SSL div-alg.xml Sprache: XML

<Chapter Label="Div-alg">
<Heading>Functions for calculating Schur indices and identifying division algebras</Heading>

<Section Label="MainSchurIndexFunctions">
<Heading>Main Schur Index and Division Algebra Functions</Heading>

<ManSection>
   <Prop Name="WedderburnDecompositionWithDivAlgParts"
         Arg="A"/>
   <Returns>
         A list of lists <C>[r,D]</C>, each representing a ring of
         <M> r \times r </M> matrices over a field or division algebra <K>D</K>.
   </Returns>

   <Description>
         The input <A>A</A> should be a group ring of a finite group over an abelian number
         field.  The function will give the same result as <Ref Attr="WedderburnDecompositionInfo" />
         if the field of coefficients for the group ring is finite.  The output is a list of
         pairs <C>[r,D]</C>, each of which indicates a simple component isomorphic to the
         ring of <M> r \times r</M> matrices over a division algebra described using the
         information in the record <C>D</C>.  This record contains information on the center,
         Schur index, and local indices of the division algebra. <P/>

<C>Local indices</C> is a list of pairs <M>[p,m]</M>, where <M>p</M> is a rational prime
(possibly 'infinity') and <M>m</M> is the local index of  the division algebra at the prime <M>p</M>.

<Example>
<![CDATA[
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 5 generators>
gap> WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
  [ 2, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ],
  [ 2, CF(3) ], [ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
]
gap> WedderburnDecompositionWithDivAlgParts(R);
[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],
  [ 2, Rationals ], [ 2, Rationals ], [ 2, Rationals ], [ 2, NF(8,[ 1, 7 ]) ],
  [ 2, CF(3) ],
  [ 2,
      rec( Center := Rationals, DivAlg := true,
          LocalIndices := [ [ 2, 2 ], [ 3, 2 ] ], SchurIndex := 2 ) ] ]
]]>
</Example>

   </Description>
</ManSection>

<ManSection>
   <Prop Name="CyclotomicAlgebraWithDivAlgPart"
         Arg="A"  />
   <Returns>
         A list of length two indicating a matrix ring of a given size over a field or
         a noncommutative division algebra.
   </Returns>

   <Description>
         The input <A>A</A> should be a cyclotomic algebra; i.e. a crossed
         product in the same form as in the output of
         <Ref Attr="WedderburnDecompositionInfo" />.  The output is in
         the form <C>[r,D]</C>, which indicates an <M> r \times r </M> matrix ring over
         the division algebra described by <C>D</C>.  <C>D</C> is either a field or a
         noncommutative division algebra described using a record giving information
         on the center, Schur index, and local indices of the division algebra.

<Example>
<![CDATA[
gap> G:=SmallGroup(240,89);
<permutation group of size 240 with 2 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 2 generators>
gap> W:=WedderburnDecompositionInfo(R);
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!

[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],
  [ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],
  [ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],
  [ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ]
gap> CyclotomicAlgebraWithDivAlgPart(W[3]);
[ 2, rec( Center := Rationals, DivAlg := true,
      LocalIndices := [ [ 5, 2 ], [ infinity, 2 ] ], SchurIndex := 2 ) ]
gap> CyclotomicAlgebraWithDivAlgPart(W[9]);
[ 2, rec( Center := NF(12,[ 1, 11 ]), DivAlg := true,
      LocalIndices := [ [ infinity, 2 ] ], SchurIndex := 2 ) ]
gap> CyclotomicAlgebraWithDivAlgPart(W[10]);
[ 3, rec( Center := NF(8,[ 1, 7 ]), DivAlg := true,
      LocalIndices := [ [ infinity, 2 ] ], SchurIndex := 2 ) ]
]]>
</Example>

   </Description>
</ManSection>

<ManSection>
   <Prop Name="SchurIndex"
         Arg="A"  />
   <Oper Name="SchurIndexByCharacter"
         Arg="F G n " />

   <Returns>
         The first of these returns the Schur index of the simple algebra <A>A</A>.  The
         second returns the Schur index of the simple component of the group ring <A>FG</A>
         corresponding to the irreducible character <C>Irr(G)[n]</C> of <A>G</A>.
   </Returns>

   <Description>
These are the main functions for computing Schur indices.  The first can be used to find
the rational Schur index of a simple component of the group ring of a finite group over
an abelian number field, or a quaternion algebra in &GAP; (see
<Ref Func="QuaternionAlgebra" BookName="ref" />) whose center is the field of rational
numbers. If <A>A</A> is a quaternion algebra over a number field other than the Rationals,
<C>fail</C> is returned.  In these cases, the quaternion algebra can be converted to a
cyclic algebra and the Schur index of the cyclic algebra can be determined through the
solution of norm equations.  Currently this functionality is not implemented in &GAP;,
but available in number theory packages such as <Package>PARI/GP</Package>.
   <P/>

The second function computes the Schur index of the cyclotomic algebra
that would occur as the simple component of the group ring <A>FG</A>
that corresponds to the irreducible character <C>Irr(G)[n]</C>.
The function uses <Ref Attr="SimpleComponentByCharacterDescent" />, which uses
the character descent algorithm to reduce to as small a group as possible. For
larger groups this is preferrable as it is less demanding on memory.  The Schur
index of the resulting cyclotomic algebra is then computed with the
<Ref Attr="SchurIndex" /> function.
   <P/>

<Example>
<![CDATA[
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 3 generators>
gap> W:=WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
  [ 1, NF(21,[ 1, 4, 16 ]), 21, [ 3, 4, 7 ] ] ]
gap> SchurIndex(W[5]);
3
gap> G:=SmallGroup(40,3);
<pc group of size 40 with 4 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> SchurIndexByCharacter(GaussianRationals,G,Irr(G)[i]);
2
gap> SchurIndexByCharacter(CF(3),G,i);
1
]]>
</Example>

<Log>
<![CDATA[
gap> G:=AtlasGroup("J2");
<permutation group of size 604800 with 2 generators>
gap> Irr(G)[20];;
gap> SchurIndexByCharacter(Rationals,G,Irr(G)[20]);
1
]]>
</Log>

   </Description>
</ManSection>

<ManSection>
   <Oper Name="WedderburnDecompositionAsSCAlgebras"
         Arg="R"  />
   <Oper Name="CyclotomicAlgebraAsSCAlgebra"
         Arg="A" />
   <Oper Name="SimpleComponentByCharacterAsSCAlgebra"
         Arg="F G n " />

   <Returns>
         The first of these returns the Wedderburn decomposition of the group ring <C>R</C>
         with each simple component presented as an algebra with structure constants in &GAP;
         (see <Ref Sect="Constructing Algebras by Structure Constants" BookName="ref"/> in
         the main &GAP; manual).  The second converts a list <C>A</C> that is output from
         <Ref Attr="WedderburnDecompositionInfo"/> into an algebra with structure constants
         in &GAP;.  The third determines an algebra with structure constants that is
         isomorphic to the simple component of the group ring of the finite group <C>G</C>
         over the field <C>F</C> that corresponds to the irreducible character <C>Irr(G)[n]</C>.
   </Returns>

   <Description>
These functions are an option for obtaining a Wedderburn decomposition or simple component
of the group ring <C>FG</C> in which the output is in the form of an algebra with structure
constants, which is more compatible with GAP's built-in operations for finite-dimensional algebras.

