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<div class="ChapSects"><a href="chap9_mj.html#X840E625A81FDAEC6">9 The basic theory behind Wedderga</a>
<div class="ContSect"> <a href="chap9_mj.html#X815ECCD97B18314B">9.1 Group rings and group algebras</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7FDD93FB79ADCC91">9.2 Semisimple group algebras</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X84BB4A6081EAE905">9.3 Wedderburn components</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X87B6505C7C2EE054">9.4 Characters and primitive central idempotents</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7A24D5407F72C633">9.5 Central simple algebras and Brauer equivalence</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7FB21779832CE1CB">9.6 Crossed Products</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X828C42CD86AF605F">9.7 Cyclic Crossed Products</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7869E2A48784C232">9.8 Abelian Crossed Products</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X80BABE5078A29793">9.9 Classical crossed products</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X84C98BB8859BBEE2">9.10 Cyclic Algebras</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X8099A8C784255672">9.11 Cyclotomic algebras</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X84A142407B7565E0">9.12 Numerical description of cyclotomic algebras</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X8310E96086509397">9.13 Idempotents given by subgroups</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7D518BAB80EDE190">9.14 Shoda pairs of a group</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7E3479527BAE5B9E">9.15 Strong Shoda pairs of a group</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X81B5CE0378DC4913">9.16 Extremely strong Shoda pairs of a group</a>

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<div class="ContSect"> <a href="chap9_mj.html#X84C694978557EFE5">9.17 Strongly monomial characters and strongly monomial groups</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7C8D47C180E0ACAD">9.18 Normally monomial characters and normally monomial groups</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X800D8C5087D79DC8">9.19 Cyclotomic Classes and Strong Shoda Pairs</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X803562E087325AF6">9.20 Theory for Local Schur Index and Division Algebra Part Calculations</a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X7B18AF347AE68020">9.21 Obtaining Algebras with structure constants as terms of the Wedderburn decomposition </a>

</div>
<div class="ContSect"> <a href="chap9_mj.html#X8472ACCF802EC188">9.22 A complete set of orthogonal primitive idempotents</a>

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<div class="ContSect"> <a href="chap9_mj.html#X856D7975810BF987">9.23 Applications to coding theory</a>

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</div>

<h3>9 The basic theory behind Wedderga</h3>

In this chapter we describe the theory that is behind the algorithms used by Wedderga.

All the rings considered in this chapter are associative and have an identity.

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

<a id="X815ECCD97B18314B" name="X815ECCD97B18314B"></a>

<h4>9.1 Group rings and group algebras</h4>

Given a group \(G\) and a ring \(R\), the group ring \(RG\) over the group \(G\) with coefficients in \(R\) is the ring whose underlying additive group is a right \(R-\)module with basis \(G\) such that the product is defined by the following rule

\[
 (gr)(hs)=(gh)(rs)
 \]

for \(r,s \in R\) and \(g, h \in G\), and extended to \(RG\) by linearity.

A group algebra is a group ring in which the coefficient ring is a field.

<a id="X7FDD93FB79ADCC91" name="X7FDD93FB79ADCC91"></a>

<h4>9.2 Semisimple group algebras</h4>

We say that a ring \(R\) is semisimple if it is a direct sum of simple left (alternatively right) ideals or equivalently if \(R\) is isomorphic to a direct product of simple algebras each one isomorphic to a matrix ring over a division ring.

By Maschke's Theorem, if \(G\) is a finite group then the group algebra \(FG\) is semisimple if and only the characteristic of the coefficient field \(F\) does not divide the order of \(G\).

In fact, an arbitrary group ring \(RG\) is semisimple if and only if the coefficient ring \(R\) is semisimple, the group \(G\) is finite and the order of \(G\) is invertible in \(R\).

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.

<a id="X84BB4A6081EAE905" name="X84BB4A6081EAE905"></a>

<h4>9.3 Wedderburn components</h4>

If \(R\) is a semisimple ring (<a href="chap9_mj.html#X7FDD93FB79ADCC91">9.2</a>) then the Wedderburn decomposition of \(R\) is the decomposition of \(R\) as a direct product of simple algebras. The factors of this Wedderburn decomposition are called Wedderburn components of \(R\). Each Wedderburn component of \(R\) is of the form \(Re\) for \(e\) a primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) of \(R\).

