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<Chapter Label="Theory">
<Heading>The basic theory behind &Wedderga;</Heading>    


In this chapter we describe the theory that is behind the algorithms used by &Wedderga;. <P/>

All the rings considered in this chapter are associative and have an identity. <P/>

We use the following notation: <M>&QQ;</M> denotes the field of rationals and <M>\mathbb F_q</M> the finite field 
of order <M>q</M>. For every positive integer <M>k</M>, we denote a complex <M>k</M>-th primitive root 
of unity by <M>\xi_k</M> and so <M>&QQ;(\xi_k)</M> is the <M>k</M>-th cyclotomic extension of <M>&QQ;</M>.

<Section Label="GroupRings">
<Heading>Group rings and group algebras</Heading>

<Index>group ring</Index>
Given a group <M>G</M> and a ring <M>R</M>, the <E>group ring</E> 
<M>RG</M> over the group <M>G</M> with coefficients in <M>R</M> is the ring whose underlying additive 
group is a right <M>R-</M>module with basis <M>G</M> such that the product is defined by the 
following rule
  <Display> 
  (gr)(hs)=(gh)(rs)
  </Display>
for <M>r,s \in R</M> and <M>g, h \in G</M>, and extended to <M>RG</M> by linearity.
<P/>

<Index>group algebra</Index>
A <E>group algebra</E> is a group ring in which the coefficient ring is a field.

</Section>

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<Section Label="Semisimple">
<Heading>Semisimple group algebras</Heading>

<Index>semisimple ring</Index>

We say that a ring <M>R</M> is semisimple if it is a direct sum of simple 
left (alternatively right) ideals or equivalently if <M>R</M> is isomorphic 
to a direct product of simple algebras each one isomorphic to a matrix ring 
over a division ring.
<P/>
 
By Maschke's Theorem, if <M>G</M> is a finite group then the group algebra 
<M>FG</M> is semisimple if and only the characteristic of the coefficient 
field <M>F</M> does not divide the order of <M>G</M>.
<P/>

In fact, an arbitrary group ring <M>RG</M> is semisimple if and only if the 
coefficient ring <M>R</M> is semisimple, the group <M>G</M> is finite and 
the order of <M>G</M> is invertible in <M>R</M>.
<P/>

Some authors use the notion semisimple ring for rings with zero Jacobson 
radical. To avoid confusion we usually refer to semisimple rings as 
semisimple artinian rings.

</Section>


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<Section Label="WedDec">
<Heading>Wedderburn decomposition</Heading>

<Index>Wedderburn decomposition</Index>
<Index>Wedderburn components</Index>

If <M>R</M> is a <E>semisimple ring</E> (<Ref Sect="Semisimple" />) then the <E>Wedderburn decomposition</E> of 
<M>R</M> is the decomposition of <M>R</M> as a direct product of simple algebras. 
The factors of this Wedderburn decomposition are called <E>Wedderburn components</E>
of <M>R</M>. 
Each Wedderburn component of <M>R</M> is of the form <M>Re</M> for <M>e</M> a <E>primitive central 
idempotent</E> (<Ref Sect="Idempotents" />) of <M>R</M>.
<P/> 

Let <M>FG</M> be a <E>semisimple group algebra</E> (<Ref Sect="Semisimple" />).
If <M>F</M> has positive characteristic, then the Wedderburn components of <M>FG</M>
are matrix algebras over finite extensions of <M>F</M>.
If <M>F</M> has zero characteristic then by the 
<E>Brauer-Witt Theorem</E> <Cite Key="Y" />, 
the <E>Wedderburn components</E> of <M>FG</M> are <E>Brauer equivalent</E> 
(<Ref Sect="Brauer" />)
to <E>cyclotomic algebras</E> (<Ref Sect="Cyclotomic" />). <P/> 

The main functions of &Wedderga; compute the Wedderburn components of a semisimple group algebra <M>FG</M>,
such that the coefficient field is either an abelian number field (i.e. a subfield of a finite cyclotomic extension 
of the rationals) or a finite field. 
In the finite case, the Wedderburn components are matrix algebras over finite fields and so can be described by the 
size of the matrices and the size of the finite field.
<P/>

In the zero characteristic case each Wedderburn component <M>A</M> is <E>Brauer equivalent</E> (<Ref Sect="Brauer"/>)
to a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />) 
and therefore <M>A</M> is a 
(possibly fractional) matrix algebra over <E>cyclotomic algebra</E> 
and can be described numerically in one of the following three forms:
  <Display>
  [n,K],
  </Display> 

  <Display>
  [n,K,k,[d,\alpha,\beta]],
  </Display>
  
  <Display>
  [n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i &lt; j \le n} ],
  </Display>
where <M>n</M> is the matrix size, <M>K</M> is the centre of <M>A</M> (a finite field extension of <M>F</M>) 
and the remaining data are integers whose interpretation is explained in <Ref Sect="NumDesc" />.
<P/>

