‣ CentralizerBlocksOfRepresentation ( rho ) | ( function ) |
Returns: List of vector space generators for the centralizer ring of \(\rho(G)\), written in the basis given by BlockDiagonalBasisOfRepresentation
(5.1-1). The matrices are given as a list of blocks.
Let \(G\) have irreducible representations \(\rho_i\) with multiplicities \(m_i\). The centralizer has dimension \(\sum_i m_i^2\) as a \(\mathbb{C}\)-vector space. This function gives the minimal number of generators required.
‣ CentralizerOfRepresentation ( arg ) | ( function ) |
Returns: List of vector space generators for the centralizer ring of \(\rho(G)\).
This gives the same result as CentralizerBlocksOfRepresentation
(6.1-1), but with the matrices given in their entirety: not as lists of blocks, but as full matrices.
‣ ClassSumCentralizer ( rho, class, cent_basis ) | ( function ) |
Returns: \(\sum_{s \in t^G} \rho(s)\), where \(t\) is a representative of the conjugacy class class of \(G\).
We require that rho is unitary. Uses the given orthonormal basis (with respect to the inner product \(\langle A, B \rangle = \mbox{Trace}(AB^*)\)) for the centralizer ring of rho to calculate the sum of the conjugacy class class quickly, i.e. without summing over the class. NOTE: Orthonormality of cent_basis and unitarity of rho are checked. See ClassSumCentralizerNC
(6.2-2) for a version of this function without checks. The checks are not very expensive, so it is recommended you use the function with checks.
‣ ClassSumCentralizerNC ( rho, class, cent_basis ) | ( function ) |
The same as ClassSumCentralizer
(6.2-1), but does not check the basis for orthonormality or the representation for unitarity.
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