GAP (RepnDecomp) - Chapter 6: Centralizer (commutant) rings

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6 Centralizer (commutant) rings
6.1 Finding a basis for the centralizer
6.1-1 CentralizerBlocksOfRepresentation
6.1-2 CentralizerOfRepresentation
6.2 Using the centralizer for computations
6.2-1 ClassSumCentralizer
6.2-2 ClassSumCentralizerNC

### 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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