Let \(FG\) be a semisimple group algebra (<a href="chap9_mj.html#X7FDD93FB79ADCC91">9.2</a>). If \(F\) has positive characteristic, then the Wedderburn components of \(FG\) are matrix algebras over finite extensions of \(F\). If \(F\) has zero characteristic then by the Brauer-Witt Theorem <a href="chapBib_mj.html#biBY">[Yam74]</a>, the Wedderburn components of \(FG\) are Brauer equivalent (<a href="chap9_mj.html#X7A24D5407F72C633">9.5</a>) to cyclotomic algebras (<a href="chap9_mj.html#X8099A8C784255672">9.11</a>).

The main functions of Wedderga compute the Wedderburn components of a semisimple group algebra \(FG\), 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.

In the zero characteristic case each Wedderburn component \(A\) is Brauer equivalent (<a href="chap9_mj.html#X7A24D5407F72C633">9.5</a>) to a cyclotomic algebra (<a href="chap9_mj.html#X8099A8C784255672">9.11</a>) and therefore \(A\) is a (possibly fractional) matrix algebra over cyclotomic algebra and can be described numerically in one of the following three forms:

\[
 [n,K],
 \]

\[
 [n,K,k,[d,\alpha,\beta]],
 \]

\[
 [n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le n} ],
 \]

where \(n\) is the matrix size, \(K\) is the centre of \(A\) (a finite field extension of \(F\)) and the remaining data are integers whose interpretation is explained in <a href="chap9_mj.html#X84A142407B7565E0">9.12</a>.

In some cases (for the zero characteristic coefficient field) the size \(n\) of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the Brauer-Witt Theorem <a href="chapBib_mj.html#biBY">[Yam74]</a> only ensures that each Wedderburn component (<a href="chap9_mj.html#X84BB4A6081EAE905">9.3</a>) of a semisimple group algebra is Brauer equivalent (<a href="chap9_mj.html#X7A24D5407F72C633">9.5</a>) to a cyclotomic algebra (<a href="chap9_mj.html#X8099A8C784255672">9.11</a>), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component \(D\) of a group algebra can be a division algebra but not a cyclotomic algebra. In this case \(M_n(D)\) is a cyclotomic algebra \(C\) for some \(n\) and therefore \(D\) can be described as \(M_{1/n}(C)\) (see last Example in <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7">2.1-1</a>)).

The main algorithm of Wedderga is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> which uses previous work by Olivieri, del Río and Simón <a href="chapBib_mj.html#biBORS">[OdRS04]</a> (see also <a href="chapBib_mj.html#biBOR">[OdR03]</a> ) for rational group algebras of strongly monomial groups (<a href="chap9_mj.html#X84C694978557EFE5">9.17</a>). The algorithms are also based upon the work of Bakshi and Maheshwary <a href="chapBib_mj.html#biBBM14">[BM14]</a> (see also <a href="chapBib_mj.html#biBBM16">[BM16]</a>) on the rational group algebras of normally monomial groups (<a href="chap9_mj.html#X7C8D47C180E0ACAD">9.18</a>).

<a id="X87B6505C7C2EE054" name="X87B6505C7C2EE054"></a>

<h4>9.4 Characters and primitive central idempotents</h4>

A primitive central idempotent of a ring \(R\) is a non-zero central idempotent \(e\) which cannot be written as the sum of two non-zero central idempotents of \(Re\), or equivalently, such that \(Re\) is indecomposable as a direct product of two non-trivial two-sided ideals.

The Wedderburn components (<a href="chap9_mj.html#X84BB4A6081EAE905">9.3</a>) of a semisimple ring \(R\) are the rings of the form \(Re\) for \(e\) running over the set of primitive central idempotents of \(R\).

Let \(FG\) be a semisimple group algebra (<a href="chap9_mj.html#X7FDD93FB79ADCC91">9.2</a>) and \(\chi\) an irreducible character of \(G\) (in an algebraic closure of \(F\)). Then there is a unique Wedderburn component \(A=A_F(\chi)\) of \(FG\) such that \(\chi(A)\ne 0\). Let \(e_F(\chi)\) denote the unique primitive central idempotent of \(FG\) in \(A_F(\chi)\), that is the identity of \(A_F(\chi)\), i.e.

\[
 A_F(\chi)=FGe_F(\chi).
 \]

The centre of \(A_F(\chi)\) is \(F(\chi)=F(\chi(g):g \in G)\), the field of character values of \(\chi\) over \(F\).