<Example>
<![CDATA[
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 3 generators>
gap> W:=WedderburnDecompositionInfo(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
  [ 1, NF(21,[ 1, 4, 16 ]), 21, [ 3, 4, 7 ] ] ]
gap> WedderburnDecompositionWithDivAlgParts(R);
[ [ 1, Rationals ], [ 1, CF(3) ], [ 1, CF(9) ], [ 3, NF(7,[ 1, 2, 4 ]) ],
  [ 1,
      rec( Center := NF(21,[ 1, 4, 16 ]), DivAlg := true,
          LocalIndices := [ [ 7, 3 ] ], SchurIndex := 3 ) ] ]
gap> WedderburnDecompositionAsSCAlgebras(R);
[ Rationals, CF(3), CF(9), ( NF(7,[ 1, 2, 4 ])^[ 3, 3 ] ),
  <algebra of dimension 9 over NF(21,[ 1, 4, 16 ])> ]
gap> CyclotomicAlgebraAsSCAlgebra(W[5]);
<algebra of dimension 9 over NF(21,[ 1, 4, 16 ])>
gap> Size(Irr(G));
15
gap> SimpleComponentByCharacterAsSCAlgebra(Rationals,G,15);
<algebra of dimension 9 over NF(21,[ 1, 4, 16 ])>

]]>
</Example>

   </Description>
</ManSection>

</Section>



<Section Label="Cyclotomic Reciprocity Functions">
<Heading>Cyclotomic Reciprocity Functions </Heading>

<ManSection>
<Oper Name="PPartOfN"
         Arg="n p" />
<Oper Name="PDashPartOfN"
         Arg="n p" />
   <Description>
  These are standard arithmetic functions required by several
  subroutines for the cyclotomic reciprocity and Schur index
  functions in <Package>Wedderga</Package>.

<Example>
<![CDATA[
gap> PPartOfN(2275,5);
25
gap> PDashPartOfN(2275,5);
91
]]>
</Example>

   </Description>
</ManSection>

<ManSection>
<Oper Name="PSplitSubextension"
         Arg="F,n,p"   />
   <Returns> The maximal subextension <C>K</C> of the cyclotomic extension <C>F(E(n))/F</C>
   for which <M>K/F</M> splits completely at the prime <M>p</M>.
   </Returns>
   <Description> This function finds the maximal subextension <C>K</C>
   of the cyclotomic extension <C>F(E(n))</C> of an abelian number field <C>F</C> for
   which both the ramification index and residue degree of <C>K/F</C> over any prime
   lying over <A>p</A> are <M>1</M>.  To do this, it finds the field fixed by an
   appropriate power of the field automorphism inducing the local Frobenius automorphism.

<Example>
<![CDATA[
gap> PSplitSubextension(Rationals,60,5);
GaussianRationals
gap> PSplitSubextension(NF(5,[1,4]),70,2);
NF(35,[ 1, 4, 9, 11, 16, 29 ])
gap> PSplitSubextension(NF(40,[1,9,11,19]),20,2);
NF(40,[ 1, 9, 11, 19])
]]>

</Example>

  </Description>
</ManSection>

<ManSection>
<Oper Name="SplittingDegreeAtP"
         Arg="F,n,p" />
<Oper Name="ResidueDegreeAtP"
         Arg="F,n,p"  />
<Oper Name="RamificationIndexAtP"
         Arg="F,n,p" />
   <Returns> The splitting degree, residue degree, and ramification index  of the extension
   <C>F(E(n))/F</C> at the prime <M>p</M>.
   </Returns>
   <Description> These functions calculate the cyclotomic reciprocity parameters
   <C>g</C>, <C>f</C>, and <C>e</C> for the extension <C>F(E(n))/F</C> at the prime
   <M>p</M> for an abelian number field <C>F</C>.  To do this, it finds the <C>p</C>-split
   subextension <C>K</C> and the <M>p</M>-dash part <M>n' of n, then calculates
   <C>g = [K:F]</C>, <C>f = [K(E(n'):K], and e = [K(E(n)):K(E(n'))]</C>.  These
   functions enable the user to calculate cyclotomic reciprocity parameters for any extension
   of abelian number fields, as the example illustrates.

<Example>
<![CDATA[
gap> F:=CF(12);
CF(12)
gap> K:=NF(120,[1,49]) # Note that F is a subfield of K, with index 4.
> ; # Then we can find e, f, and g for the extension K/F at the prime 5.
NF(120,[ 1, 49 ])
gap> RamificationIndexAtP(F,120,5); RamificationIndexAtP(K,120,5); last2/last;
4
2
2
gap> ResidueDegreeAtP(F,120,5); ResidueDegreeAtP(K,120,5); last2/last;
1
1
1
gap> SplittingDegreeAtP(F,120,5); SplittingDegreeAtP(K,120,5); last2/last;
2
1
2
]]>
</Example>
</Description>
</ManSection>

</Section>



<Section Label="GlobalSplittingAndCharacterDescentFunctions">
<Heading>Global Splitting and Character Descent Functions </Heading>

<ManSection>
   <Oper Name="GlobalSplittingOfCyclotomicAlgebra"
         Arg="A" />
   <Oper Name="KillingCocycle"
         Arg="A" />
   <Oper Name="AntiSymMatUpMat"
         Arg="x" />
   <Oper Name="CyclotomicExtensionGenerator"
         Arg="K,F"/>
   <Oper Name="ReducingCyclotomicAlgebra"
         Arg="A" />

