<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 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 G 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.
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 < 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" /> (see also <Cite Key="OR" /> ) for rational group
algebras of <E>strongly monomial groups</E> (<Ref Sect="StMon" />). The algorithms
are also based upon the work of Bakshi and Maheshwary <Cite Key="BM14" />
(see also <Cite Key="BM16" />) on the rational group algebras of <E>normally
monomial groups</E> (<Ref Sect="NorMon" />).
</Section>
<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 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/>
<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>.
<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 < 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>
<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 < 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 < 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 < 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 < j \le n} ].
</Display>
<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/>
<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/>
<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 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 < 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 < 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>
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 Label="SPDef">
<Heading>Shoda pairs of a group</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 Label="SSPDef">
<Heading>Strong Shoda pairs of a group</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>. A set of representatives of strong Shoda pairs of
<M>G</M> is termed as a complete irredundant set of strong Shoda pairs of <M>G</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'), where d=[&QQ; (\xi): &QQ;(\chi)]/[F(\xi):F(\chi)] and
<M>t' is the restriction to Gal(F(\xi)/F(\chi)) of t
(a cocycle of <M>N/K = Gal(&QQ; (\xi)/&QQ; (\chi))</M>). <P/>
<Section Label="ESSPDef">
<Heading>Extremely strong Shoda pairs of a group</Heading>
<Index> extremely strong Shoda pair</Index>
An <E> extremely 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>K</M> is normal in <M>G</M>, <P/>
* <M>K/H</M> is cyclic and a maximal abelian subgroup of <M>N/H</M>, where
<M>N</M> is the normalizer of <M>H</M> in <M>G</M>.
<P/>
Let <M>(K,H)</M> be an extremely strong Shoda pair of <M>G</M>.
Then <M>(K,H)</M> is a strong Shoda pair (<Ref Sect="SSPDef" />) of <M>G</M>, with
<M>K</M> normal in <M>G</M> <Cite Key="BM14" />, so that 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>
<M>e_{&QQ; }(\chi)</M> of <M>&QQ; G</M>
realized by <M>(K,H)</M> is <M>e(G,K,H)</M> (<Ref Sect="Idempotents" />)
and one can describe the associated simple algebra (<Ref Sect="SSPDef" />).
<Index>equivalent extremely strong Shoda pairs</Index>
Two <E>extremely strong Shoda pairs</E>
of <M>G</M> are said to be <E>equivalent</E> if they are equivalent
as strong Shoda pairs (<Ref Sect="SSPDef"/>). A set of representatives of extremely
strong Shoda pairs of <M>G</M> is called a <E>complete irredundant set</E> of extremely
strong Shoda pairs of <M>G</M> <Cite Key="BM14" />. <P/>
If <M>G</M> is a normally monomial group (<Ref Sect="NorMon" />), then the set of
primitive central idempotents of the rational group algebra realized by strong
Shoda pairs of <M>G</M> is same as the one realized by extremely strong
Shoda pairs of <M>G</M> <Cite Key="BM14" />. The algorithm to compute a
complete irredundant set of extremely strong Shoda pairs of <M>G</M> has been
explained in <Cite Key="BM16" />.
<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>
<Section Label="NorMon">
<Heading>Normally monomial characters and normally monomial groups</Heading>
<Index Key="NMCh">normally monomial character</Index>
Let <M>G</M> be a finite group and <M>\chi</M> be an irreducible character of <M>G</M>.<P/>
One says that <M>\chi</M> is <E>normally monomial</E> if there is a normal subgroup
<M>K</M> of <M>G</M> such that <M>\chi</M> is induced from a linear character of
<M>K</M>.<P/>
<Index Key="NMG">normally monomial group</Index>
The group <M>G</M> is <E>normally monomial</E>
if every irreducible character of <M>G</M> is
normally monomial. Bakshi and Maheshwary proved that if <M>G</M> is a normally monomial
group, then for every irreducible character <M>\chi</M> of <M>G</M>, there exists an extremely strong Shoda pair
<M>(K,H)</M> of <M>G</M> (<Ref Sect="ESSPDef"/>) such that <M>\chi=\theta^G</M>, where
<M>\theta</M> is a linear character of <M>K</M> with kernel <M>H</M> <Cite Key="BM14" />.<P/>
<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>
<!-- ****************theory for div-alg functions******************** -->
<Alt Only="LaTeX">\pagebreak</Alt>.