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

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

If the irreducible character \(\chi\) of \(G\) takes values in \(F\) then

\[
 e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1}) g.
 \]

In general one has

\[
 e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).
 \]

<a id="X7A24D5407F72C633" name="X7A24D5407F72C633"></a>

<h4>9.5 Central simple algebras and Brauer equivalence</h4>

Let \(K\) be a field. A central simple \(K\)-algebra is a finite dimensional \(K\)-algebra with center \(K\) which has no non-trivial proper ideals. Every central simple \(K\)-algebra is isomorphic to a matrix algebra \(M_n(D)\) where \(D\) is a division algebra (which is finite-dimensional over \(K\) and has centre \(K\)). The division algebra \(D\) is unique up to \(K\)-isomorphisms.

Two central simple \(K\)-algebras \(A\) and \(B\) are said to be Brauer equivalent, or simply equivalent, if there is a division algebra \(D\) and two positive integers \(m\) and \(n\) such that \(A\) is isomorphic to \(M_m(D)\) and \(B\) is isomorphic to \(M_n(D)\).

<a id="X7FB21779832CE1CB" name="X7FB21779832CE1CB"></a>

<h4>9.6 Crossed Products</h4>

Let \(R\) be a ring and \(G\) a group.

Intrinsic definition. A crossed product <a href="chapBib_mj.html#biBP">[Pas89]</a> of \(G\) over \(R\) (or with coefficients in \(R\)) is a ring \(R*G\) with a decomposition into a direct sum of additive subgroups

\[
 R*G = \bigoplus_{g \in G} A_g
 \]

such that for each \(g,h\) in \(G\) one has:

* \(A_1=R\) (here \(1\) denotes the identity of \(G\)),

* \(A_g A_h = A_{gh}\) and

* \(A_g\) has a unit of \(R*G\).

Extrinsic definition. Let \(Aut(R)\) denote the group of automorphisms of \(R\) and let \(R^*\) denote the group of units of \(R\).

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

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

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

The crossed product <a href="chapBib_mj.html#biBP">[Pas89]</a> of \(G\) over \(R\) (or with coefficients in \(R\)), action \(a\) and twisting \(t\) is the ring

\[
 R*_a^t G = \bigoplus_{g\in G} u_g R
 \]

where \(\{u_g : g\in G \}\) is a set of symbols in one-to-one correspondence with \(G\), with addition and multiplication defined by

\[
 (u_g r) + (u_g s) = u_g(r+s), \quad (u_g r)(u_h s) = u_{gh} t(g,h) r^h s
 \]

for \(g,h \in G\) and \(r,s\in R\), and extended to \(R*_a^t G\) by linearity.

The associativity of the product defined is a consequence of conditions (1) and (2) <a href="chapBib_mj.html#biBP">[Pas89]</a>.

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

Conversely, let \(R*G=\bigoplus_{g\in G} A_g\) be an (intrinsic) crossed product and select for each \(g\in G\) a unit \(u_g\in A_g\) of \(R*G\). This is called a basis of units for the crossed product \(R*G\). Then the maps \(a:G \rightarrow Aut(R)\) and \(t:G\times G \rightarrow R^*\) given by

\[
 r^g = u_g^{-1} r u_g, \quad t(g,h) = u_{gh}^{-1} u_g u_h \quad (g,h \in G, r \in R)
 \]

satisfy conditions (1) and (2) and \(R*G = R*_a^t G\).

The choice of a basis of units \(u_g \in A_g\) determines the action \(a\) and twisting \(t\). If \(\{u_g \in A_g : g \in G \}\) and \(\{v_g \in A_g : g \in G \}\) are two sets of units of \(R*G\) then \(v_g = u_g r_g\) for some units \(r_g\) of \(R\). 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.

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

<a id="X828C42CD86AF605F" name="X828C42CD86AF605F"></a>

<h4>9.7 Cyclic Crossed Products</h4>

Let \(R*G=\bigoplus_{g \in G} A_g\) be a crossed product (<a href="chap9_mj.html#X7FB21779832CE1CB">9.6</a>) and assume that \(G = \langle g \rangle \) is cyclic. Then the crossed product can be given using a particularly nice description.

Select a unit \(u\) in \(A_{g}\), and let \(a\) be the automorphism of \(R\) given by \(r^a = u^{-1} r u\).

