|
#! @Chapter Miscellaneous useful functions
#! @Section Space-efficient representation of tensors of matrices
#! Suppose we have representations of $G$, $\rho$ and $\tau$, with
#! degree $n$ and $m$. If we would like to construct the tensor
#! product representation of $G$, $\rho \otimes \tau$, the usual way
#! to do it would be to take the Kronecker product of the
#! matrices. This means we now have to store very large $nm \times nm$
#! matrices for each generator of $G$.
#! This can be avoided by storing the tensor of matrices as pairs,
#! essentially storing $A \otimes B$ as a pair $(A,B)$ and
#! implementing group operations on top of these, along with some
#! representation-theoretic functions.
#! It is only possible to guarantee an economical representation for
#! pure tensors, i.e. matrices of the form $A \otimes B$. These are
#! closed under group operations, so it is natural to define a group
#! structure.
DeclareCategory( "IsTensorProductOfMatricesObj", IsMultiplicativeElementWithInverse );
CategoryCollections( IsTensorProductOfMatricesObj );
#! Position $i$ in this representation stores the matrix $A_i$ in the
#! tensor product $A_1 \otimes A_2$.
DeclareRepresentation( "IsTensorProductPairRep", IsPositionalObjectRep, [1, 2] );
#! Position 1 stores the full Kronecker product of the matrices, this
#! is very space inefficient and supposed to be used as a last resort.
DeclareRepresentation( "IsTensorProductKroneckerRep", IsPositionalObjectRep, [1] );
DeclareGlobalFunction( "TensorProductOfMatricesObj" );
#! More convenient constructor for a tensor product (automatically
#! handles family)
DeclareGlobalFunction( "TensorProductOfMatrices" );
DeclareGlobalFunction( "TensorProductOfMatricesFamily" );
DeclareGlobalFunction( "TensorProductOfRepresentations" );
DeclareGlobalFunction( "KroneckerProductOfRepresentations" );
#! This uses the multiplicativity of characters when taking tensor
#! products to avoid having to compute the trace of a big matrix.
DeclareGlobalFunction( "CharacterOfTensorProductOfRepresentations" );
[ Dauer der Verarbeitung: 0.19 Sekunden
(vorverarbeitet)
]
|
