  
  [1m[4m[31m7. The basic theory behind [1mWedderga[1m[4m[31m[0m
  
  In this chapter we describe the theory that is behind the algorithms used by
  [1mWedderga[0m.
  
  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 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.
  
  
  [1m[4m[31m7.1 Group rings and group algebras[0m
  
  Given  a  group  G  and  a  ring  R, the [22m[36mgroup ring[0m 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 [22m[36mgroup algebra[0m is a group ring in which the coefficient ring is a field.
  
  
  [1m[4m[31m7.2 Semisimple group algebras[0m
  
  A  ring  R  is  [22m[36msemisimple  artinian[0m  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 the group algebra FG is semisimple artinian if and only
  if  the  group G is finite and the characteristic of the coefficient field F
  does not divide the order of G.
  
  
  [1m[4m[31m7.3 Wedderburn decomposition[0m
  
  If  R  is  a [22m[36msemisimple ring[0m ([1m7.2[0m) then the [22m[36mWedderburn decomposition[0m of R is
  the  decomposition  of R as a direct product of simple algebras. The factors
  of this Wedderburn decomposition are called [22m[36mWedderburn components[0m of R. Each
  Wedderburn  component  of  R  is  of  the  form Re for e a [22m[36mprimitive central
  idempotent[0m ([1m7.4[0m) of R.
  
  Let   FG   be   a   [22m[36msemisimple  group  algebra[0m  ([1m7.2[0m).  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
  [22m[36mBrauer-Witt  Theorem[0m  [Y74],  the  [22m[36mWedderburn  components[0m  of  FG are [22m[36mBrauer
  equivalent[0m ([1m7.5[0m) to [22m[36mcyclotomic algebras[0m ([1m7.11[0m).
  
  The  main  functions  of  [1mWedderga[0m  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 components A is [22m[36mBrauer
  equivalent[0m  ([1m7.5[0m)  to  a  [22m[36mcyclotomic  algebra[0m  ([1m7.11[0m)  and  therefore A is a
  (possibly  fractional)  matrix  algebras over [22m[36mcyclotomic algebras[0m 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 [1m7.12[0m.
  
  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 [22m[36mBrauer-Witt Theorem[0m [Y74]
  only  ensures  that  each  [22m[36mWedderburn  component[0m ([1m7.3[0m) of a semisimple group
  algebra  is  Brauer equivalent ([1m7.5[0m) to a [22m[36mcyclotomic algebra[0m ([1m7.11[0m), 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 [1m[34mWedderburnDecomposition[0m ([1m2.1-1[0m)).
  
  The main algorithm of [1mWedderga[0m is based in a computational oriented proof of
  the  Brauer-Witt  Theorem  due  to  Olteanu  [O] which uses previous work by
  Olivieri,  del Río and Simón [ORS04] for rational group algebras of [22m[36mstrongly
  monomial groups[0m ([1m7.16[0m).
  
  
  [1m[4m[31m7.4 Characters and primitive central idempotents[0m
  
  A  [22m[36mprimitive central idempotent[0m 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  [22m[36mWedderburn components[0m ([1m7.3[0m) of a semisimple ring R are the rings of the
  form Re for e running on the set of primitive central idempotents of R.
  
  Let FG be [22m[36msemisimple group algebra[0m ([1m7.2[0m) 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 [22m[36mfield of character
  values[0m of chi over F.
  
  The map chi -> 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 -> 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
  
  \[
       e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G}
       \chi(g\inv) g.
  \]
  
  In general one has
  
  \[
       e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).
  \]
  
  
  [1m[4m[31m7.5 Central simple algebras and Brauer equivalence[0m
  
  Let  K  be  a  field.  A  [22m[36mcentral  simple  K-algebra[0m is a finite dimensional
  K-algebra  with  center  K  which  has  not 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 [22m[36mBrauer equivalent[0m, or
  simple  [22m[36mequivalent[0m,  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).
  
  
  [1m[4m[31m7.6 Crossed Products[0m
  
  Let R be a ring and G a group.
  
  [1m[46mIntrinsecal  definition[0m.  A  [22m[36mcrossed  product[0m  [P89]  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 RG.
  
  [1m[46mExtrinsecal  definition[0m.  Let  Aut(R) denote the group of automorphisms of R
  and let R^* denote the group of units of R.
  
