Quelle _Chapter_Testing.xml Sprache: XML

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


<Chapter Label="Chapter_Testing">
<Heading>Methods for testing</Heading>

By the Chinese Remainder Theorem, it suffices to test irreps of prime power level,
so those are the irreps handled by the functions in this section.
<Section Label="Chapter_Testing_Section_Testing">
<Heading>Testing</Heading>

<ManSection>
  <Func Arg="p,lambda" Name="SL2WithConjClasses" />
<Returns>the group <Math>\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</Math> with conjugacy classes set to the format we use.
</Returns>
<Description>
<P/>
</Description>
</ManSection>

<ManSection>
  <Func Arg="S,T,p,lambda" Name="SL2ChiST" />
<Returns>a list representing a character of <Math>\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</Math>.
</Returns>
<Description>
Converts the modular data <Math>(S,T)</Math>, which must have level dividing <Math>p^\lambda</Math>,
into a character of <Math>\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</Math>, presented in a
form matching the conjugacy classes used in <Code>SL2WithConjClasses</Code>.
</Description>
</ManSection>

<ManSection>
  <Func Arg="p,lambda" Name="SL2TestPositions" />
<Returns>a boolean.
</Returns>
<Description>
Constructs and tests all non-trivial irreps of level dividing <Math>p^\lambda</Math> by checking their
positions in <Code>Irr(G)</Code> (see <URL> <Link>https://www.gap-system.org/Manuals/doc/ref/chap71.html#X873B3CC57E9A5492</Link><LinkText>Section 71.8-2 of the GAP Manual</LinkText></URL>).
Note that this function will print information on the irreps involved if <Code>InfoSL2Reps</Code> is set to level 1 or higher; see Section <Ref Sect="Chapter_Introduction_Section_Usage"/>.
</Description>
</ManSection>

<ManSection>
  <Func Arg="p,lambda" Name="SL2TestSymmetry" />
<Returns>a boolean.
</Returns>
<Description>
Constructs and tests all irreps of level <Math>p^\lambda</Math>, confirming that the
<Math>S</Math>-matrix is symmetric and unitary and the <Math>T</Math> matrix is diagonal.
Note that this function will print information on the irreps involved if <Code>InfoSL2Reps</Code> is set to level 1 or higher; see Section <Ref Sect="Chapter_Introduction_Section_Usage"/>.
</Description>
</ManSection>

</Section>

</Chapter>

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