In some cases (for the zero characteristic coefficient field) the size <M>n</M> 
of the matrix algebras is not a positive integer but a positive rational number. 
This is a consequence of the fact that the <E>Brauer-Witt Theorem</E> 
<Cite Key="Y" /> only ensures that each <E>Wedderburn component</E> 
(<Ref Sect="WedDec" />) of a semisimple group algebra is Brauer equivalent 
(<Ref Sect="Brauer" />) to a <E>cyclotomic algebra</E> (<Ref Sect="Cyclotomic" />), 
but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. 
For example, a Wedderburn component <M>D</M> of a group algebra can be a division 
algebra but not a cyclotomic algebra. In this case <M>M_n(D)</M> is a cyclotomic 
algebra <M>C</M> for some <M>n</M> and therefore <M>D</M> can be described
as <M>M_{1/n}(C)</M> (see last Example in <Ref Attr="WedderburnDecomposition" />).
<P/> 

The main algorithm of &Wedderga; is based on a computational oriented proof of the Brauer-Witt Theorem
due to Olteanu <Cite Key="O" /> which uses previous work by Olivieri, del Río and Simón <Cite Key="ORS" /> 
for rational group algebras of <E>strongly monomial groups</E> 
(<Ref Sect="StMon" />). 

</Section>

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<Section Label="Idempotents">
<Heading>Characters and primitive central idempotents</Heading>

<Index>primitive central idempotent</Index>
<Index>field of character values</Index>

A <E>primitive central idempotent</E> of a ring <M>R</M> is a non-zero central idempotent 
<M>e</M> which cannot be written as the sum of two non-zero central idempotents 
of <M>Re</M>, or equivalently, such that <M>Re</M> is indecomposable as a direct 
product of two non-trivial two-sided ideals.  <P/>

The <E>Wedderburn components</E> (<Ref Sect="WedDec" />) of a semisimple ring 
<M>R</M> are the rings of the form <M>Re</M> for <M>e</M> running over the set 
of primitive central idempotents of <M>R</M>. <P/>

Let <M>FG</M> be a <E>semisimple group algebra</E> (<Ref Sect="Semisimple" />)
and <M>\chi</M> an irreducible character 
of <M>G</M> (in an algebraic closure of <M>F</M>).
Then there is a unique Wedderburn component <M>A=A_F(\chi)</M> of <M>FG</M> such that <M>\chi(A)\ne 0</M>. 
Let <M>e_F(\chi)</M> denote the unique primitive central idempotent of <M>FG</M> in <M>A_F(\chi)</M>, 
that is the identity of <M>A_F(\chi)</M>, i.e. 
    <Display>
    A_F(\chi)=FGe_F(\chi).
    </Display>

The centre of <M>A_F(\chi)</M> is <M>F(\chi)=F(\chi(g):g \in G)</M>, the <E>field of character values</E> of 
<M>\chi</M> over <M>F</M>. <P/>

The map <M>\chi \mapsto A_F(\chi)</M> defines a surjective map from the set of irreducible characters of 
<M>G</M> (in an algebraic closure of <M>F</M>) onto the set of Wedderburn components of <M>FG</M>. 
<P/>

Equivalently, the map <M>\chi \mapsto e_F(\chi)</M> defines a surjective map from the set of irreducible characters 
of <M>G</M> (in an algebraic closure of <M>F</M>) onto the set of primitive central idempontents of <M>FG</M>. <P/>

If the irreducible character <M>\chi</M> of <M>G</M> takes values in <M>F</M> then 
  <Display>
  e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1}) g.
  </Display><P/>  
In general one has 
  <Display>
  e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).
  </Display><P/>  

</Section>


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<Section Label="Brauer">
<Heading>Central simple algebras and Brauer equivalence</Heading>

Let <M>K</M> be a field.
<Index Key="central simple algebra">central simple algebra</Index>
A <E>central simple <M>K</M>-algebra</E> is a finite dimensional <M>K</M>-algebra 
with center <M>K</M> which has no non-trivial proper ideals. 
Every central simple <M>K</M>-algebra is isomorphic to a matrix algebra <M>M_n(D)</M>
where <M>D</M> is a division algebra (which is finite-dimensional over <M>K</M> and has centre <M>K</M>).
The division algebra <M>D</M> is unique up to <M>K</M>-isomorphisms.
<P/>

<Index Key="Brauer equivalence">(Brauer) equivalence</Index>
<Index Key="equivalence (Brauer)">equivalence (Brauer)</Index>
Two central simple <M>K</M>-algebras <M>A</M> and <M>B</M> are said to be <E>Brauer equivalent</E>, 
or simply <E>equivalent</E>, if there is a division algebra <M>D</M> and two positive integers <M>m</M> and <M>n</M> 
such that <M>A</M> is isomorphic to <M>M_m(D)</M> and <M>B</M> is isomorphic to <M>M_n(D)</M>.
<P/>

</Section>


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<Section Label="CrossedProd">
<Heading>Crossed Products</Heading>

<Index Key="Crossed Product">Crossed Product</Index>
Let <M>R</M> be a ring and <M>G</M> a group. <P/>

<B>Intrinsic definition</B>.