<Section Label="DivAlgTheory">
<Heading>Theory for Local Schur Index and Division Algebra Part Calculations</Heading>
(By Allen Herman, May 2013. Updated October 2014.) <P/>
The division algebra parts of simple algebras in the Wedderburn
Decomposition of the group algebra of a finite group over an abelian number
field <M>F</M> correspond to elements of the Schur Subgroup <M>S(F)</M> of
the Brauer group of <M>F</M>. Like all classes in the Brauer group of an
algebraic number field <M>F</M>, the division algebra part of a
representative of a given Brauer class is determined up to <M>F</M>-algebra
isomorphism by its list of local Hasse invariants at all primes (i.e.
places) of <M>F</M>. The local invariant at a prime <M>P</M> of <M>F</M> is
a lowest terms fraction <M>r/m_P</M> whose denominator is the local Schur
index <M>m_P</M> of the simple algebra at the prime <M>q</M> (see <Cite
Key="R" />). For division algebras whose Brauer class lies in the Schur
Subgroup of an abelian number field <M>F</M>, the local indices at any of
the primes <M>P</M> lying over the same rational prime <M>p</M> are equal
to the same positive integer <M>m_p</M>, and the numerator of the local
invariants among these primes are uniformly distributed among the integers
<M>r</M> coprime to <M>m_p</M> <Cite Key="BS"/>. <P/>
The local Schur index functions in wedderga produce a list of the nontrivial
local indices of the division algebra part of the simple algebra at all
rational primes. The Schur index of the simple algebra over <M>F</M> is the
least common multiple <M>m</M> of these local indices, and the dimension of
the division algebra part of the simple algebra over <M>F</M> is
<M>m^2</M>. While not sufficient to identify these division algebras up to
ring isomorphism in general, this list of local indices does identify the
division algebra up to ring isomorphism whenever there is no pair of local
indices at odd primes that are greater than 2. (This is at least the case
for groups of order less than 3^2*7*13.) So it gives the information
desired in most basic situations, and allows one to distinguish almost all
pairs of simple components of group algebras. <P/>
Wedderga's functions compute local indices for generalized quaternion
algebras defined over the rationals and cyclotomic algebras defined over
any abelian number field. Special shortcut functions are available for
cyclic cyclotomic algebras. There are also versions of the functions that
compute the local and global Schur index of a character of a finite group
over a given abelian number field. The steps in the general character-
theoretic method involve 1) a Brauer-Witt reduction to a cyclic-by-abelian
group, 2) use of the Frobenius-Schur indicator to compute the local index
at infinity, 3) computing the <M>p</M>-local index for an ordinary
irreducible character <M>\chi</M> of a <M>p</M>-solvable group using the
values of an irreducible Brauer character in the same <M>p</M>-block
in cases where the <M>p</M>-defect group of <M>\chi</M> is cyclic, and
4) use of Riese and Schmid's characterization of dyadic Schur groups
(<Cite Key="Sch"/> and <Cite Key="RSch"/>) to handle the exceptional cases
where step 3) is not available.
