Quelle _Chapter_Centralizer_commutant_rings.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Centralizer_commutant_rings">
<Heading>Centralizer (commutant) rings</Heading>

<Section Label="Chapter_Centralizer_commutant_rings_Section_Finding_a_basis_for_the_centralizer">
<Heading>Finding a basis for the centralizer</Heading>

<ManSection>
  <Func Arg="rho" Name="CentralizerBlocksOfRepresentation" />
<Returns>List of vector space generators for the centralizer ring  of <Math>\rho(G)</Math>, written in the basis given by <Ref
  Func="BlockDiagonalBasisOfRepresentation" />.  The matrices are
  given as a list of blocks.
</Returns>
<Description>
Let <Math>G</Math> have irreducible representations <Math>\rho_i</Math> with
multiplicities <Math>m_i</Math>. The centralizer has dimension <Math>\sum_i m_i^2</Math>
as a <Math>\mathbb{C}</Math>-vector space. This function gives the minimal
number of generators required.
<P/>
<P/>
<#Include Label="Example_CentralizerBlocksOfRepresentation">

<P/>
</Description>
</ManSection>

<ManSection>
  <Func Arg="arg" Name="CentralizerOfRepresentation" />
<Returns>List of vector space generators for the centralizer ring  of <Math>\rho(G)</Math>.
</Returns>
<Description>
This gives the same result as <Ref
Func="CentralizerBlocksOfRepresentation" />, but with the matrices
given in their entirety: not as lists of blocks, but as full
matrices.
<P/>
<P/>
<#Include Label="Example_CentralizerOfRepresentation">

<P/>
</Description>
</ManSection>

</Section>

<Section Label="Chapter_Centralizer_commutant_rings_Section_Using_the_centralizer_for_computations">
<Heading>Using the centralizer for computations</Heading>

<ManSection>
  <Func Arg="rho, class, cent_basis" Name="ClassSumCentralizer" />
<Returns><Math>\sum_{s \in t^G} \rho(s)</Math>, where <Math>t</Math> is a representative  of the conjugacy class <A>class</A> of <Math>G</Math>.
</Returns>
<Description>
We require that <A>rho</A> is unitary. Uses the given
orthonormal basis (with respect to the inner product <Math>\langle A, B
\rangle = \mbox{Trace}(AB^*)</Math>) for the centralizer ring of
<A>rho</A> to calculate the sum of the conjugacy class <A>class</A>
quickly, i.e. without summing over the class.
<P/>
NOTE: Orthonormality of <A>cent_basis</A> and unitarity of
<A>rho</A> are checked. See <Ref Func="ClassSumCentralizerNC" />
for a version of this function without checks. The checks are not
very expensive, so it is recommended you use the function with
checks.
<P/>
<P/>
<#Include Label="Example_ClassSumCentralizer">

<P/>
</Description>
</ManSection>

<ManSection>
  <Func Arg="rho, class, cent_basis" Name="ClassSumCentralizerNC" />
<Description>
The same as <Ref Func="ClassSumCentralizer" />, but
does not check the basis for orthonormality or the representation
for unitarity.
<P/>
<P/>
<#Include Label="Example_ClassSumCentralizerNC">

<P/>
</Description>
</ManSection>

</Section>

</Chapter>

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