   <Returns>
If the input cyclotomic crossed product algebra <C>A</C> admits a rescaling of
  basis elements by roots of unity that makes it equivalent to one with a trivial
  factor set, the first function here returns a full matrix algebra over the same
  center with the same dimension as <C>A</C>.  In general the function will output
  a cyclotomic algebra equivalent to <C>A</C> whose factor set does not admit a
  partial splitting by a simple rescaling of basis elements by roots of unity.
  <Ref Attr="KillingCocycle"/> is a subroutine used by
  <Ref Attr="GlobalSplittingOfCyclotomicAlgebra"/> function
  to produce equivalent factor sets that have more zeroes,
  <Ref Attr="AntiSymMatUpMat"/> is a subroutine it uses to manipulate the factor
  set information in anti-symmetric matrix form.
  <Ref Attr="ReducingCyclotomicAlgebra"/> is another subroutine used by the
  <Ref Attr="GlobalSplittingOfCyclotomicAlgebra"/> that either returns an isomorphic
  cyclotomic algebra whose associated Galois group has fewer generators than
  the input, or returns fail.  <Ref Attr="CyclotomicExtensionGenerator"/> is a
  subroutine called by <Ref Attr="ReducingCyclotomicAlgebra"/> which returns a
  positive integer <C>m</C> if the field <C>K</C> is the cyclotomic extension of
  <C>F</C> obtained by adjoining a primitive <C>m</C>-th root of <C>1</C>, and
  returns fail otherwise.
   </Returns>

   <Description>
        The input <C>A</C> must be a list representing a cyclotomic algebra in the
        same form as in the output of <Ref Attr="WedderburnDecompositionInfo"/> or
        <Ref Attr="SimpleAlgebraByCharacterInfo"/>.  If the output has length 5,
        the function will first search using <Ref Attr="KillingCocycle"/> and
        <Ref Attr="AntiSymMatUpMat"/> for a rescaling of basis elements that
        makes the factor set trivial, and in general finds one that produces as
        many zeroes as possible in the factor set information.  If there is a
        generator that produces a split tensor factor with a cyclotomic algebra
        <C>B</C> defined over a root of unity of smaller order, then
        <Ref Attr="ReducingCyclotomicAlgebra"/> reduces to <C>B</C>.  These
        steps are repeated until such tensor factorizations are no longer
        available.
      <P/>

<Example>
<![CDATA[
gap> A := [ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 0 ] ], [ [ 0 ] ] ];;
gap> GlobalSplittingOfCyclotomicAlgebra(A);
[ 8, Rationals ]
gap> A := [ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 10 ] ], [ [ 0 ] ] ];;
gap> GlobalSplittingOfCyclotomicAlgebra(A);
[ 2, Rationals, 10, [ 4, 3, 5 ] ]
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
  <Oper Name="CharacterDescent"
      Arg="F,G,n,e,H" />
   <Oper Name="GlobalCharacterDescent"
         Arg="F,G,n" />
   <Oper Name="SimpleComponentByCharacterDescent"
         Arg="F,G,n" />

   <Returns>
  The first function returns <C>[r,F1,H,k]</C> where <C>r</C> is a positive
  rational, <C>F1</C> is an extension field of <C>F</C>, <C>H</C> is the input
  subgroup <C>H</C>, and <C>k</C> is the position in <C>Irr(H)</C> of the
  irreducible character that admits a Global Character Descent reduction; i.e.
  an irreducible character of <C>H</C> that has the same local indices as
  <C>Irr(G)[n]</C>.

  The second function returns the limit of the Character Descent reductions,
  it returns the last <Ref Attr="CharacterDescent"/> output obtained before
  reaching a state where no maximal subgroup of <C>G</C> admits a further
  reduction by <Ref Attr="CharacterDescent"/>.

  The third function returns the simple component of <C>FG</C> resulting from
  <Ref Attr="GlobalCharacterDescent"/> that is produced by applying
  <Ref Attr="SimpleComponentOfGroupRingByCharacter"/> and adjusting the matrix
  degree.
  </Returns>

   <Description>
        The <Ref Attr="CharacterDescent"/> function tries to find an irreducible
        character <C>psi</C>=<C>Irr(H)[k]</C> of <C>H</C> for which
        <C>[F(chi,psi),F(chi)]</C> and <C>(chi_H,psi)</C> are both coprime to
        <C>|G|</C>.  This character <C>psi</C> will have the same local indices
        over <C>F</C> as <C>chi</C> by a theorem of Yamada.
        The <Ref Attr="GlobalCharacterDescent"/> function iterates a search for
        these global character descent reductions over irreducible characters of
        maximal subgroups of <C>G</C>, and when it finds one it replaces <C>G</C>,
        <C>F(chi)</C>, and <C>chi</C> by <C>H</C>, <C>F(chi,psi)</C>, and <C>psi</C>,
        and then begins a search over maximal subgroups of <C>H</C>.  It terminates
        when no maximal subgroup admits a global reduction, and returns the
        relevant matrix degree factor along with the last <C>H</C> and <C>psi</C>
        it found.  <Ref Attr="SimpleComponentByCharacterDescent"/> implements an
        algorithm that returns the simple component of <C>FG</C> associated with
        <C>Irr(G)[n]</C> obtained using the global character descent algorithm.
      <P/>

<Example>
<![CDATA[
gap> G:=PSU(3,3);
<permutation group of size 6048 with 2 generators>
gap> sc := SimpleComponentByCharacterDescent(Rationals,G,8);;
gap> sc{[1..3]}; # the 4th entry is [ 2, 5, 3 ] or [ 2, 5, 9 ]
[ 21/2, GaussianRationals, 12 ]
gap> SchurIndex(sc);
1
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
  <Oper Name="GaloisRepsOfCharacters"
      Arg="F,G" />

   <Returns>
  A list of positive integers <C>n</C> that indicates the positions in <C>Irr(G)</C>
  of the Galois conjugacy classes of irreducible characters of <C>G</C> over the
  field <C>F</C>.
  </Returns>

   <Description>
        <Ref Attr="GaloisRepsOfCharacters"/> finds a list of representatives of
        the distinct Galois conjugacy classes of irreducible characters of <C>G</C>
        over <C>F</C>.  It runs through the irreducible characters and determines
        if a given irreducible is Galois conjugate over <C>F</C> to any of the
        previous ones, and if not it adds the position of that character to the
        list.
      <P/>
<Example>
<![CDATA[
gap> G:=SmallGroup(63,1);
<pc group of size 63 with 3 generators>
gap> GaloisRepsOfCharacters(Rationals,G);
[ 1, 2, 4, 10, 12 ]
gap> GaloisRepsOfCharacters(CF(9),G);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13 ]
gap> GaloisRepsOfCharacters(NF(7,[1,2,4]),G);
[ 1, 2, 4, 10, 11, 12, 14 ]
gap> GaloisRepsOfCharacters(CF(63),G);
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ]
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
  <Oper Name="WedderburnDecompositionByCharacterDescent"
      Arg="F,G" />