A <E>crossed product</E> <Cite Key="P" /> of <M>G</M> over <M>R</M> (or with coefficients in <M>R</M>) is a ring <M>R*G</M> with 
a decomposition into a direct sum of additive subgroups
  <Display>
  R*G = \bigoplus_{g \in G} A_g
  </Display>
such that for each <M>g,h</M> in <M>G</M> one has:<P/>
* <M>A_1=R</M> (here <M>1</M> denotes the identity of <M>G</M>),<P/>
* <M>A_g A_h = A_{gh}</M> and <P/>
* <M>A_g</M> has a unit of <M>R*G</M>.<P/>

<B>Extrinsic definition</B>.

Let <M>Aut(R)</M> denote the group of automorphisms of <M>R</M> and let <M>R^*</M> denote the group of units of 
<M>R</M>.<P/>

Let <M>a:G \rightarrow Aut(R)</M> and <M>t:G \times G \rightarrow R^*</M> 
be mappings satisfying the following conditions 
for every <M>g</M>, <M>h</M> and <M>k</M> in <M>G</M>: 
<P/>

(1) <M>a(gh)^{-1} a(g) a(h)</M> is the inner automorphism of <M>R</M> induced by 
<M>t(g,h)</M> (i.e. the automorphism <M>x\mapsto t(g,h)^{-1} x t(g,h)</M>) and <P/>

(2) <M>t(gh,k) t(g,h)^k = t(g,hk) t(h,k)</M>, where for 
<M>g \in G</M> and <M>x \in R</M> we denote <M>a(g)(x)</M> by <M>x^g</M>.
<P/>

The <E>crossed product</E> <Cite Key="P" /> of <M>G</M> over <M>R</M> (or with coefficients in <M>R</M>), 
action <M>a</M> and twisting <M>t</M> is the ring 
  <Display>
  R*_a^t G = \bigoplus_{g\in G} u_g R
  </Display> 
where <M>\{u_g : g\in G \}</M> is a set of symbols in one-to-one correspondence with <M>G</M>,  with addition and multiplication defined by 
  <Display> 
   (u_g r) + (u_g s) = u_g(r+s), \quad (u_g r)(u_h s) = u_{gh} t(g,h) r^h s
  </Display>
for <M>g,h \in G</M> and <M>r,s\in R</M>, and extended to <M>R*_a^t G</M> by linearity.<P/>

The associativity of the product defined is a consequence of conditions (1) and (2) <Cite Key="P" />.<P/>

<B>Equivalence of the two definitions</B>.

Obviously the crossed product of <M>G</M> over <M>R</M> defined using the extrinsic definition is a crossed product 
of <M>G</M> over <M>u_1 R</M> in the sense of the first definition. 
Moreover, there is <M>r_0</M> in <M>R^*</M> such that <M>u_1r_0</M> is the identity of <M>R*_a^t G</M> and the map 
<M>r \mapsto u_1 r_0 r </M> is a ring isomorphism <M>R \rightarrow u_1R </M>. <P/>

<Index Key="Basis of units">Basis of units (for crossed product)</Index>
Conversely, let <M>R*G=\bigoplus_{g\in G} A_g</M> be an (intrinsic) crossed product and select for each <M>g\in G</M> a 
unit <M>u_g\in A_g</M> of <M>R*G</M>. 
This is called a <E>basis of units for the crossed product</E> <M>R*G</M>. 
Then the maps <M>a:G \rightarrow Aut(R)</M> and <M>t:G\times G \rightarrow R^*</M> given by 
  <Display>
  r^g = u_g^{-1} r u_g, \quad t(g,h) = u_{gh}^{-1} u_g u_h \quad (g,h \in G, r \in R)
  </Display>
satisfy conditions (1) and (2) and <M>R*G = R*_a^t G</M>. <P/>

The choice of a basis of units <M>u_g \in A_g</M> determines the action <M>a</M> and twisting <M>t</M>. 
If <M>\{u_g \in A_g : g \in G \}</M> and <M>\{v_g \in A_g : g \in G \}</M> are two sets of units of <M>R*G</M>
then <M>v_g = u_g r_g</M> for some units <M>r_g</M> of <M>R</M>. 
Changing the basis of units results in a change of the action and the twisting and so changes the extrinsic definition of
the crossed product but it does not change the intrinsic crossed product. <P/> 

It is customary to select <M>u_1=1</M>. In that case <M>a(1)</M> is the identity map of <M>R</M> and 
<M>t(1,g)=t(g,1)=1</M> for each <M>g</M> in <M>G</M>. 