  Let  a:G  ->  Aut(R)  and  t:G  times  G  ->  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-> 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  [22m[36mcrossed product[0m [P89] 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  : gin G is a set of symbols in one-to-one correspondence with G,
  with the 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,sin R, and extended to R*_a^t G by linearity.
  
  The  associativity of the product defined is a consequence of conditions (1)
  and (2) [P89].
  
  [1m[46mEquivalence  of the two definitions[0m. Obviously the crossed product of G over
  R  defined  using  the extrinsecal 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 -> u_1 r_0 r is a
  ring isomorphism R -> u_1R.
  
  Conversely,  let  R*G=bigoplus_gin G A_g be an (intrinsecal) crossed product
  and select for each gin G a unit u_gin A_g of R*G. This is called a [22m[36mbasis of
  units  for the crossed product[0m R*G. Then the maps a:G -> Aut(R) and t:Gtimes
  G -> 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  election  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  action  an twisting and so change the
  extrinsecal  definition  of  the  crossed product but it does not change the
  intrinsecal 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.
  
  
  [1m[4m[31m7.7 Cyclic Crossed Products[0m
  
  Let R*G=bigoplus_g in G A_g be a [22m[36mcrossed product[0m ([1m7.6[0m) and assume that G = <
  g  >  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:
  
  \[
       [ R , [ d , a , b ] ]
  \]
  
  
  [1m[4m[31m7.8 Abelian Crossed Products[0m
  
  Let  R*G=bigoplus_g in G A_g be a [22m[36mcrossed product[0m ([1m7.6[0m) 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:
  
  \[
       G = < g_1 > \times \cdots \times < g_n >
  \]
  
  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 an 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 vanish if g_i has infinite order.
  
  Therefore R*G is completely determined by the following data:
  
  \[
       [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n}
       ].
  \]
  
  
  [1m[4m[31m7.9 Classical crossed products[0m
  
  A  [22m[36mclassical  crossed  product[0m 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/E and a is the
  natural  action  of  G  on  L. Then t is a 2-cocycle and the [22m[36mcrossed product[0m
  ([1m7.6[0m)  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 [R03].
  
  
  [1m[4m[31m7.10 Cyclic Algebras[0m
  
  A [22m[36mcyclic algebra[0m is a [22m[36mclassical crossed product[0m ([1m7.9[0m) (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  ([1m7.6[0m)  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)
  
  \[
       (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].
  \]
  
  
  [1m[4m[31m7.11 Cyclotomic algebras[0m
  
  A   [22m[36mcyclotomic   algebra[0m  over  F  is  a  [22m[36mclassical  crossed  product[0m  ([1m7.9[0m)
  (F(xi)/F,t),  where F is a field, xi is a root of unity on an extension of F
  and t(g,h) is a root of unity for every g and h in Gal(F(xi)/F).
  
  The  [22m[36mBrauer-Witt Theorem[0m [Y74] asserts that every [22m[36mWedderburn component[0m ([1m7.3[0m)
  of  a  group  algebra  is  [22m[36mBrauer  equivalent[0m  ([1m7.5[0m)  (over its centre) to a
  cyclotomic algebra.
  
  
  [1m[4m[31m7.12 Numerical description of cyclotomic algebras[0m
  
  Let  A=(F(xi)/F,t)  be  a [22m[36mcyclotomic algebra[0m ([1m7.11[0m), 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 [22m[36mabelian crossed product[0m ([1m7.8[0m).
  
  Then the n-by-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:
  
  \[
       \alpha^d \equiv 1 \; \text{mod} \; k, \quad \beta(\alpha-1) \equiv
       0 \; \text{mod} \; k
  \]
  
  *  If  G  is  abelian  but  not cyclic then M_n(A) can be described with the
  following data (see [1m7.8[0m):
  
  \[
       [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j
       \le m} ]
  \]
  
  representing the n-by-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,...,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
  
  \[
       \xi^{g_i} = \xi^{\alpha_i}.
  \]
  
  
  [1m[4m[31m7.13 Idempotents given by subgroups[0m
  
  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  widehatH  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
  
  \[
       \varepsilon(K,H) = \prod_{L} (\widehat{H}-\widehat{L}).
  \]
  
  where  L runs on the minimal normal subgroups of K containing H properly. By
  convention,  varepsilon(K,K)=widehatK.  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  ([1m7.14[0m) of G then there is a non-zero rational
  number a such that ae(G,K,H)) is a [22m[36mprimitive central idempotent[0m ([1m7.4[0m) of the
  rational group algebra Q G. If (K,H) is a strong Shoda pair ([1m7.15[0m) 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 [1m7.17[0m).
  