   <Returns>
     A Wedderburn decomposition of <C>FG</C> that is obtained by applying the
    global character descent algorithm to representatives of the Galois
    conjugacy classes of irreducible characters of <C>G</C> over <C>F</C>.
  </Returns>

   <Description>
        <Ref Attr="WedderburnDecompositionByCharacterDescent"/> is available as
        an option to <Ref Attr="WedderburnDecompositionInfo"/> that can be more
        effective for larger groups, especially ones with relatively few
        irreducible characters.  As long as <C>GAP</C> is able to compute
        maximal subgroups and restrict irreducible characters of <C>G</C> to
        them, this function should eventually return a Wedderburn decomposition.
        As it operates on one irreducible character at a time, it is less
        effective on Abelian groups, which have far too many irreducible characters,
        and on most groups in GAP's Small Groups Library, where memory limitations
        are not a factor and it is possible for the Shoda pair algorithms can
        deal with many irreducible characters at the same time.
      <P/>
<Example>
<![CDATA[
gap> G:=GL(3,3);
GL(3,3)
gap> Size(G);
11232
gap> WedderburnDecompositionByCharacterDescent(Rationals,G);
[ [ 1, Rationals ], [ 1, Rationals ], [ 12, Rationals ], [ 12, Rationals ],
[ 13, Rationals ], [ 13, Rationals ], [ 16, NF(13,[ 1, 3, 9 ]) ],
[ 16, NF(13,[ 1, 3, 9 ]) ], [ 26, Rationals ], [ 26, Rationals ],
[ 26, NF(8,[ 1, 3 ]) ], [ 26, NF(8,[ 1, 3 ]) ], [ 27, Rationals ],
[ 27, Rationals ], [ 39, Rationals ], [ 39, Rationals ] ]
]]>
</Example>

   </Description>
</ManSection>

</Section>



<Section Label="LocIndsOfCyclicCyclotomicAlgs">
<Heading>Local index functions for Cyclic Cyclotomic Algebras</Heading>

<ManSection>
   <Oper Name="LocalIndicesOfCyclicCyclotomicAlgebra"
         Arg="A" />
   <Returns>
A list of the pairs <C>[p,m]</C> indicating the nontrivial local indices <C>m</C> at
the primes <C>p</C> of the cyclic cyclotomic algebra indicated by <C>A</C>.
   </Returns>
   <Description>
        The input <C>A</C> must be a list representing a cyclic cyclotomic algebra in the
        same form as in the output of <Ref Attr="WedderburnDecompositionInfo"/> or
        <Ref Attr="SimpleAlgebraByCharacterInfo"/>.  This function computes the local
        Schur indices at rational primes <M>p</M> using the specialized functions for
        cyclic cyclotomic algebras described in this section. <P/>

<Example>
<![CDATA[
gap> A:=[1,Rationals,6,[2,5,3]];
[ 1, Rationals, 6, [ 2, 5, 3 ] ]
gap> LocalIndicesOfCyclicCyclotomicAlgebra(A);
[ [ 3, 2 ], [ infinity, 2 ] ]
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
   <Oper Name="LocalIndexAtInfty"
         Arg="A"  />
   <Oper Name="LocalIndexAtTwo"
         Arg="A"  />
   <Oper Name="LocalIndexAtOddP"
         Arg="A,p" />

   <Returns>
  These return the local index of the cyclic cyclotomic algebra <A>A</A> at the indicated
  rational prime.
   </Returns>

   <Description>
         The input <C>A</C> must be a cyclic cyclotomic algebra;
         that is, a list of the form <C>[r,F,n,[a,b,c]]</C> that indicates a cyclic
         cyclotomic crossed product algebra.  This is a special case of the output of
         <Package>wedderga</Package>'s
         <Ref Attr="WedderburnDecompositionInfo"/> or
         <Ref Attr="SimpleAlgebraByCharacterInfo"/>.
         For the <Ref Oper="LocalIndexAtOddP"/> function, <A>p</A> must be an odd prime.
         The functions <Ref Attr="PPartOfN"/> and <Ref Attr="PDashPartOfN"/> are standard
         (and self-explanatory) arithmetic functions for a positive integer <M>n</M> and prime <M>p</M>.
         <P/>

         These functions determine the local index of a cyclic cyclotomic algebra at the
         rational primes <C>'infinity'</C>, <M>2</M>, or odd primes <M>p</M>, respectively.
         The first two functions check for a relationship of <M>A</M> to a nonsplit real
         or 2-adic quaternion algebra.
<Ref Attr="LocalIndexAtOddP"/> calculates the local index at <M>p</M> by counting the number
of roots of unity coprime to <M>p</M> found in the <M>p</M>-adic completion, and using a
formula due to Janusz.  <P/>

<Example>
<![CDATA[
gap> A:=[1,CF(4),20,[4,13,15]];
[ 1, GaussianRationals, 20, [ 4, 13, 15 ] ]
gap> LocalIndexAtOddP(A,5);
4
gap> A:=[1,NF(8,[1,7]),8,[2,7,4]];
[ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ]
gap> LocalIndexAtInfty(A);
2
gap> A:=[1,CF(7),28,[2,15,14]];
[ 1, CF(7), 28, [ 2, 15, 14 ] ]
gap> LocalIndexAtTwo(A);
2
]]>
</Example>

   </Description>
</ManSection>

</Section>



<Section Label="LocIndsOfCyclotomicAlgs">
<Heading>Local index functions for Non-Cyclic Cyclotomic Algebras</Heading>

<ManSection>
   <Oper Name="LocalIndicesOfCyclotomicAlgebra"
         Arg="A" />
  <Returns> A list of pairs <C>[p,m]</C> indicating the nontrivial local indices <C>m</C>
  at the primes <C>p</C> of the cyclic cyclotomic algebra indicated by <C>A</C>.
  </Returns>

  <Description> The input <C>A</C> should be a cyclotomic algebra; i.e. a list of length
  2, 4, or 5 in the form of the output by <Package>Wedderga</Package>'s -Info
  functions.  If the cyclotomic algebra <A>A</A> is represented by a list of length 2,
  the local indices are all <M>1</M>, so the function will return an empty list.  If the
  cyclotomic algebra <A>A</A> is given by a list of length 4, then it represents a cyclic
  cyclotomic algebra, so the function <Ref Oper="LocalIndicesOfCyclicCyclotomicAlgebra"/>
  is utilized.