</Section>


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<Section Label="CyclicCP">
<Heading>Cyclic Crossed Products</Heading>


<Index Key="Cyclic Crossed Product">Cyclic Crossed Product</Index>
Let <M>R*G=\bigoplus_{g \in G} A_g</M> be a <E>crossed product</E> 
(<Ref Sect="CrossedProd" />) and assume that <M>G = \langle g \rangle </M> is cyclic. 
Then the crossed product can be given using a particularly nice description. <P/>

Select a unit <M>u</M> in <M>A_{g}</M>, and let <M>a</M> be the automorphism 
of <M>R</M> given by <M>r^a = u^{-1} r u</M>.  
<P/>

If <M>G</M> is infinite then set <M>u_{g^k} = u^k</M> for every integer <M>k</M>.
Then   
  <Display>
    R*G = R[ u | ru = u r^a ],
  </Display>
a skew polynomial ring. Therefore in this case <M>R*G</M> is determined
by 
<Display>
[ R, a ].
</Display>

If <M>G</M> is finite of order <M>d</M> then set <M>u_{g^k} = u^k</M>
for <M>0 \le k &lt; d</M>. Then <M> b = u^d \in R </M> and
  <Display>
    R*G = R[ u | ru = u r^a, u^d = b ]
  </Display>
Therefore, <M>R*G</M> is completely determined by the following data:
  <Display>
    [ R , [ d , a , b ] ]
  </Display>
  
</Section>

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<Section Label="AbelianCP">
<Heading>Abelian Crossed Products</Heading>


<Index Key="Abelian Crossed Product">Abelian Crossed Product</Index>
Let <M>R*G=\bigoplus_{g \in G} A_g</M> be a <E>crossed product</E> 
(<Ref Sect="CrossedProd" />) and assume that <M>G</M> is abelian. 
Then the crossed product can be given using a simple description. <P/>

Express <M>G</M> as a direct sum of cyclic groups:
  <Display>
  G = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle
  </Display>
and for each <M>i=1,\dots,n</M> select a unit <M>u_i</M> in <M>A_{g_i}</M>. <P/>

Each element <M>g</M> of <M>G</M> has a unique expression 
  <Display>
  g = g_1^{k_1} \cdots g_n^{k_n},
  </Display>
where <M>k_i</M> is an arbitrary integer, if <M>g_i</M> has infinite order, 
and <M>0 \le k_i &lt; d_i</M>, 
if <M>g_i</M> has finite order <M>d_i</M>.
Then one selects a basis for the crossed product by taking
  <Display>
  u_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}.
  </Display>
<P/>

* For each <M>i=1,\dots, n</M>, let <M>a_i</M> be the automorphism of <M>R</M> given by 
<M>r^{a_i} = u_i^{-1} r u_i</M>. 
<P/>

* For each <M>1 \le i &lt; j \le n</M>, let <M>t_{i,j} = u_j^{-1} u_i^{-1} u_j u_i \in R</M>. <P/>

* If <M>g_i</M> has finite order <M>d_i</M>, let <M>b_i=u_i^{d_i} \in R</M>. <P/>

Then 
  <Display>
  R*G = R[u_1,\dots,u_n | ru_i = u_i r^{a_i}, u_j u_i = t_{ij} u_i u_j, u_i^{d_i} = b_i (1 \le i &lt; j \le n) ],
  </Display>
where the last relation vanishes if <M>g_i</M> has infinite order.<P/>

Therefore <M>R*G</M> is completely determined by the following data:
  <Display>
  [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i &lt; j \le n} ].
  </Display>
  
</Section>

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<Section Label="ClassCP">
<Heading>Classical crossed products</Heading>

<Index Key="ClassicalCP">Classical Crossed Product</Index>
A <E>classical crossed product</E> is a crossed product <M>L*_a^t G</M>, where <M>L/K</M> is a finite Galois 
extension, <M>G=Gal(L/K)</M> is the Galois group of <M>L/K</M> and 
<M>a</M> is the natural action of <M>G</M> on <M>L</M>.