Our approach to rational quaternion algebras is the standard one given, for
example, in <Cite Key="Pi"/>. The Legendre symbol operation in GAP is used
to determine the local index at odd primes. The local index of the
generalized quaternion algebra <M>(a,b)</M> over <M>Q</M> at the infinite
prime will be <M>2</M> if both <M>a</M> and <M>b</M> are negative, and
otherwise <M>1</M>. We avoid the complicated case of quadratic reciprocity
when working over Q by using the Hasse-Brauer-Albert-Noether Theorem (<Cite
Key="R"/>, pg. 276): since we know the other primes of <M>Q</M> where the
local index is <M>2</M>, it determines the local index at the prime
<M>2</M>. For generalized quaternion algebras over number fields <M>F</M>
other than <M>Q</M>, we have to convert to cyclic or cyclic cyclotomic
algebras and use the other local index functions, or appeal to a number
theory system outside of GAP that can solve norm equations. <P/>
There are three shortcut functions used to compute local indices of cyclic
cyclotomic algebras, which wedderga's -Info functions produce in the form
<M>[r,F,n,[a,b,c]]</M>. The local index at infinity is calculated by
determining if the real completion of the corresponding algebra will produce a real quaternion algebra. In order to do this, <M>F</M> must be a
real subfield, <M>n</M> must be strictly greater than <M>2</M>, and
<M>E(n)^c</M> (which has to be a root of unity in <M>F</M>) must be
<M>-1</M>. These facts can be checked directly, so this is faster than
calculating the character table of the group and checking the value of a
Frobenius-Schur indicator.
The shortcut to calculate the local index of a cyclic cyclotomic algebra at
an odd prime makes direct use of the following lemma of Janusz: If
<M>E_p/F_p</M> is a Galois extension of <M>p</M>-local fields with
ramification index <M>e</M>, and <M>z</M> is a root of unity with order
prime to <M>p</M>, then <M>z</M> is a norm in <M>E_p/F_p</M> if and only
if it is the <M>e</M>-th power of a root of unity in <M>F</M>.
(<Cite Key="J"/>, pg. 535). It follows that in order to calculate the local
index at <M>p</M> of a cyclic cyclotomic algebra <M>[r,F,n,[a,b,c]]</M>, we
first determine the splitting degree, residue degree, and ramification
index <M>e</M> of the extension <M>F(\zeta_n)/F</M> at <M>p</M>. Comparing
the behaviour of the Galois automorphism <M>\sigma_b</M> to the behaviour
of the Frobenius automorphism at <M>p</M> allows us to determine the order
of the largest root of unity <M>z</M> with order coprime to <M>p</M> in the
<M>p</M>-completion <M>F_p</M>. The local index <M>m_p</M> is then the
least power of <M>E(n)^c</M> that lies in the group generated by
<M>z^e</M>. <P/> Calculation of the local index at the prime <M>2</M> makes
use of the following consequence of (<Cite Key="J"/>, Theorem 5): A cyclic
cyclotomic algebra <M>[r,F_2,n,[a,b,c]]</M> over a <M>2</M>-local field
<M>F_2</M> that is a subfield of a cyclotomic extension of the rational
<M>2</M>-local field <M>Q_2</M> has Schur index at most <M>2</M>. It has
Schur index <M>2</M> if and only if <M>4</M> divides <M>n</M>,
<M>F_2(\zeta_4)</M> is totally ramified of degree 2, the Galois automorphism <M>\sigma_b</M> of <M>F_2(\zeta_n)/F_2</M> inverts all
<M>2</M>-power roots of unity in <M>F_2(\zeta_n)</M>, the order of
<M>E(n)^c</M> is 2 times an odd number, and <M>(F_2:Q_2)</M> is odd. The
same approach to cyclotomic reciprocity makes it possible to check all of
these conditions in the <M>2</M>-local situation. <P/>
The wedderga function that computes the <M>p</M>-local index of an ordinary
irreducible character <M>\chi</M> of a finite non-nilpotent cyclic-by-
abelian group <M>G</M> is based directly on a theorem of Benard
<Cite Key="B"/> that applies whenever the <M>p</M>-defect group of
<M>\chi</M> is cyclic. We have to restrict our application of it to groups
whose orders are small because the &GAP; records for irreducible Brauer characters are only available in these cases. In order to use this approach effectively, we developed a function that computes the defect group of the block containing a given ordinary irreducible character <M>\chi</M>. This function makes use of the Min half of Brauer's Min-Max theorem (see Theorem 4.4 of ), and thus is able to find the defect group directly from the ordinary character table. It is thus available for nonsolvable groups, even in cases where &GAP;'s Brauer character records are not available. We are indebted to Michael Geline and Friederich Ladisch for discussions concerning the calculation of defect groups in &GAP;. The current algorithm we use is based on an approach suggested by Ladisch.