If the cyclotomic algebra <C>A</C> is presented as a list of length 5 or more,
the function first applies <Ref Oper="GlobalSplittingOfCyclotomicAlgebra"/> to
reduce the length as much as possible.  If this does not reduce the length to 4 or
less, it applies the character descent algorithm to try to reduce it again with
Clifford theory: it determines the group and character <C>chi</C> that faithfully
represent the algebra using <Ref Oper="DefiningGroupOfCyclotomicAlgebra"/> and
<Ref Oper="DefiningCharacterOfCyclotomicAlgebra"/>, then applies
<Ref Oper="SimpleComponentByCharacterDescent"/>.  It repeats this until it cannot
reduce the length of cyclotomic algebra any longer.  If the length is 4 it will
apply the local index functions for cyclic cyclotomic algebras to compute the
local indices at each prime dividing the order of the group.  If the length
is 5 or more, it applies the character-theoretic local Schur index functions to
the output <C>[G,chi]</C> of <Ref Oper="DefiningGroupAndCharacterOfCyclotAlg"/>.
It uses the Frobenius-Schur indicator of <C>chi</C> to determine the local index
at infinity (see <Ref Oper="LocalIndexAtInftyByCharacter"/>).  For local indices
at odd primes and sometimes for the prime <M>2</M>, the defect group of the block
containing <C>chi</C> will be cyclic, so the local index can be found using the
values of a Brauer character by a theorem of Benard (see
<Ref Oper="LocalIndexAtPByBrauerCharacter"/>.)   Sometimes for the prime 2 the
defect group is not necessarily cyclic, so in these cases we appeal to the
classification of dyadic Schur groups by Schmid and Riese
(see <Ref Oper="LocalIndexAtTwoByCharacter"/>).

<Example>
<![CDATA[
gap> G:=SmallGroup(480,600);
<pc group of size 480 with 7 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
27
gap> W[27];
[ 1, NF(5,[ 1, 4 ]), 60, [ [ 2, 11, 0 ], [ 2, 19, 30 ], [ 2, 31, 30 ] ],
  [ [ 0, 45 ], [ 15 ] ] ]
gap> LocalIndicesOfCyclotomicAlgebra(W[27]);
[ [ infinity, 2 ] ]
]]>
</Example>

<Example>
<![CDATA[
gap> G:=SmallGroup(160,82);
<pc group of size 160 with 6 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
14
gap> W[14];
[ 1, Rationals, 20, [ [ 2, 11, 0 ], [ 4, 3, 0 ] ], [ [ 5 ] ] ]
gap> LocalIndicesOfCyclotomicAlgebra(W[14]);
[ [ 2, 2 ], [ 5, 2 ] ]
]]>
</Example>

  </Description>
</ManSection>

<ManSection>
   <Oper Name="RootOfDimensionOfCyclotomicAlgebra"
         Arg="A" />
   <Returns> A positive integer representing the square root of the dimension of the
   cyclotomic algebra over its center. </Returns>
   <Description>

<Example>
<![CDATA[
gap> A:=[3,Rationals,12,[[2,5,3],[2,7,0]],[[3]]];
[ 3, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
gap> RootOfDimensionOfCyclotomicAlgebra(A);
12
]]>
</Example>

   </Description>
</ManSection>

<ManSection>
   <Oper Name="DefiningGroupOfCyclotomicAlgebra"
         Arg="A"  />
   <Oper Name="DefiningCharacterOfCyclotomicAlgebra"
         Arg="A"  />
   <Oper Name="DefiningGroupAndCharacterOfCyclotAlg"
         Arg="A"  />
   <Returns> These functions return a finite group <C>G</C> and a positive integer
   <C>n</C> (or, in the case to the third function, an irreducible character <C>n</C>
   of <C>G</C> for which the simple component of a group algebra over <C>G</C> over
   the center of the cyclotomic algebra <C>A</C> corresponding to the character
   <C>Irr(G)[n]</C> (resp. <C>n</C>) will be (rationally) isomorphic to <C>A</C>.
  </Returns>

<Oper Name="SimpleComponentOfGroupRingByCharacter"
         Arg="F,G,n" />
<Returns>
A list that describes the algebraic structure of the simple component of the group
algebra <C>FG</C> which corresponds to the irreducible character <C>Irr(G)[n]</C> (or <C>n</C>). </Returns>

   <Description> This function is an alternative to
<C>SimpleAlgebraByCharacterInfo(GroupRing(F,G),</C> <C>Irr(G)[n]);</C>.
It is used in subroutines of local index functions when we need to work over a field
larger than the field of character values.

<Example>
<![CDATA[
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> R:=GroupRing(Rationals,G);
<algebra-with-one over Rationals, with 5 generators>
gap> W:=WedderburnDecompositionInfo(R);;
gap> A:=W[10];
[ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
gap> g:=DefiningGroupOfCyclotomicAlgebra(A);;
gap> IdSmallGroup(g);
[ 48, 15 ]
gap> n:=DefiningCharacterOfCyclotomicAlgebra(A);
12
gap> SimpleComponentOfGroupRingByCharacter(Rationals,G,n)
> ;#Note:this cyclotomic algebra is isomorphic to the other by a change of basis.
[ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ]
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
   <Oper Name="LocalIndexAtInftyByCharacter"
         Arg="F G n"  />
<Returns> The local index at an infinite prime of the field <A>F</A> of the irreducible
character <C>Irr(G)[n]</C> (or <C>n</C>) of the finite group <A>G</A>.
</Returns>

<Description> This function computes the Frobenius-Schur indicator of the irreducible
character <C>Irr(G)[n]</C>, and uses it to calculate the local index at <C>infinity</C>
of the corresponding simple component of <A>FG</A>.

<Example>
<![CDATA[
gap> G:=SmallGroup(48,16);
<pc group of size 48 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtInftyByCharacter(Rationals,G,i);
2
gap> LocalIndexAtInftyByCharacter(CF(3),G,Irr(G)[i]);
1

]]>
</Example>

</Description>
</ManSection>

<ManSection>
   <Oper Name="DefectGroupOfConjugacyClassAtP"
         Arg="G c p" />
   <Oper Name="DefectGroupsOfPBlock"
         Arg="G n p"  />
   <Oper Name="DefectOfCharacterAtP"
         Arg="G n p" />
<Returns>
      The first of these functions returns a defect group of the <C>c</C>-th conjugacy
      class of the finite group <C>G</C> at the prime <C>p</C>.  The second returns the
      conjugacy class of <C>p</C>-subgroups of <C>G</C> that consists of defect groups
      for the <C>p</C>-block containing the ordinary irreducible character <C>Irr(G)[n]</C>.
      The last of these functions returns the nonnegative integer <C>d</C> for which
      <C>p^d</C> is the order of a <C>p</C>-defect group for <C>Irr(G)[n]</C>.
</Returns>
<Description>
The <C>p</C>-defect group of a given conjugacy class of <C>G</C> is a <C>p</C>-Sylow
subgroup of the centralizer in <C>G</C> of any representative of the class.
A defect group for a <C>p</C>-block of <C>G</C> is a minimal <C>p</C>-subgroup that is a
defect group for a defect class of the block.  By Brauer's Min-Max theorem, this will
occur for at least one <C>p</C>-regular class of <C>G</C>.  The function
<Ref Attr="DefectGroupsOfPBlock"/> identifies the defect classes for the block containing
<C>Irr(G)[n]</C>, finds the one whose defect group has minimal order, and returns the
conjugacy class of the defect group of this class.  The function <Ref Attr="DefectOfCharacterAtP"/>
gives the logarithm base <C>p</C> of the order of a defect group of the <C>p</C>-block
containing the character <C>Irr(G)[n]</C>.