Then <M>t</M> is a <M>2</M>-cocycle and the <E>crossed product</E> 
(<Ref Sect="CrossedProd" />) <M>L*_a^t G</M> is denoted by <M>(L/K,t)</M>. 
The crossed product <M>(L/K,t)</M> is known to be a central simple <M>K</M>-algebra  <Cite Key="R" />. <P/> 


</Section>

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<Section Label="CycAlg">
<Heading>Cyclic Algebras</Heading>

<Index Key="Cyclic Algebra">Cyclic Algebra</Index>
A <E>cyclic algebra</E> is a <E>classical crossed product</E> 
(<Ref Sect="ClassCP" />) <M>(L/K,t)</M> where <M>L/K</M> 
is a finite cyclic field extension. 
The cyclic algebras have a very simple form. <P/>

Assume that <M>Gal(L/K)</M> is generated by <M>g</M> and has order <M>d</M>.
Let <M>u=u_g</M> be the basis unit (<Ref Sect="CrossedProd" />) of the crossed product 
corresponding to <M>g</M> and take the remaining basis units for the crossed 
product by setting <M>u_{g^i} = u^i</M>, (<M> i = 0, 1, \dots, d-1 </M>). 
Then <M>a = u^n \in K</M>. The cyclic algebra is usually denoted by <M>(L/K,a)</M> and one has the following 
description of <M>(L/K,t)</M>
  <Display>
  (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].
  </Display>
<P/>



</Section>

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<Section Label="Cyclotomic">
<Heading>Cyclotomic algebras</Heading>

<Index Key="Cyclotomic algebra">Cyclotomic algebra</Index>
A <E>cyclotomic algebra</E> over <M>F</M> is a 
<E>classical crossed product</E> (<Ref Sect="ClassCP" />)
<M>(F(\xi)/F,t)</M>, 
where <M>F</M> is a field, <M>\xi</M> is a root of unity in an extension of <M>F</M> 
and <M>t(g,h)</M> is a root of unity for every <M>g</M> and <M>h</M> in 
<M>Gal(F(\xi)/F)</M>. <P/>

The <E>Brauer-Witt Theorem</E> <Cite Key="Y" /> asserts that every <E>Wedderburn component</E> (<Ref Sect="WedDec" />)
of a group algebra is <E>Brauer equivalent</E> (<Ref Sect="Brauer" />) 
(over its centre) to a cyclotomic algebra. <P/> 

</Section>

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<Section Label="NumDesc">
<Heading>Numerical description of cyclotomic algebras</Heading>

Let <M>A=(F(\xi)/F,t)</M> be a <E>cyclotomic algebra</E> 
(<Ref Sect="Cyclotomic" />), 
where <M>\xi=\xi_k</M> is a <M>k</M>-th root of unity. 
Then the Galois group <M>G=Gal(F(\xi)/F)</M> is abelian and therefore one can obtain a simplified form for the
description of cyclotomic algebras as for any 
<E>abelian crossed product</E> (<Ref Sect="AbelianCP" />). <P/>

Then the <M>n \times n</M> matrix algebra <M>M_n(A)</M> can be described numerically in one of the following forms: <P/>

* If <M>F(\xi)=F</M>, (i.e. <M>G=1</M>) then <M>A=M_n(F)</M> and thus the only data needed to describe <M>A</M> 
are the matrix size  <M>n</M> and the field <M>F</M>:
  <Display>
  [n,F]
  </Display>  
  <P/>

* If <M>G</M> is cyclic (but not trivial) of order <M>d</M> then <M>A</M> is a cyclic cyclotomic algebra
  <Display>
  A = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ]
  </Display>
and so <M>M_n(A)</M> can be described with the following data 
  <Display>
  [n,F,k,[d,\alpha,\beta]],
  </Display>
where the integers <M>k</M>, <M>d</M>, <M>\alpha</M> and <M>\beta</M>
satisfy the following conditions:
  <Display>
  \alpha^d \equiv 1 \; mod \; k, \quad 
  \beta(\alpha-1) \equiv 0 \; mod \; k.
  </Display>
<P/>

* If <M>G</M> is abelian but not cyclic then <M>M_n(A)</M> 
can be described with the following data 
(see <Ref Sect="AbelianCP" />):
  <Display>
  [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i &lt; j \le m} ]
  </Display>
representing the <M>n \times n</M> matrix ring over the following algebra:
  <Display>
  A = F(\xi)[ u_1, \ldots, u_m \mid 
  \xi u_i = u_i \xi^{\alpha_i}, \quad
  u_i^{d_i}=\xi^{\beta_i}, \quad
  u_s u_r = \xi^{\gamma_{rs}} u_r u_s, \quad
  i = 1, \ldots, m, \quad
  0 \le r &lt; s \le m ]
  </Display>
where 
<P/> 
* <M>\{g_1,\ldots,g_m\}</M> is an independent set of generators of <M>G</M>, <P/>
* <M>d_i</M> is the order of <M>g_i</M>, <P/>
* <M>\alpha_i</M>, <M>\beta_i</M> and <M>\gamma_{rs}</M> are integers, and
  <Display> 
    \xi^{g_i} = \xi^{\alpha_i}.
  </Display>

</Section>


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<Section Label="IdempotentsSbgps">
<Heading>Idempotents given by subgroups</Heading>