If \(G\) is infinite then set \(u_{g^k} = u^k\) for every integer \(k\). Then

\[
 R*G = R[ u | ru = u r^a ],
 \]

a skew polynomial ring. Therefore in this case \(R*G\) is determined by

\[
[ R, a ].
\]

If \(G\) is finite of order \(d\) then set \(u_{g^k} = u^k\) for \(0 \le k < d\). Then \( b = u^d \in R \) and

\[
 R*G = R[ u | ru = u r^a, u^d = b ]
 \]

Therefore, \(R*G\) is completely determined by the following data:

\[
 [ R , [ d , a , b ] ]
 \]

<a id="X7869E2A48784C232" name="X7869E2A48784C232"></a>

<h4>9.8 Abelian Crossed Products</h4>

Let \(R*G=\bigoplus_{g \in G} A_g\) be a crossed product (<a href="chap9_mj.html#X7FB21779832CE1CB">9.6</a>) and assume that \(G\) is abelian. Then the crossed product can be given using a simple description.

Express \(G\) as a direct sum of cyclic groups:

\[
 G = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle
 \]

and for each \(i=1,\dots,n\) select a unit \(u_i\) in \(A_{g_i}\).

Each element \(g\) of \(G\) has a unique expression

\[
 g = g_1^{k_1} \cdots g_n^{k_n},
 \]

where \(k_i\) is an arbitrary integer, if \(g_i\) has infinite order, and \(0 \le k_i < d_i\), if \(g_i\) has finite order \(d_i\). Then one selects a basis for the crossed product by taking

\[
 u_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}.
 \]

* For each \(i=1,\dots, n\), let \(a_i\) be the automorphism of \(R\) given by \(r^{a_i} = u_i^{-1} r u_i\).

* For each \(1 \le i < j \le n\), let \(t_{i,j} = u_j^{-1} u_i^{-1} u_j u_i \in R\).

* If \(g_i\) has finite order \(d_i\), let \(b_i=u_i^{d_i} \in R\).

Then

\[
 R*G = R[u_1,\dots,u_n | ru_i = u_i r^{a_i}, u_j u_i = t_{ij} u_i u_j, u_i^{d_i} = b_i (1 \le i < j \le n) ],
 \]

where the last relation vanishes if \(g_i\) has infinite order.

Therefore \(R*G\) is completely determined by the following data:

\[
 [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n} ].
 \]

<a id="X80BABE5078A29793" name="X80BABE5078A29793"></a>

<h4>9.9 Classical crossed products</h4>

A classical crossed product is a crossed product \(L*_a^t G\), where \(L/K\) is a finite Galois extension, \(G=Gal(L/K)\) is the Galois group of \(L/K\) and \(a\) is the natural action of \(G\) on \(L\). Then \(t\) is a \(2\)-cocycle and the crossed product (<a href="chap9_mj.html#X7FB21779832CE1CB">9.6</a>) \(L*_a^t G\) is denoted by \((L/K,t)\). The crossed product \((L/K,t)\) is known to be a central simple \(K\)-algebra <a href="chapBib_mj.html#biBR">[Rei03]</a>.

<a id="X84C98BB8859BBEE2" name="X84C98BB8859BBEE2"></a>

<h4>9.10 Cyclic Algebras</h4>

A cyclic algebra is a classical crossed product (<a href="chap9_mj.html#X80BABE5078A29793">9.9</a>) \((L/K,t)\) where \(L/K\) is a finite cyclic field extension. The cyclic algebras have a very simple form.

Assume that \(Gal(L/K)\) is generated by \(g\) and has order \(d\). Let \(u=u_g\) be the basis unit (<a href="chap9_mj.html#X7FB21779832CE1CB">9.6</a>) of the crossed product corresponding to \(g\) and take the remaining basis units for the crossed product by setting \(u_{g^i} = u^i\), (\( i = 0, 1, \dots, d-1 \)). Then \(a = u^n \in K\). The cyclic algebra is usually denoted by \((L/K,a)\) and one has the following description of \((L/K,t)\)

\[
 (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].
 \]

<a id="X8099A8C784255672" name="X8099A8C784255672"></a>

<h4>9.11 Cyclotomic algebras</h4>

A cyclotomic algebra over \(F\) is a classical crossed product (<a href="chap9_mj.html#X80BABE5078A29793">9.9</a>) \((F(\xi)/F,t)\), where \(F\) is a field, \(\xi\) is a root of unity in an extension of \(F\) and \(t(g,h)\) is a root of unity for every \(g\) and \(h\) in \(Gal(F(\xi)/F)\).