<Example>
<![CDATA[
gap> G:=SmallGroup(72,21);
<pc group of size 72 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> D:=DefectGroupOfConjugacyClassAtP(G,i,3);
Group([ f4, f5 ])
gap> IsCyclic(last);
false
gap> D:=DefectGroupsOfPBlock(G,Irr(G)[i],3);
Group( [ f4, f5 ] )^G
gap> IsCyclic(Representative(D));
false
gap> DefectOfCharacterAtP(G,Irr(G)[i],3);
2
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
   <Oper Name="LocalIndexAtPByBrauerCharacter"
         Arg="F G n p"  />
   <Oper Name="FinFieldExt"
         Arg="F G p n m"  />

   <Returns>
      The first returns the local index at the rational prime <M>p</M> of the simple
      component of the group ring <C>FG</C> that corresponds to <C>Irr(G)[n]</C>.
      The second returns the degree of a certain extension of finite fields of
      <C>p</C>-power order.
    </Returns>

   <Description>
         The input of <Ref Attr="LocalIndexAtPByBrauerCharacter"/> must be an abelian
         number field <A>F</A>, a finite group <A>G</A>, and the number <A>n</A> of an
         ordinary irreducible character <C>Irr(G)[n]</C>, and <A>p</A> a prime divisor
         of the order of <A>G</A>. Since this function is intended to be used for faithful
         characters of groups that are the defining groups of non-cyclic cyclotomic algebras
         that result from <Package>Wedderga</Package>'s Info functions, it is expected that
         <C>G</C> is a non-nilpotent cyclic-by-abelian group, and <C>Irr(G)[n]</C> is a
         faithful character.  The Brauer character table records of such groups can be
         accessed in <K>GAP</K> (provided <C>G</C> is sufficiently small).   <P/>

The local index calculation uses Benard's theorem, which shows that the local index at
<A>p</A> of the simple component of the rational group algebra <A>QG</A> corresponding
to the character <C>Irr(G)[n]</C> is the degree of the extension of the residue field
of the center given by adjoining an irreducible <A>p</A>-Brauer character
<C>IBr(G,p)[m]</C> lying in the same block, provided the defect group of the block
is cyclic.  If the defect group of the block is not cyclic, the resulting calculation
is unreliable, and the function will output a list whose second term is the warning
label <C>"DGnotCyclic"</C>.  The degree of this finite field extension is calculated
by <Ref Attr="FinFieldExt"/>.  It determines the local index relative to the field
<C>F</C> by dividing the local index at <M>p</M> over the rationals by a constant
determied using a theorem of Yamada. <P/>

<Example>
<![CDATA[
gap> G:=SmallGroup(80,28);
<pc group of size 80 with 5 generators>
gap> T:=CharacterTable(G);;
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> S:=T mod 5;
BrauerTable( <pc group of size 80 with 5 generators>, 5 )
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,i,5);
2
gap> FinFieldExt(Rationals,G,5,i,9);
2
]]>
</Example>

<Example>
<![CDATA[
gap> G:=SmallGroup(72,20);
<pc group of size 72 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,Irr(G)[i],3);
[ 2, "DGnotCyclic" ]
gap> LocalIndexAtPByBrauerCharacter(Rationals,G,i,2);
1
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
   <Oper Name="LocalIndexAtOddPByCharacter"
         Arg="F G n p" />
   <Oper Name="LocalIndexAtTwoByCharacter"
         Arg="F G n"  />
   <Oper Name="IsDyadicSchurGroup"
         Arg="G"   />

<Returns>
The first two function determines the local index at the given prime <M>p</M> of
the simple component of <A>FG</A> corresponding to the irreducible character
<C>Irr(G)[n]</C>.  The third one returns <C>'true'</C> if <C>G</C> is a dyadic
Schur group, and otherwise <C>'false'</C> .
</Returns>

<Description>
<Ref Oper="LocalIndexAtOddPByCharacter"/> and
<Ref Oper="LocalIndexAtTwoByCharacter"/> first determine a cyclotomic algebra representing
the simple component of <A>FG</A> corresponding to the character <C>Irr(G)[n]</C>.  They
then extend the field <M>F</M> to <M>K</M>, where <M>K</M> is the maximal <M>p</M>-split
subextension of <C>F(E(n))/F</C>, and recalculates the simple component of <C>KG</C>
corresponding to <C>Irr(G)[n]</C>.  It then uses the <C>DefiningGroup...</C> functions
to reduce to a faithful character of a possibly smaller cyclic-by-abelian group.  If the
simple component for this character is given in <Package>Wedderga</Package> as a list of
length 2 or 4, they make use of <Ref Oper="LocalIndexAtOddP"/> or <Ref Oper="LocalIndexAtTwo"/>
as appropriate.  If the simple component over <M>F</M> has length 5, it checks if the
defect group of the <M>p</M>-block containing <C>Irr(G)[n]</C> is cyclic.  If this is
definitely so, they use <Ref Oper="LocalIndexAtPByBrauerCharacter"/> to calculate the
<M>p</M>-local index.  Exceptions can occur when <M>p</M> is <M>2</M>.  When the defect
group is not necessarily cyclic, <Ref Oper="LocalIndexAtTwoByCharacter"/> makes use of
<Ref Oper="IsDyadicSchurGroup"/>, which checks if a quasi-elementary group has a faithful
irreducible character <C>2</C>-local index <C>2</C>, then verifies that <M>K</M> does not
split the simple component generated by this character.
<P/>

These functions are designed for faithful characters of groups that
faithfully represent cyclotomic algebras, and so should be used with
caution in other situations.