<Index><M>\varepsilon(K,H)</M></Index>
<Index><M>e(G,K,H)</M></Index>
<Index><M>e_C(G,K,H)</M></Index>

Let <M>G</M> be a finite group and <M>F</M> a field whose characteristic does not 
divide the order of <M>G</M>. If <M>H</M> is a subgroup of <M>G</M> then set
<Display>
\widehat{H} = |H|^{-1}\sum_{x \in H} x.
</Display>
The element <M>\widehat{H}</M> is an idempotent of <M>FG</M> which is central in
<M>FG</M> if and only if <M>H</M> is normal in <M>G</M>. 
<P/>

If <M>H</M> is a proper normal subgroup of a subgroup <M>K</M> of  <M>G</M>
then set <Display> \varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L})
</Display> where <M>L</M> runs on the normal subgroups of <M>K</M> which are
minimal among the normal subgroups of <M>K</M> containing <M>N</M> properly. By
convention, <M>\varepsilon(K,K)=\widehat{K}</M>. The element
<M>\varepsilon(K,H)</M> is an idempotent of <M>FG</M>. 
<P/>

If <M>H</M> and <M>K</M> are subgroups of <M>G</M> such that <M>H</M> is normal in 
<M>K</M> then <M>e(G,K,H)</M> denotes the sum of all different <M>G</M>-conjugates 
of <M>\varepsilon(K,H)</M>.
The element <M>e(G,K,H)</M> is central in <M>FG</M>. 
In general it is not an idempotent but if the 
different conjugates of <M>\varepsilon(K,H)</M> are orthogonal then 
<M>e(G,K,H)</M> is a central idempotent of <M>FG</M>.
<P/>

If <M>(K,H)</M> is a Shoda Pair (<Ref Sect="SPDef" />) of <M>G</M> then there is
a non-zero rational number <M>a</M> such that <M>ae(G,K,H))</M> is a <E>primitive central idempotent</E>  
(<Ref Sect="Idempotents" />) of the rational group algebra <M>&QQ; G</M>. 
If <M>(K,H)</M> is a strong Shoda pair (<Ref Sect="SSPDef" />) of <M>G</M> then
<M>e(G,K,H)</M> is a primitive central idempotent of <M>&QQ; G</M>. 
<P/>

Assume now that <M>F</M> is a finite field of order <M>q</M>, 
<M>(K,H)</M> is a strong Shoda pair of <M>G</M> and 
<M>C</M> is a cyclotomic class of <M>K/H</M> containing a 
generator of <M>K/H</M>. Then <M>e_C(G,K,H)</M> is a primitive 
central idempotent of <M>FG</M> (see <Ref Sect="CyclotomicClass" />). 
<P/>

</Section>

<!-- ********************************************************* -->

<Section Label="SPDef">
<Heading>Shoda pairs</Heading>

Let <M>G</M> be a finite group.
<Index>Shoda pair</Index>
A <E>Shoda pair</E> of <M>G</M> is a pair <M>(K,H)</M> of subgroups of <M>G</M>
for which there is a linear character <M>\chi</M> of <M>K</M> with
kernel <M>H</M> such that the induced character <M>\chi^G</M> in <M>G</M> is 
irreducible.
By <Cite Key="S" /> or <Cite Key="ORS" />, <M>(K,H)</M> is a Shoda pair if and only
if the following conditions hold:<P/>
* <M>H</M> is normal in <M>K</M>, <P/>
* <M>K/H</M> is cyclic and <P/> 
* if <M>K^g \cap K \subseteq H</M> for some <M>g \in G</M> then <M>g \in K</M>. <P/>

<Index>primitive central idempotent realized by a Shoda pair</Index>
If <M>(K,H)</M> is a Shoda pair and <M>\chi</M> is a linear character of <M>K\le G</M> with kernel <M>H</M> 
then the <E>primitive central idempotent</E>  (<Ref Sect="Idempotents" />)
of <M>&QQ; G</M> 
associated to the irreducible character <M>\chi^G</M> is of the form 
<M>e=e_&QQ; (\chi^G)=a e(G,K,H)</M> for some <M>a \in &QQ; </M> <Cite Key="ORS" /> 
(see <Ref Sect="IdempotentsSbgps" /> for the definition of <M>e(G,K,H)</M>).
In that case we say that <M>e</M> is the <E>primitive central idempotent realized 
by the Shoda pair</E> <M>(K,H)</M> of <M>G</M>. <P/>

A group <M>G</M> is monomial, that is every irreducible character of <M>G</M> is 
monomial, if and only if every primitive central idempotent of <M>&QQ; G</M> is 
realizable by a Shoda pair of <M>G</M>.
<P/>