</Section>
<!-- *Realizing simple components as algebras with structure constants* -->
<Section Label="SCAlgebras">
<Heading> Obtaining Algebras with structure constants as terms of the Wedderburn decomposition </Heading>
Some users may find it desirable to have an alternative description for
the components of the Wedderburn decomposition of a group ring as algebras with structure constants, because the operations for algebras in &GAP; are designed for algebras with structure constants. We have provided such an algorithm that converts the output of
<Ref Oper="WedderburnDecompositionInfo"/> into algebras with structure constants. Matrix rings over fields are converted directly. For components that are cyclotomic algebras, it calculates their defining group and defining character using those &Wedderga; operations, then uses
<Ref Oper="IrreducibleRepresentationsDixon" BookName="ref"/> to obtain matrix generators of an algebra isomorphic to the simple component corresponding to the character over a suitable field. An algebra with structure constants version of this is finally obtained by applying
<Ref Oper="IsomorphismSCAlgebra" BookName="ref"/> to this algebra.
</Section>
<Section Label="TheoryPI">
<Heading>A complete set of orthogonal primitive idempotents</Heading>
When <M>R</M> is a semisimple ring, then every left ideal <M>L</M> of <M>R</M> is of the form <M>L=Re</M>,
where <M>e</M> is an idempotent of <M>R</M>. Therefore, we can use the idempotents to characterize the decompositions
of semisimple rings as a direct sum of minimal left ideals. In particular, let <M>R=\oplus_{i=1}^t L_i</M> be a decomposition
of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family <M>\{e_1,\dots,e_t\}</M> of elements
of <M>R</M> such that: each <M>e_i\neq 0</M> is an idempotent element, if <M>i\neq j</M>, then <M>e_ie_j=0</M>,
<M>1=e_1+\cdots+e_t</M> and each <M>e_i</M> cannot be written as <M>e_i=e_i'+e_i'', where e_i',e_i''</M>
are idempotents such that <M>e_i',e_i''\neq 0 and e_i'e_i''=0</M>, <M>1\leq i\leq </M>.
Conversely, if there exists a family of idempotents <M>\{e_1,\dots,e_t\}</M> satisfying the previous conditions,
then the left ideals <M>L_i=Re_i</M> are minimal and <M>R=\oplus_{i=1}^t L_i</M>. Such a set of idempotents is called
a <E>complete set of orthogonal primitive idempotents</E> of the ring <M>R</M>. Such a set is not uniquely determined.
<Index>Complete set of orthogonal primitive idempotents</Index> <P/>
Let <M>\mathbb F</M> be a finite field and <M>G</M> a finite nilpotent group such that <M>\mathbb F G</M> is semisimple.
Let <M>(H,K)</M> be a strong Shoda pair of <M>G</M>, <M>C\in\mathcal{C}(H/K)</M> and set <M>e_C=e_C(G,H,K)</M>,
<M>\varepsilon_C=\varepsilon_C(H,K)</M>, <M>H/K=\langle\overline{a}\rangle</M>, <M>E=E_G(H/K)</M>.
Let <M>E_2/K</M> and <M>H_2/K=\langle\overline{a_2}\rangle</M> (respectively <M>E_{2'}/K and
<M>H_{2'}/K=\langle\overline{a_{2'}}\rangle</M>) denote the 2-parts (respectively 2'-parts) of E/K and H/K
respectively. Then <M>\langle\overline{a_{2'}}\rangle has a cyclic complement \langle\overline{b_{2'}}\rangle</M>
in <M>E_{2'}/K.
Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra
<M>F G</M> (<Ref Sect="CyclotomicClass" />), a complete set of orthogonal primitive idempotents of <M>\mathbb F Ge_C</M> is described
(see <Cite Key="OV" />) as the set of conjugates of <M>\beta_{e_C}=\widetilde{b_{2'}}\beta_2\varepsilon_C by the elements of
<M>T_{e_C}=T_{2'}T_2T_E, where T_{2'}=\{1,a_{2'},a_{2'}^2,\dots,a_{2'}^{[E_{2'}:H_{2'}]-1}\}, T_E
denotes a right transversal of <M>E</M> in <M>G</M> and <M>\beta_2</M> and <M>T_2</M> are given according to the cases below.