<Example>
<![CDATA[
gap> G:=SmallGroup(48,15);
<pc group of size 48 with 5 generators>
gap> i:=First([1..Length(Irr(G))],i->Size(KernelOfCharacter(Irr(G)[i]))=1);;
gap> LocalIndexAtOddPByCharacter(Rationals,G,Irr(G)[i],3);
2
gap> LocalIndexAtTwoByCharacter(Rationals,G,Irr(G)[i]);
2
gap> LocalIndexAtTwoByCharacter(CF(3),G,Irr(G)[i]);
1
]]>
</Example>

</Description>
</ManSection>

</Section>



<Section Label="LocalIndicesOfRationalQuaternionAlgebras">
<Heading>Local index functions for Rational Quaternion Algebras</Heading>

<ManSection>
   <Oper Name="LocalIndicesOfRationalQuaternionAlgebra"
         Arg="A"  />
   <Oper Name="LocalIndicesOfRationalSymbolAlgebra"
         Arg="a b" />
   <Oper Name="LocalIndicesOfTensorProductOfQuadraticAlgs"
         Arg="L M" />
<Oper Name="GlobalSchurIndexFromLocalIndices"
         Arg="L"  />

   <Returns>
     The first of these functions return a list of pairs <C>[p,m]</C> indicating that
     <C>m</C> is the local index at the prime <M>p</M> for the given quaternion algebra.
     The second does the same for <C>QuaternionAlgebra(Rationals,a,b)</C>.  The third
     returns a list of local indices computed from two given lists of local indices,
     and the fourth returns the least common multiple of the local indices in the
     given list of local indices.
   </Returns>

   <Description>
         For the first function, the input must be a quaternion algebra over the rationals,
         output from <C>QuaternionAlgebra(Rationals,a,b)</C>.  For the first function,
         <M>a</M> and <M>b</M> can be any pair of integers, and for the second rational
         symbol algebra version, <M>a</M> and <M>b</M> are restricted to be either -1 or
         positive prime integers, otherwise it will return <K>fail</K>. The input of the
         third function is a pair of lists of <M>p</M>-local indices in which the maximum
         local index at any prime is at most 2.  The input of the fourth function is a list
         of pairs <C>[p,m]</C> in which each prime that appears only appears in one of the
         pairs, and the <C>m</C>'s that appear are all positive integers.

<Ref Oper="LocalIndicesOfRationalQuaternionAlgebra"/> first factors the algebra as a
  tensor product of rational quaternion algebras, obtaining suitable pairs <C>a</C> and
  <C>b</C> to which  <Ref Oper="LocalIndicesOfRationalSymbolAlgebra"/> can be applied.
  The local indices are calculated using well-known formulas involving the Legendre Symbol.
  The local indices of the original algebra are then determined  using
  <Ref Oper="LocalIndicesOfTensorProductOfQuadraticAlgs"/>, which takes a pair of lists
  of local indices of quadratic algebras - for which the maximum local index at any
  prime <M>p</M> is 2, and finds the list of local indices of the tensor product of
  two algebras with these local indices.
<P/>
   <Ref Oper="GlobalSchurIndexFromLocalIndices"/> simply computes the least common multiple
   of the local indices at each prime that occurs in the list.

<Example>
<![CDATA[
gap> LocalIndicesOfRationalSymbolAlgebra(-1,-1);
[ [ infinity, 2 ], [ 2, 2 ] ]
gap> LocalIndicesOfRationalSymbolAlgebra(3,-1);
[ [ 2, 2 ], [ 3, 2 ] ]
gap> LocalIndicesOfRationalSymbolAlgebra(-3,2);
fail
gap> LocalIndicesOfRationalSymbolAlgebra(3,7);
[ [ 2, 2 ], [ 7, 2 ] ]
gap> A:=QuaternionAlgebra(Rationals,-30,-15);
<algebra-with-one of dimension 4 over Rationals>
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[ [ 5, 2 ], [ infinity, 2 ] ]
gap> A:=QuaternionAlgebra(CF(5),3,-2);
<algebra-with-one of dimension 4 over CF(5)>
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
fail
]]>
</Example>

  </Description>
</ManSection>

<ManSection>
   <Oper Name="IsRationalQuaternionAlgebraADivisionRing"
         Arg="A" />
   <Returns> If the rational quaternion algebra is a noncommutative division ring,
   <K>true</K> is returned, and if otherwise, <K>false</K>.
   </Returns>

   <Description> The input <M>A</M> must be a quaternion algebra over the rationals,
   as output from <C>QuaternionAlgebra(Rationals,a,b)</C>. <M>a</M> and <M>b</M> must
   be rational integers.  When applied to other algebras, it returns <K>fail</K>.  <P/>

The function calculates the rational Schur index of the algebra using
<Ref Oper="LocalIndicesOfRationalQuaternionAlgebra" />, and returns <K>true</K> if
the rational Schur index of the algebra is <C>2</C>, and <K>false</K> if the rational
Schur index is <C>1</C>. <P/>

This function should be preferred over <K>GAP</K>'s
when dealing with rational quaternion algebras, since the result of latter function only
depends on the local index at infinity for quaternion algebras, and makes no use of the
local indices at the finite primes.

<Example>
<![CDATA[
gap> A:=QuaternionAlgebra(Rationals,-30,-15);
<algebra-with-one of dimension 4 over Rationals>
gap> IsRationalQuaternionAlgebraADivisionRing(A);
true
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[ [ 5, 2 ], [ infinity, 2 ] ]
gap> A:=QuaternionAlgebra(Rationals,3,-2);
<algebra-with-one of dimension 4 over Rationals>
gap> IsRationalQuaternionAlgebraADivisionRing(A);
false
gap> LocalIndicesOfRationalQuaternionAlgebra(A);
[  ]
]]>
</Example>

   </Description>
</ManSection>

</Section>



<Section Label="Cyclic">
<Heading>Functions involving Cyclic Algebras</Heading>

  Cyclic algebras are represented in <Package>Wedderga</Package> as lists of length 3,
  in the form <C>[F,K,[c]]</C>, which stands for a cyclic crossed product algebra of
  the form <C>(K/F,c)</C>, with <C>K/F</C> a cyclic galois extension of abelian number
  fields, and <C>c</C> an element of <C>F</C> determining the factor set.  Schur
  indices of cyclic algebras can be determined through the solution of inverse norm
  equations in general.  Though currently algorithms for this are not available in
  &GAP;, algorithms have been implemented in some computational number theory software
  systems such as <Package>PARI/GP</Package>.
  <P/>

The functions in this section allow one to convert cyclotomic algebras into cyclic
algebras (or possibly as tensor products of two cyclic algebras), to convert
generalized quaternion algebras into quadratic algebras (i.e. cyclic algebras for
a Galois extension of degree 2), to convert quadratic algebras into generalized
quaternion algebras, and to convert cyclic algebras into cyclic cyclotomic algebras,
whenever possible.