</Section>

<!-- ********************************************************* -->

<Section Label="SSPDef">
<Heading>Strong Shoda pairs</Heading>

<Index>strong Shoda pair</Index>
A <E>strong Shoda pair</E> of <M>G</M> is a pair <M>(K,H)</M> of
subgroups of <M>G</M> satisfying the following conditions:<P/>
* <M>H</M> is normal in <M>K</M> and <M>K</M> is normal in the normalizer <M>N</M> of <M>H</M> in 
  <M>G</M>, <P/>
* <M>K/H</M> is cyclic and a maximal abelian subgroup of <M>N/H</M> and <P/>
* for every <M>g \in G\setminus N</M> ,  <M>\varepsilon(K,H)\varepsilon(K,H)^g=0</M>.
(See <Ref Sect="IdempotentsSbgps" /> for the definition of <M>\varepsilon(K,H)</M>).
<P/>

Let <M>(K,H)</M> be a strong Shoda pair of <M>G</M>. 
Then <M>(K,H)</M> is a Shoda pair (<Ref Sect="SPDef" />) of <M>G</M>. 
Thus there is a linear character <M>\theta</M> of <M>K</M> with kernel <M>H</M> 
such that the induced character <M>\chi=\chi(G,K,H)=\theta^G</M> is irreducible. 
Moreover the <E>primitive central idempotent</E> (<Ref Sect="Idempotents" />)
<M>e_{&QQ; }(\chi)</M> of <M>&QQ; G</M> 
realized by <M>(K,H)</M> is <M>e(G,K,H)</M>, see <Cite Key="ORS" />. <P/>

<Index>equivalent strong Shoda pairs</Index>
Two <E>strong Shoda pairs</E> (<Ref Sect="SSPDef"/>) <M>(K_1,H_1)</M> and <M>(K_2,H_2)</M> of <M>G</M> 
are said to be <E>equivalent</E> if the characters
<M>\chi(G,K_1,H_1)</M> and <M>\chi(G,K_2,H_2)</M> are Galois conjugate, or equivalently if 
<M>e(G,K_1,H_1)=e(G,K_2,H_2)</M>.<P/>

The advantage of strong Shoda pairs over Shoda pairs is that 
one can describe the simple algebra
<M>FGe_F(\chi)</M> as a matrix algebra of a <E>cyclotomic algebra</E> 
(<Ref Sect="Cyclotomic" />, see <Cite Key="ORS" /> for <M>F=&QQ; </M> and 
<Cite Key="O" /> for the general case). <P/>

More precisely, <M>&QQ; Ge(G,K,H)</M> is isomorphic to <M>M_n(&QQ; (\xi)*_a^t N/K)</M>, 
where <M>\xi</M> is a <M>[K:H]</M>-th root of unity, <M>N</M> is the normalizer of 
<M>H</M> in <M>G</M>, <M>n=[G:N]</M> and <M>&QQ; (\xi)*_a^t N/K</M> is a <E>crossed product</E> (see <Ref Sect="CrossedProd" />) with action <M>a</M> and twisting <M>t</M> given as follows: <P/>
Let <M>x</M> be a fixed generator of <M>K/H</M> and 
<M>\varphi : N/K \rightarrow N/H</M> a fixed left inverse of the canonical projection 
<M>N/H\rightarrow N/K</M>. Then 
  <Display>
  \xi^{a(r)} = \xi^i, \mbox{ if }  x^{\varphi(r)}= x^i
  </Display>
and 
  <Display>
  t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j,
  </Display>
for <M>r,s \in N/K</M> and integers <M>i</M> and <M>j</M>, see <Cite Key="ORS" />. 
Notice that the cocycle is the one given by the natural extension
  <Display>
  1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1 
  </Display>
where <M>K/H</M> is identified with the multiplicative group generated by <M>\xi</M>. 
Furthermore the centre of the algebra is <M>&QQ; (\chi)</M>, the field of character values over <M>&QQ; </M>,
and <M>N/K</M> is isomorphic to <M>Gal(&QQ; (\xi)/&QQ; (\chi))</M>.
<P/>

If the rational field is changed to an arbitrary ring <M>F</M> of characteristic <M>0</M> then the Wedderburn 
component <M>A_F(\chi)</M>, where <M>\chi = \chi(G,K,H)</M> is isomorphic to <M>F(\chi)\otimes_{&QQ; (\chi)}A_&QQ; (\chi)</M>.
Using the description given above of <M>A_&QQ; (\chi)=&QQ; G e(G,K,H)</M> one can easily describe <M>A_F(\chi)</M> as
<M>M_{nd}(F(\xi)/F(\chi),t')</M>, where <M>d=[&QQ; (\xi): &QQ;(\chi)]/[F(\xi):F(\chi)]</M> and 
<M>t'</M> is the restriction to <M>Gal(F(\xi)/F(\chi))</M> of <M>t</M> 
(a cocycle of <M>N/K = Gal(&QQ; (\xi)/&QQ;  (\chi))</M>). <P/>