The Brauer-Witt Theorem <a href="chapBib_mj.html#biBY">[Yam74]</a> asserts that every Wedderburn component (<a href="chap9_mj.html#X84BB4A6081EAE905">9.3</a>) of a group algebra is Brauer equivalent (<a href="chap9_mj.html#X7A24D5407F72C633">9.5</a>) (over its centre) to a cyclotomic algebra.

<a id="X84A142407B7565E0" name="X84A142407B7565E0"></a>

<h4>9.12 Numerical description of cyclotomic algebras</h4>

Let \(A=(F(\xi)/F,t)\) be a cyclotomic algebra (<a href="chap9_mj.html#X8099A8C784255672">9.11</a>), where \(\xi=\xi_k\) is a \(k\)-th root of unity. Then the Galois group \(G=Gal(F(\xi)/F)\) is abelian and therefore one can obtain a simplified form for the description of cyclotomic algebras as for any abelian crossed product (<a href="chap9_mj.html#X7869E2A48784C232">9.8</a>).

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

* If \(F(\xi)=F\), (i.e. \(G=1\)) then \(A=M_n(F)\) and thus the only data needed to describe \(A\) are the matrix size \(n\) and the field \(F\):

\[
 [n,F]
 \]

* If \(G\) is cyclic (but not trivial) of order \(d\) then \(A\) is a cyclic cyclotomic algebra

\[
 A = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ]
 \]

and so \(M_n(A)\) can be described with the following data

\[
 [n,F,k,[d,\alpha,\beta]],
 \]

where the integers \(k\), \(d\), \(\alpha\) and \(\beta\) satisfy the following conditions:

\[
 \alpha^d \equiv 1 \; mod \; k, \quad
 \beta(\alpha-1) \equiv 0 \; mod \; k.
 \]

* If \(G\) is abelian but not cyclic then \(M_n(A)\) can be described with the following data (see <a href="chap9_mj.html#X7869E2A48784C232">9.8</a>):

\[
 [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ]
 \]

representing the \(n \times n\) matrix ring over the following algebra:

\[
 A = F(\xi)[ u_1, \ldots, u_m \mid
 \xi u_i = u_i \xi^{\alpha_i}, \quad
 u_i^{d_i}=\xi^{\beta_i}, \quad
 u_s u_r = \xi^{\gamma_{rs}} u_r u_s, \quad
 i = 1, \ldots, m, \quad
 0 \le r < s \le m ]
 \]

where

* \(\{g_1,\ldots,g_m\}\) is an independent set of generators of \(G\),

* \(d_i\) is the order of \(g_i\),

* \(\alpha_i\), \(\beta_i\) and \(\gamma_{rs}\) are integers, and

\[
 \xi^{g_i} = \xi^{\alpha_i}.
 \]

<a id="X8310E96086509397" name="X8310E96086509397"></a>

<h4>9.13 Idempotents given by subgroups</h4>

Let \(G\) be a finite group and \(F\) a field whose characteristic does not divide the order of \(G\). If \(H\) is a subgroup of \(G\) then set

\[
\widehat{H} = |H|^{-1}\sum_{x \in H} x.
\]

The element \(\widehat{H}\) is an idempotent of \(FG\) which is central in \(FG\) if and only if \(H\) is normal in \(G\).

If \(H\) is a proper normal subgroup of a subgroup \(K\) of \(G\) then set

\[ \varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L})
\]

where \(L\) runs on the normal subgroups of \(K\) which are minimal among the normal subgroups of \(K\) containing \(N\) properly. By convention, \(\varepsilon(K,K)=\widehat{K}\). The element \(\varepsilon(K,H)\) is an idempotent of \(FG\).

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

If \((K,H)\) is a Shoda Pair (<a href="chap9_mj.html#X7D518BAB80EDE190">9.14</a>) of \(G\) then there is a non-zero rational number \(a\) such that \(ae(G,K,H))\) is a primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) of the rational group algebra \(ℚ G\). If \((K,H)\) is a strong Shoda pair (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) of \(G\) then \(e(G,K,H)\) is a primitive central idempotent of \(ℚ G\).

Assume now that \(F\) is a finite field of order \(q\), \((K,H)\) is a strong Shoda pair of \(G\) and \(C\) is a cyclotomic class of \(K/H\) containing a generator of \(K/H\). Then \(e_C(G,K,H)\) is a primitive central idempotent of \(FG\) (see <a href="chap9_mj.html#X800D8C5087D79DC8">9.19</a>).

