8,6,3)+S(8,5,4)+S(8,5,3,1)+S(6,5,3,3)+S(5,5,4,3)+S(5,5,3,3,1)
gap> ERegulars(x);
[ 9, 5, 3 ]  [ 8, 6, 3 ]  [ 8, 5, 4 ]  [ 8, 5, 3, 1 ]  
[ 6, 5, 3, 3 ]  [ 5, 5, 4, 3 ]  [ 5, 5, 3, 3, 1 ]  
gap> H.P(x);
P(9,5,3)+P(8,6,3)+P(8,5,4)+P(8,5,3,1)

This example shows why these functions are useful: given a projective module x, as above, and the list of e--regular partitions in x we know the possible indecomposable direct summands of x.

Note that it is not necessary to specify what e is when calling this function because x ``knows'' the value of e.

The function ERegulars can also be applied to a decomposition matrix d; in this case it returns the unitriangular submatrix of d whose rows and columns are indexed by the e--regular partitions.

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

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