<Enum>
<Item> If <M>H_2/K</M> has a complement <M>M_2/K</M> in <M>E_2/K</M> then <M>\beta_2=\widetilde{M_2}</M>. Moreover, if <M>M_2/K</M>
is cyclic, then there exists <M>b_2\in E_2</M> such that <M>E_2/K</M> is given by the following presentation
<Display>\langle \overline{a_2},\overline{b_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=\overline{b_2}\hspace{1pt}^{2^k}=1,
\overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r \rangle,</Display>
and if <M>M_2/K</M> is not cyclic, then there exist <M>b_2,c_2\in E_2</M> such that <M>E_2/K</M> is given by the
following presentation
<Display>\langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=
\overline{b_2}\hspace{1pt}^{2^k}=\overline{c_2}\hspace{1pt}^2=1,
\overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,
\overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1}, [\overline{b_2},\overline{c_2}]=1 \rangle,</Display>
with <M>r \equiv 1 {\rm \,mod\,} 4</M> (or equivalently <M>\overline{a_2}\hspace{1pt}^{2^{n-2}}</M> is central in <M>E_2/K</M>). Then
<Enum>
<Item> <M>T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1}\}</M>, if <M>\overline{a_2}\hspace{1pt}^{2^{n-2}}</M> is central in <M>E_2/K</M>
(unless <M>n\leq 1</M>) and <M>M_2/K</M> is cyclic; and </Item>
<Item> <M>T_2=\{1,a_2,a_2^2,\dots,a_2^{d/2-1},a_2^{2^{n-2}},a_2^{2^{n-2}+1},\dots,a_2^{2^{n-2}+d/2-1}\}</M>, where <M>d=[E_2:H_2]</M>,
otherwise. </Item>
</Enum>
</Item>
<Item> If <M>H_2/K</M> has no complement in <M>E_2/K</M>, then there exist <M>b_2,c_2\in E_2</M> such that <M>E_2/K</M> is given by
the following presentation
<Display> \langle \overline{a_2},\overline{b_2},\overline{c_2}\mid \overline{a_2}\hspace{1pt}^{2^n}=
\overline{b_2}\hspace{1pt}^{2^k}=1, \overline{c_2}\hspace{1pt}^2=\overline{a_2}\hspace{1pt}^{2^{n-1}},
\overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,\\
\overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1},[\overline{b_2},\overline{c_2}]=1 \rangle, </Display>
with <M>r\equiv 1 {\rm \,mod\,} 4</M>. In this case, <M>\beta_2=\widetilde{b_2}\frac{1+xa_2^{2^{n-2}}+ya_2^{2^{n-2}}c_2}{2}</M> and
<Display>T_2=\{1,a_2,a_2^2,\dots, a_2^{2^k-1},c_2,c_2a_2,c_2a_2^2,\dots,c_2a_2^{2^k-1}\},</Display>
with <M>x,y\in\mathbb F</M>, satisfying <M>x^2+y^2=-1</M> and <M>y\neq 0</M>.
</Item>
</Enum>
When <M>G</M> is not nilpotent, we can still use the following description in some specific cases.
Let <M>G</M> be a finite group and <M>\mathbb F</M> a finite field of order <M>s</M> such that <M>s</M> is coprime to the
order of <M>G</M>. Let <M>(H,K)</M> be a strong Shoda pair of <M>G</M> such that <M>\tau(gH,g'H)=1 for all
<M>g,g'\in E=E_G(H/K), and let C\in\mathcal{C}(H/K). Let \varepsilon=\varepsilon_C(H,K) and e=e_C(G,H,K)
(<Ref Sect="CyclotomicClass" />). Let <M>w</M> be a normal element of <M>\mathbb F_{s^o}/\mathbb F_{s^{o/[E:H]}}</M> (with <M>o</M>
the multiplicative order of <M>s</M> modulo <M>[H:K]</M>) and <M>B</M> the normal basis determined by <M>w</M>.