<ManSection>
   <Oper Name="DecomposeCyclotomicAlgebra"
         Arg="A"   />
   <Returns> Two lists, each representing a cyclic algebra over the center of <M>A</M>,
   whose tensor product is isomorphic to the cyclotomic algebra described by <M>A</M>.
   </Returns>
   <Description>
         The input must be list representing a cyclotomic algebra of length 5 whose Galois
         group has <C>2</C> generators.  This is represented in <Package>Wedderga</Package>
         as a list of the form <C>[r,F,n,[[m1,k1,l1],[m2,k2,l2]],[[d]]]</C>.
         (Longer presentations of cyclotomic algebras do occur in <Package>Wedderga</Package>
         output.  Currently we do not have a general decomposition algorithm for them.)  <P/>

  For these algebras, the extension <C>F(E(n))/F</C> is the tensor product of two
  disjoint extensions <C>K1</C> and <C>K2</C> of <C>F</C>, and the program adjusts
  one of the factor sets (corresponding to <M>l1</M> or <M>l2</M>) so that <M>d</M>
  becomes <C>0</C>. After this adjustment, the algebra is then the tensor product
  of cyclic algebras of the form <C>[F,K1,[c1]]</C> and <C>[F,K2,[c2]]</C> provided
  <C>c1</C> and <C>c2</C> lie in <C>F</C>.  If the latter condition is not satisfied,
  the string <Q>fails</Q> is appended to the output.  (We have not encountered this
  problem among the group algebras of small groups we have tested so far.)

<Example>
<![CDATA[
gap> G:=SmallGroup(96,35);
<pc group of size 96 with 6 generators>
gap> W:=WedderburnDecompositionInfo(GroupRing(Rationals,G));;
gap> Size(W);
12
gap> A:=W[12];
[ 1, NF(8,[ 1, 7 ]), 24, [ [ 2, 7, 12 ], [ 2, 17, 9 ] ], [ [ 3 ] ] ]
gap> DecomposeCyclotomicAlgebra(A);
[ [ NF(8,[ 1, 7 ]), CF(8), [ -1 ] ],
  [ NF(8,[ 1, 7 ]), NF(24,[ 1, 7 ]), [ E(8)+2*E(8)^2+E(8)^3 ] ] ]
]]>
</Example>
   </Description>
</ManSection>

<ManSection>
   <Oper Name="ConvertCyclicAlgToCyclicCyclotomicAlg"
         Arg="A" />
   <Returns> A list of the form <C>[1,F,n,[a,b,c]]</C> which represents a cyclic
   cyclotomic algebra. </Returns>
   <Description> This function converts a cyclic algebra given by a list
   <C>[F,F(E(n)),[E(n)^c]]</C> to an isomorphic cyclic cyclotomic algebra represented
   as the list <C>[1,F,n,[a,b,c]]</C>.
   </Description>

   <Oper Name="ConvertQuadraticAlgToQuaternionAlg"
         Arg="A" />
   <Returns> A generalized quaternion algebra.
   </Returns>
   <Description>
     The input should be a list of the form <C>[F,K,[c]]</C> where the field <C>K</C>
     must be obtained by adjoining the square root of a nonsquare element <C>d</C>
     of <C>F</C>.  The function then returns the quaternion algebra given in <K>GAP</K>
     by <C>QuaternionAlgebra(F,d,c);</C>.

<Example>
<![CDATA[
gap> A:=[NF(24,[1,11]),CF(24),[-1]];
[ NF(24,[ 1, 11 ]), CF(24), [ -1 ] ]
gap> ConvertCyclicAlgToCyclicCyclotomicAlg(A);
[ 1, NF(24,[ 1, 11 ]), 24, [ 2, 11, 12 ] ]
gap> LocalIndicesOfCyclicCyclotomicAlgebra(last);
[  ]
gap> ConvertQuadraticAlgToQuaternionAlg(A);
<algebra-with-one of dimension 4 over NF(24,[ 1, 11 ])>
gap> b:=Basis(last);; b[1]^2; b[2]^2; b[3]^2; b[4]^2;
e
(-1)*e
(-1)*e
(-1)*e
gap> b[2]*b[3]+b[3]*b[2];
0*e
]]>
</Example>

   </Description>
</ManSection>

<ManSection>
   <Oper Name="ConvertQuaternionAlgToQuadraticAlg"
         Arg="A" />
   <Returns> A list of the form <C>[F,K,[c]]</C> representing a cyclic algebra for which
   the degree of the extension <C>K/F</C> is <C>2</C>.
   </Returns>
   <Description>
   The input must be a quaternion algebra whose center is an abelian number field <M>F</M>,
   presented as in the output from <C>QuaternionAlgebra( F, a, b )</C>, with <M>a</M>,
   <M>b</M> in <M>F</M>. It returns a list <C>[F,F(ER(a)),[b]]</C> representing the cyclic
   algebra isomorphic to <M>A</M>.
   </Description>

<Oper Name="ConvertCyclicCyclotomicAlgToCyclicAlg"
         Arg="A" />
   <Returns> A list of the form <C>[F,K,[c]]</C>.
   </Returns>
   <Description>The input should be a list <C>[r,F,n,[a,b,c]]</C> representing a matrix
   ring over a cyclic cyclotomic algebra.  The function returns the list
   <C>[F,F(E(n)),[E(n)^c]]</C>, which represents a cyclic algebra that is Morita equivalent
   to the given cyclic cyclotomic algebra.

<Example>
<![CDATA[
gap> A:=QuaternionAlgebra(CF(5),-3,-1);
<algebra-with-one of dimension 4 over CF(5)>
gap> ConvertQuaternionAlgToQuadraticAlg(A);
[ CF(5), CF(15), [ -1 ] ]
gap> ConvertCyclicAlgToCyclicCyclotomicAlg(last);
[ 1, CF(5), 30, [ 2, 11, 15 ] ]
gap> SchurIndex(last);
1
gap> ConvertCyclicCyclotomicAlgToCyclicAlg(last2);
[ 1, [ CF(5), CF(15), [ -1 ] ] ]
gap> ConvertQuadraticAlgToQuaternionAlg(last[2]);
<algebra-with-one of dimension 4 over CF(5)>
gap> b:=Basis(last);; b[1]^2; b[2]^2; b[3]^2; b[4]^2;
e
(-3)*e
(-1)*e
(-3)*e
]]>
</Example>

   </Description>
</ManSection>

</Section>

</Chapter>

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