<Section Label="sec:BM">
<Heading>Minimal polynomials of sequences</Heading>
<#Include Label="InvModCoeffs">
<#Include Label="BerlekampMassey">
</Section>
<Section Label="sec:BrauerChar">
<Heading>Brauer characters with respect to different lifts</Heading>
Let <M>G</M> be a finite group, <M>g \in G</M>, and <M>\rho: G \to
GL(d,p^n)</M>be a representation over a finite field. The Brauer character
value <M>\chi(g)</M> of <M>\rho</M> at <M>g</M> is defined as the
sum of the eigenvalues of <M>\rho(g)</M> in the algebraic closure of
<M>\mathbb{F}_p</M> lifted to complex roots of unity.
<P/>
The lift used by <Ref BookName="Reference" Oper="BrauerCharacterValue"/>
and in the computation of many Brauer character tables (available through
the <Package>CTblLib</Package> package) is defined by Conway polynomials
(see <Ref BookName="Reference" Func="ConwayPolynomial"/>): They define the
primitive root <C>Z(q)</C> in <C>GF(q)</C> which is mapped to <M>\exp(2
\pi i / (q-1))</M> (that is <C>E(q-1)</C> in &GAP;).
<P/>
Another lift is defined by the function <Ref
Func="StandardCyclicGenerator"/> provided by this package. Here,
<C>StandardCyclicGenerator(F, m)</C> is mapped to <M>\exp(2 \pi i / m)</M>
(that is <C>E(m)</C> in &GAP;).
<P/>
The following function translates between these two lifts.
<#Include Label="StandardValuesBrauerCharacter">
The inverse of a lift is used to reduce character values in characteristic
<M>0</M> modulo a prime <M>p</M>. Choosing a lift is equivalent
to choosing a <M>p</M>-modular system. &GAP; has the function <Ref
BookName="Reference" Attr="FrobeniusCharacterValue"/> which computes this
reduction with respect to the lift defined by Conway polynomials.
<P/>
Here is the corresponding function with respect to the lift constructed in
this package.
<#Include Label="FrobeniusCharacterValues">
</Section>
<Section Label="sec:FactorData">
<Heading>Known factorizations of multiplicative group orders</Heading>
<#Include Label="CANFACT">
</Section>
<Section Label="sec:Tests">
<Heading>Some loops for <Package>StandardFF</Package></Heading>
<#Include Label="TestLoops">
</Section>
<Section Label="sec:nondoc">
<Heading>Undocumented features</Heading>
We mention some features of this package which may be temporary, vanish or
changed.
<P/>
A directory <F>ntl</F> contains some simple standalone programs
which use the library NTL <Cite Key="NTL"/>. There is a function
<C>StandardIrreducibleCoeffListNTL(K, d, a)</C> which can be used instead
of <C>StandardIrreducibleCoeffListNTL(K, d, a)</C> when <C>K</C> is a
prime field. This gives a good speedup for not too small <C>d</C>, say
<C>d</C> <M>>500</M>.
</Section>
</Chapter>
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