<a id="X7D518BAB80EDE190" name="X7D518BAB80EDE190"></a>

<h4>9.14 Shoda pairs of a group</h4>

Let \(G\) be a finite group. A Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) for which there is a linear character \(\chi\) of \(K\) with kernel \(H\) such that the induced character \(\chi^G\) in \(G\) is irreducible. By <a href="chapBib_mj.html#biBS">[Sho33]</a> or <a href="chapBib_mj.html#biBORS">[OdRS04]</a>, \((K,H)\) is a Shoda pair if and only if the following conditions hold:

* \(H\) is normal in \(K\),

* \(K/H\) is cyclic and

* if \(K^g \cap K \subseteq H\) for some \(g \in G\) then \(g \in K\).

If \((K,H)\) is a Shoda pair and \(\chi\) is a linear character of \(K\le G\) with kernel \(H\) then the primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) of \(ℚ G\) associated to the irreducible character \(\chi^G\) is of the form \(e=e_ℚ (\chi^G)=a e(G,K,H)\) for some \(a \in ℚ \) <a href="chapBib_mj.html#biBORS">[OdRS04]</a> (see <a href="chap9_mj.html#X8310E96086509397">9.13</a> for the definition of \(e(G,K,H)\)). In that case we say that \(e\) is the primitive central idempotent realized by the Shoda pair \((K,H)\) of \(G\).

A group \(G\) is monomial, that is every irreducible character of \(G\) is monomial, if and only if every primitive central idempotent of \(ℚ G\) is realizable by a Shoda pair of \(G\).

<a id="X7E3479527BAE5B9E" name="X7E3479527BAE5B9E"></a>

<h4>9.15 Strong Shoda pairs of a group</h4>

A strong Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) satisfying the following conditions:

* \(H\) is normal in \(K\) and \(K\) is normal in the normalizer \(N\) of \(H\) in \(G\),

* \(K/H\) is cyclic and a maximal abelian subgroup of \(N/H\) and

* for every \(g \in G\setminus N\) , \(\varepsilon(K,H)\varepsilon(K,H)^g=0\). (See <a href="chap9_mj.html#X8310E96086509397">9.13</a> for the definition of \(\varepsilon(K,H)\)).

Let \((K,H)\) be a strong Shoda pair of \(G\). Then \((K,H)\) is a Shoda pair (<a href="chap9_mj.html#X7D518BAB80EDE190">9.14</a>) of \(G\). Thus there is a linear character \(\theta\) of \(K\) with kernel \(H\) such that the induced character \(\chi=\chi(G,K,H)=\theta^G\) is irreducible. Moreover the primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) \(e_{ℚ }(\chi)\) of \(ℚ G\) realized by \((K,H)\) is \(e(G,K,H)\), see <a href="chapBib_mj.html#biBORS">[OdRS04]</a>.

Two strong Shoda pairs (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) \((K_1,H_1)\) and \((K_2,H_2)\) of \(G\) are said to be equivalent if the characters \(\chi(G,K_1,H_1)\) and \(\chi(G,K_2,H_2)\) are Galois conjugate, or equivalently if \(e(G,K_1,H_1)=e(G,K_2,H_2)\). A set of representatives of strong Shoda pairs of \(G\) is termed as a complete irredundant set of strong Shoda pairs of \(G\).

The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra \(FGe_F(\chi)\) as a matrix algebra of a cyclotomic algebra (<a href="chap9_mj.html#X8099A8C784255672">9.11</a>, see <a href="chapBib_mj.html#biBORS">[OdRS04]</a> for \(F=ℚ \) and <a href="chapBib_mj.html#biBO">[Olt07]</a> for the general case).

More precisely, \(ℚ Ge(G,K,H)\) is isomorphic to \(M_n(ℚ (\xi)*_a^t N/K)\), where \(\xi\) is a \([K:H]\)-th root of unity, \(N\) is the normalizer of \(H\) in \(G\), \(n=[G:N]\) and \(ℚ (\xi)*_a^t N/K\) is a crossed product (see <a href="chap9_mj.html#X7FB21779832CE1CB">9.6</a>) with action \(a\) and twisting \(t\) given as follows:

Let \(x\) be a fixed generator of \(K/H\) and \(\varphi : N/K \rightarrow N/H\) a fixed left inverse of the canonical projection \(N/H\rightarrow N/K\). Then

\[
 \xi^{a(r)} = \xi^i, \mbox{ if } x^{\varphi(r)}= x^i
 \]

and

\[
 t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j,
 \]

for \(r,s \in N/K\) and integers \(i\) and \(j\), see <a href="chapBib_mj.html#biBORS">[OdRS04]</a>. Notice that the cocycle is the one given by the natural extension

\[
 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1
 \]

where \(K/H\) is identified with the multiplicative group generated by \(\xi\). Furthermore the centre of the algebra is \(ℚ (\chi)\), the field of character values over \(ℚ \), and \(N/K\) is isomorphic to \(Gal(ℚ (\xi)/ℚ (\chi))\).