</Section>

<!-- ********************************************************* -->

<Section Label="StMon">
<Heading>Strongly monomial characters and strongly monomial groups</Heading>

<Index Key="SMCh">strongly monomial character</Index>
Let <M>G</M> be a finite group an <M>\chi</M> an irreducible character of <M>G</M>.<P/>

One says that <M>\chi</M> is <E>strongly monomial</E> if there is a <E>strong Shoda pair</E> (<Ref Sect="SSPDef"/>)
<M>(K,H)</M> of <M>G</M> and a linear character <M>\theta</M> of <M>K</M> of <M>G</M> with kernel <M>H</M> 
such that <M>\chi=\theta^G</M>.<P/>

<Index Key="SMG">strongly monomial group</Index>  
The group <M>G</M> is <E>strongly monomial</E> 
if every irreducible character of <M>G</M> is 
strongly monomial. <P/>

Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that 
every abelian-by-supersolvable group is strongly monomial  <Cite Key="ORS" />.
The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups
was explained in <Cite Key="OR" />. 
This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <Cite Key="BR" /> 
(see Section <Ref Sect="CyclotomicClass" />). 
Finally, Olteanu <Cite Key="O" /> shows how to compute the <E>Wedderburn decomposition</E> (<Ref Sect="WedDec" />)
of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of <M>G</M> but also 
the strong Shoda pairs of the subgroups of <M>G</M>. 

</Section>

<!-- ********************************************************* -->

<Alt Only="LaTeX">\pagebreak</Alt>.

<Section Label="CyclotomicClass">
<Heading>Cyclotomic Classes and Strong Shoda Pairs</Heading>

Let <M>G</M> be a finite group and <M>F</M> a finite field of order <M>q</M>, 
coprime to the order of <M>G</M>. <P/>

<Index>cyclotomic class</Index>
Given a positive integer <M>n</M>, coprime to <M>q</M>, 
the <M>q</M>-<E>cyclotomic classes</E> modulo <M>n</M> are 
the set of residue classes module <M>n</M> of the form
  <Display>
  \{i,iq,iq^2,iq^3, \dots \}
  </Display>
The <M>q</M>-cyclotomic classes module <M>n</M> form a partition of the set of residue classes module <M>n</M>. <P/>

<Index>generating cyclotomic class</Index>
A <E>generating cyclotomic class </E> module <M>n</M> is a cyclotomic class containing 
a generator of the additive group of residue classes module <M>n</M>, or equivalently 
formed by integers coprime to <M>n</M>. <P/>

Let <M>(K,H)</M> be a strong Shoda pair (<Ref Sect="SSPDef" />) 
of <M>G</M> and set <M>n=[K:H]</M>. 
Fix a primitive <M>n</M>-th root of unity <M>\xi</M> in some extension of <M>F</M>
and an element <M>g</M> of <M>K</M> such that <M>gH</M> is a generator of <M>K/H</M>.
Let <M>C</M> be a generating <M>q</M>-cyclotomic class modulo <M>n</M>. Then set 
    <Display>
    \varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1} tr(\xi^{-ci})g^i,
    </Display>
where <M>c</M> is an arbitrary element of <M>C</M> and <M>tr</M> is the trace map 
of the field extension <M>F(\xi)/F</M>. Then <M>\varepsilon_C(K,H)</M> does not 
depend on the choice of <M>c \in C</M> and is a <E>primitive central idempotent</E>  
(<Ref Sect="Idempotents" />) of <M>FK</M>. <P/>

<Index>primitive central idempotent realized by a strong Shoda pair and a cyclotomic class</Index>
Finally, let <M>e_C(G,K,H)</M> denote the sum of the different <M>G</M>-conjugates of
<M>\varepsilon_C(K,H)</M>. 
Then <M>e_C(G,K,H)</M> is a <E>primitive central idempotent</E>  
(<Ref Sect="Idempotents" />) of <M>FG</M> <Cite Key="BR" />. 
We say that <M>e_C(G,K,H)</M> is the primitive central idempotent realized by the 
strong Shoda pair <M>(K,H)</M> of the group <M>G</M> and the cyclotomic class <M>C</M>.
<P/>

If <M>G</M> is <E>strongly monomial</E> (<Ref Sect="StMon" />) then every primitive central idempotent of <M>FG</M> 
is realizable by some <E>strong Shoda pair</E> (<Ref Sect="SSPDef"/>) of <M>G</M> and some cyclotomic class <M>C</M> 
<Cite Key="BR" />. As in the zero characteristic case, this explain how to compute the <E>Wedderburn decomposition</E> 
(<Ref Sect="WedDec" />) of <M>FG</M> for a finite semisimple algebra of a strongly monomial group 
(see <Cite Key="BR" /> for details). 

For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer 
characters. 
<P/>

</Section>

</Chapter>
