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#! @BeginChunk Example_LinearRepresentationIsomorphism
#! @BeginExample
G := SymmetricGroup(4);;
irreps := IrreducibleRepresentations(G);;
# rho and tau are isomorphic - they just have a different block order
rho := DirectSumOfRepresentations([irreps[1], irreps[3], irreps[3]]);;
tau := DirectSumOfRepresentations([irreps[3], irreps[1], irreps[3]]);;
# tau2 is just tau with a basis change - still isomorphic
B := RandomInvertibleMat(5);;
tau2 := ComposeHomFunction(tau, x -> B^-1 * x * B);;
# using the default implementation
M := LinearRepresentationIsomorphism(rho, tau);;
IsLinearRepresentationIsomorphism(M, rho, tau);
#! true
M := LinearRepresentationIsomorphism(tau, tau2);;
IsLinearRepresentationIsomorphism(M, tau, tau2);
#! true
# using the kronecker sum implementation
M := LinearRepresentationIsomorphism(tau, tau2 : use_kronecker);;
IsLinearRepresentationIsomorphism(M, tau, tau2);
#! true
# using the orbit sum implementation
M := LinearRepresentationIsomorphism(tau, tau2 : use_orbit_sum);;
IsLinearRepresentationIsomorphism(M, tau, tau2);
#! true
# two distinct irreps are not isomorphic
M := LinearRepresentationIsomorphism(irreps[1], irreps[2]);
#! fail
#! @EndExample
#! @EndChunk
#! @BeginChunk Example_LinearRepresentationIsomorphismSlow
#! @BeginExample
# Following on from the previous example
M := LinearRepresentationIsomorphismSlow(rho, tau);;
IsLinearRepresentationIsomorphism(M, rho, tau);
#! true
#! @EndExample
#! @EndChunk
#! @BeginChunk Example_AreRepsIsomorphic
#! @BeginExample
# Following on from the previous examples
# Some isomorphic representations
AreRepsIsomorphic(rho, tau);
#! true
AreRepsIsomorphic(rho, tau2);
#! true
# rho isn't iso to irreps[1] since rho is irreps[1] plus some other stuff
AreRepsIsomorphic(rho, irreps[1]);
#! false
#! @EndExample
#! @EndChunk
#! @BeginChunk Example_IsLinearRepresentationIsomorphism
#! @BeginExample
# We have already seen this function used heavily in previous examples.
# If two representations are isomorphic, the following is always true:
IsLinearRepresentationIsomorphism(LinearRepresentationIsomorphism(rho, tau), rho, tau);
#! true
# Note: this test is sensitive to ordering:
IsLinearRepresentationIsomorphism(LinearRepresentationIsomorphism(rho, tau), tau, rho);
#! false
#! @EndExample
#! @EndChunk
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