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

<a id="X81B5CE0378DC4913" name="X81B5CE0378DC4913"></a>

<h4>9.16 Extremely strong Shoda pairs of a group</h4>

An extremely strong Shoda pair of \(G\) is a pair \((K,H)\) of subgroups of \(G\) satisfying the following conditions:

* \(K\) is normal in \(G\),

* \(K/H\) is cyclic and a maximal abelian subgroup of \(N/H\), where \(N\) is the normalizer of \(H\) in \(G\).

Let \((K,H)\) be an extremely strong Shoda pair of \(G\). Then \((K,H)\) is a strong Shoda pair (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) of \(G\), with \(K\) normal in \(G\) <a href="chapBib_mj.html#biBBM14">[BM14]</a>, so that there is a linear character \(\theta\) of \(K\) with kernel \(H\) such that the induced character \(\chi=\chi(G,K,H)=\theta^G\) is irreducible. Moreover, the primitive central idempotent \(e_{ℚ }(\chi)\) of \(ℚ G\) realized by \((K,H)\) is \(e(G,K,H)\) (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) and one can describe the associated simple algebra (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>). Two extremely strong Shoda pairs of \(G\) are said to be equivalent if they are equivalent as strong Shoda pairs (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>). A set of representatives of extremely strong Shoda pairs of \(G\) is called a complete irredundant set of extremely strong Shoda pairs of \(G\) <a href="chapBib_mj.html#biBBM14">[BM14]</a>.

If \(G\) is a normally monomial group (<a href="chap9_mj.html#X7C8D47C180E0ACAD">9.18</a>), then the set of primitive central idempotents of the rational group algebra realized by strong Shoda pairs of \(G\) is same as the one realized by extremely strong Shoda pairs of \(G\) <a href="chapBib_mj.html#biBBM14">[BM14]</a>. The algorithm to compute a complete irredundant set of extremely strong Shoda pairs of \(G\) has been explained in <a href="chapBib_mj.html#biBBM16">[BM16]</a>.

<a id="X84C694978557EFE5" name="X84C694978557EFE5"></a>

<h4>9.17 Strongly monomial characters and strongly monomial groups</h4>

Let \(G\) be a finite group an \(\chi\) an irreducible character of \(G\).

One says that \(\chi\) is strongly monomial if there is a strong Shoda pair (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) \((K,H)\) of \(G\) and a linear character \(\theta\) of \(K\) of \(G\) with kernel \(H\) such that \(\chi=\theta^G\).

The group \(G\) is strongly monomial if every irreducible character of \(G\) is strongly monomial.

Strong Shoda pairs where firstly introduced by Olivieri, del Río and Simón who proved that every abelian-by-supersolvable group is strongly monomial <a href="chapBib_mj.html#biBORS">[OdRS04]</a>. The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in <a href="chapBib_mj.html#biBOR">[OdR03]</a>. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in <a href="chapBib_mj.html#biBBR">[BdR07]</a> (see Section <a href="chap9_mj.html#X800D8C5087D79DC8">9.19</a>). Finally, Olteanu <a href="chapBib_mj.html#biBO">[Olt07]</a> shows how to compute the Wedderburn decomposition (<a href="chap9_mj.html#X84BB4A6081EAE905">9.3</a>) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of \(G\) but also the strong Shoda pairs of the subgroups of \(G\).

<a id="X7C8D47C180E0ACAD" name="X7C8D47C180E0ACAD"></a>

<h4>9.18 Normally monomial characters and normally monomial groups</h4>

Let \(G\) be a finite group and \(\chi\) be an irreducible character of \(G\).

One says that \(\chi\) is normally monomial if there is a normal subgroup \(K\) of \(G\) such that \(\chi\) is induced from a linear character of \(K\).

The group \(G\) is normally monomial if every irreducible character of \(G\) is normally monomial. Bakshi and Maheshwary proved that if \(G\) is a normally monomial group, then for every irreducible character \(\chi\) of \(G\), there exists an extremely strong Shoda pair \((K,H)\) of \(G\) (<a href="chap9_mj.html#X81B5CE0378DC4913">9.16</a>) such that \(\chi=\theta^G\), where \(\theta\) is a linear character of \(K\) with kernel \(H\) <a href="chapBib_mj.html#biBBM14">[BM14]</a>.

