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6 Centralizer (commutant) rings
 6.1 Finding a basis for the centralizer
 6.2 Using the centralizer for computations

6 Centralizer (commutant) rings

6.1 Finding a basis for the centralizer

6.1-1 CentralizerBlocksOfRepresentation
‣ 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.

6.1-2 CentralizerOfRepresentation
‣ 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.

6.2 Using the centralizer for computations

6.2-1 ClassSumCentralizer
‣ 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.

6.2-2 ClassSumCentralizerNC
‣ 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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