71.58 Specialized

Specialized(x [,q]);
Specialized(d [,q]);

Given an element of the Fock space x (see Specht), or a crystallized decomposition matrix (see CrystalDecompositionMatrix), Specialized returns the corresponding element of the Grothendieck ring or the corresponding decomposition matrix of the Hecke algebra respectively. By default the indeterminate v is specialized to 1; however v can be specialized to any (integer) q by supplying a second argument.

gap> H:=Specht(2);; x:=H.Pq(6,2);
S(6,2)+v*S(6,1,1)+v*S(5,3)+v^2*S(5,1,1,1)+v*S(4,3,1)+v^2*S(4,2,2)
+(v^3 + v)*S(4,2,1,1)+v^2*S(4,1,1,1,1)+v^2*S(3,3,1,1)+v^3*S(3,2,2,1)
+v^3*S(3,1,1,1,1,1)+v^3*S(2,2,2,1,1)+v^4*S(2,2,1,1,1,1)
gap> Specialized(x);
S(6,2)+S(6,1,1)+S(5,3)+S(5,1,1,1)+S(4,3,1)+S(4,2,2)
+2*S(4,2,1,1)+S(4,1,1,1,1)+S(3,3,1,1)+S(3,2,2,1)+S(3,1,1,1,1,1)
+S(2,2,2,1,1)+S(2,2,1,1,1,1)
gap> Specialized(x,2);
S(6,2)+2*S(6,1,1)+2*S(5,3)+4*S(5,1,1,1)+2*S(4,3,1)+4*S(4,2,2)+10*S(4,2,1,1)
+4*S(4,1,1,1,1)+4*S(3,3,1,1)+8*S(3,2,2,1)+8*S(3,1,1,1,1,1)+8*S(2,2,2,1,1)
+16*S(2,2,1,1,1,1) 

An example of Specialize being applied to a crystallized decomposition matrix can be found in CrystalDecompositionMatrix. This function requires the package ``specht'' (see RequirePackage).

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