.

<a id="X800D8C5087D79DC8" name="X800D8C5087D79DC8"></a>

<h4>9.19 Cyclotomic Classes and Strong Shoda Pairs</h4>

Let \(G\) be a finite group and \(F\) a finite field of order \(q\), coprime to the order of \(G\).

Given a positive integer \(n\), coprime to \(q\), the \(q\)-cyclotomic classes modulo \(n\) are the set of residue classes module \(n\) of the form

\[
 \{i,iq,iq^2,iq^3, \dots \}
 \]

The \(q\)-cyclotomic classes module \(n\) form a partition of the set of residue classes module \(n\).

A generating cyclotomic class module \(n\) is a cyclotomic class containing a generator of the additive group of residue classes module \(n\), or equivalently formed by integers coprime to \(n\).

Let \((K,H)\) be a strong Shoda pair (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) of \(G\) and set \(n=[K:H]\). Fix a primitive \(n\)-th root of unity \(\xi\) in some extension of \(F\) and an element \(g\) of \(K\) such that \(gH\) is a generator of \(K/H\). Let \(C\) be a generating \(q\)-cyclotomic class modulo \(n\). Then set

\[
 \varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1} tr(\xi^{-ci})g^i,
 \]

where \(c\) is an arbitrary element of \(C\) and \(tr\) is the trace map of the field extension \(F(\xi)/F\). Then \(\varepsilon_C(K,H)\) does not depend on the choice of \(c \in C\) and is a primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) of \(FK\).

Finally, let \(e_C(G,K,H)\) denote the sum of the different \(G\)-conjugates of \(\varepsilon_C(K,H)\). Then \(e_C(G,K,H)\) is a primitive central idempotent (<a href="chap9_mj.html#X87B6505C7C2EE054">9.4</a>) of \(FG\) <a href="chapBib_mj.html#biBBR">[BdR07]</a>. We say that \(e_C(G,K,H)\) is the primitive central idempotent realized by the strong Shoda pair \((K,H)\) of the group \(G\) and the cyclotomic class \(C\).

If \(G\) is strongly monomial (<a href="chap9_mj.html#X84C694978557EFE5">9.17</a>) then every primitive central idempotent of \(FG\) is realizable by some strong Shoda pair (<a href="chap9_mj.html#X7E3479527BAE5B9E">9.15</a>) of \(G\) and some cyclotomic class \(C\) <a href="chapBib_mj.html#biBBR">[BdR07]</a>. As in the zero characteristic case, this explain how to compute the Wedderburn decomposition (<a href="chap9_mj.html#X84BB4A6081EAE905">9.3</a>) of \(FG\) for a finite semisimple algebra of a strongly monomial group (see <a href="chapBib_mj.html#biBBR">[BdR07]</a> for details). For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.

.

<a id="X803562E087325AF6" name="X803562E087325AF6"></a>

<h4>9.20 Theory for Local Schur Index and Division Algebra Part Calculations</h4>

(By Allen Herman, May 2013. Updated October 2014.)

The division algebra parts of simple algebras in the Wedderburn Decomposition of the group algebra of a finite group over an abelian number field \(F\) correspond to elements of the Schur Subgroup \(S(F)\) of the Brauer group of \(F\). Like all classes in the Brauer group of an algebraic number field \(F\), the division algebra part of a representative of a given Brauer class is determined up to \(F\)-algebra isomorphism by its list of local Hasse invariants at all primes (i.e. places) of \(F\). The local invariant at a prime \(P\) of \(F\) is a lowest terms fraction \(r/m_P\) whose denominator is the local Schur index \(m_P\) of the simple algebra at the prime \(q\) (see <a href="chapBib_mj.html#biBR">[Rei03]</a>). For division algebras whose Brauer class lies in the Schur Subgroup of an abelian number field \(F\), the local indices at any of the primes \(P\) lying over the same rational prime \(p\) are equal to the same positive integer \(m_p\), and the numerator of the local invariants among these primes are uniformly distributed among the integers \(r\) coprime to \(m_p\) <a href="chapBib_mj.html#biBBS">[BS72]</a>.

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 \(F\) is the least common multiple \(m\) of these local indices, and the dimension of the division algebra part of the simple algebra over \(F\) is \(m^2\). 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.

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