|
# Block diagonalizing representations
# Takes a rho that goes to a matrix group only. Returns a basis change
# matrix which, when used on a given rho(g) (matrix), block
# diagonalises rho(g) such that each block corresponds to an irrep.
BasisChangeMatrix@ := function(rho)
# Base change matrix from new basis to standard basis, acts on the left
return TransposedMat(BlockDiagonalBasisOfRepresentation(rho));
end;
# Gives the nice basis for rho
InstallGlobalFunction( BlockDiagonalBasisOfRepresentation, function(rho)
return ComputeUsingMethod@(rho).basis;
end );
# Takes a representation going to a matrix group and gives you an
# isomorphic representation where the images are block-diagonal with
# each block corresponding to an irreducible representation
InstallGlobalFunction( BlockDiagonalRepresentation, function(rho)
return ComputeUsingMethod@(rho).diagonal_rep;
end );
# Calculates a matrix P such that X = P^-1 Y P
BasisChangeMatrixSimilar@ := function(X, Y)
local A, B;
# We find the rational canonical form conjugation matrices
A := RationalCanonicalFormTransform(X);
B := RationalCanonicalFormTransform(Y);
# Now A^-1 X A = B^-1 Y B, so P = BA^-1
return B * A^-1;
end;
InstallGlobalFunction( BasisChangeMatrixSimilar, BasisChangeMatrixSimilar@ );
[ Dauer der Verarbeitung: 0.29 Sekunden
(vorverarbeitet)
]
|