Let <M>\psi</M> be the isomorphism between <M>\mathbb F E \varepsilon</M> and the matrix algebra
<M>M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})</M> with respect to the basis <M>B</M> as stated in Corollary 29.8 in <Cite Key="R" />.
Let <M>P,A\in M_{[E:H]}(\mathbb F_{s^{o/[E:H]}})</M> be the matrices
<Display>
P= \left( \begin{array}{rrrrrr}
1 & 1 & 1 & \cdots & 1 & 1\\
1 & -1 & 0 & \cdots & 0 & 0\\
1 & 0 & -1 & \cdots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
1 & 0 & 0 & \cdots & -1 & 0\\
1 & 0 & 0 & \cdots & 0 & -1\\
\end{array} \right)
\quad {\rm and } \quad
A= \left( \begin{array}{ccccc}
0 & 0 & \cdots & 0 & 1\\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & \cdots & 0 & 0\\
0 & 0 & \cdots & 1 & 0\\
\end{array} \right).
</Display>
Then
<Display>
\{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\}
</Display>
is a complete set of orthogonal primitive idempotents of <M>\mathbb F G e</M> where <M>x_e=\psi^{-1}(PAP^{-1})</M>,
<M>T_1</M> is a transversal of <M>H</M> in <M>E</M> and <M>T_2</M> is a right transversal of <M>E</M> in <M>G</M>
(<Cite Key="OV2" />). By <M>\widehat{T_1}</M> we denote the element <M>\frac{1}{|T_1|}\sum_{t\in T_1}{t}</M> in <M>\mathbb F G</M>.
</Section>
<Section Label="codes">
<Heading>Applications to coding theory</Heading>
<Index>linear code</Index>
A <E>linear code</E> of length <M>n</M> and rank <M>k</M> is a linear subspace <M>C</M> with dimension <M>k</M>
of the vector space <M>\mathbb F_q^n</M>. The standard basis of <M>\mathbb F_q^n</M> is denoted by <M>E=\{e_1,...,e_n\}</M>.
The vectors in <M>C</M> are called codewords, the size of a code is the number of codewords and equals <M>q^k</M>.
The distance of a code is the minimum distance between distinct codewords, i.e. the number of elements in which they differ. <P/>
<Index>group code</Index>
For any group <M>G</M>, we denote by <M>\mathbb F_q G</M> the group algebra over <M>G</M> with coefficients in <M>\mathbb F_q</M>.
If <M>G</M> is a group of order <M>n</M> and <M>C\subseteq \mathbb F_q^n</M> is a linear code, then we say that <M>C</M>
is a left <M>G</M>-code (respectively a <M>G</M>-code) if there is a bijection <M>\phi:E\rightarrow G</M> such that the linear
extension of <M>\phi</M> to an isomorphism <M>\phi:\mathbb F_q^n\rightarrow \mathbb F_q G</M> maps <M>C</M> to a left ideal
(respectively a two-sided ideal) of <M>\mathbb F_q G</M>. A left <E>group code</E> (respectively a group code) is a linear code which
is a left <M>G</M>-code (respectively a <M>G</M>-code) for some group <M>G</M>. <P/>
Since left ideals in <M>\mathbb F_q G</M> are generated by idempotents, there is a one-one relation between (sums of)
primitive idempotents of <M>\mathbb F_q G</M> and left <M>G</M>-codes over <M>\mathbb F_q</M>.<P/>
Note that each element <M>c</M> in <M>\mathbb F_q G</M> is of the form <M>c=\sum_{i=1}^n f_i g_i</M>,
where we fix an ordering <M>\{g_1,g_2,...,g_n \}</M> of the group elements of <M>G</M> and <M>f_i\in \mathbb F_q</M>.
If one looks at <M>c</M> as a codeword, one writes <M>[f_1 f_2 ... f_n]</M>. <P/>
</Section>
</Chapter>
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