  
  [1m[4m[31m7.14 Shoda pairs[0m
  
  Let G be a finite group. A [22m[36mShoda pair[0m 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 [S33] or [ORS04], (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 Kle G with kernel
  H  then  the  [22m[36mprimitive  central  idempotent[0m  ([1m7.4[0m) 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  [ORS04] (see [1m7.13[0m for the definition of e(G,K,H)). In that case we
  say  that  e  is the [22m[36mprimitive central idempotent realized by the Shoda pair[0m
  (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.
  
  
  [1m[4m[31m7.15 Strong Shoda pairs[0m
  
  A  [22m[36mstrong  Shoda  pair[0m 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\ N , varepsilon(K,H)varepsilon(K,H)^g=0. (See [1m7.13[0m for
  the definition of varepsilon(K,H)).
  
  Let  (K,H) be a strong Shoda pair of G. Then (K,H) is a Shoda pair ([1m7.14[0m) 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
  [22m[36mprimitive  central  idempotent[0m  ([1m7.4[0m)  e_Q (chi) of Q G realized by (K,H) is
  e(G,K,H), see [ORS04].
  
  Two  [22m[36mstrong  Shoda  pairs[0m ([1m7.15[0m) (K_1,H_1) and (K_2,H_2) of G are said to be
  [22m[36mequivalent[0m  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).
  
  The  advantage of strong Shoda pairs with respect to Shoda pairs is that one
  can  describe  the  simple  algebra  FGe_F(chi)  as  a  matrix  algebra of a
  [22m[36mcyclotomic algebra[0m ([1m7.11[0m, see [ORS04] for F=Q and [O] 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 [22m[36mcrossed product[0m (see [1m7.6[0m) with action a and twisting t
  given as follows:
  
  Let  x  be  a  fixed  generator  of K/H and varphi : N/K -> N/H a fixed left
  inverse of the canonical projection N/H-> 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 [ORS04]. 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  identify  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 by 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))).
  
  
  [1m[4m[31m7.16 Strongly monomial characters and strongly monomial groups[0m
  
  Let G be a finite group an chi an irreducible character of G.
  
  One  says  that  chi  is  [22m[36mstrongly  monomial[0m if there is a [22m[36mstrong Shoda pair[0m
  ([1m7.15[0m)  (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  [22m[36mstrongly  monomial[0m 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
  [ORS04].  The  algorithm to compute the Wedderburn decomposition of rational
  group  algebras  for  strongly monomial groups was explained in [OR03]. This
  method  was  extended  for semisimple finite group algebras by Broche Cristo
  and  del  Río  in [BR] (see Section [1m7.17[0m). Finally, Olteanu [O] shows how to
  compute  the [22m[36mWedderburn decomposition[0m ([1m7.3[0m) 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.
  
  .
  
  
  [1m[4m[31m7.17 Cyclotomic Classes and Strong Shoda Pairs[0m
  
  Let  G  be  a finite group and F a finite field of order q, coprime with the
  order of G.
  
  Given  a positive integer n, coprime with q, the q-[22m[36mcyclotomic classes[0m 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  [22m[36mgenerating  cyclotomic class [0m module n is a cyclotomic class containing a
  generator of the additive group of residue classes module n, or equivalently
  formed by integers coprime with n.
  
  Let  (K,H)  be  a  strong  Shoda  pair  ([1m7.15[0m)  of  G and set n=[K:H]. Fix a
  primitive  n-th root of the 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 election of
  c in C and it is a [22m[36mprimitive central idempotent[0m ([1m7.4[0m) 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 [22m[36mprimitive central idempotent[0m ([1m7.4[0m)
  of  FG  [BR].  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 [22m[36mstrongly monomial[0m ([1m7.16[0m) then every primitive central idempotent of
  FG  is  realizable by some [22m[36mstrong Shoda pair[0m ([1m7.15[0m) of G and some cyclotomic
  class  C  [BR].  As  in  the  zero  characteristic case, this explain how to
  compute  the  [22m[36mWedderburn  decomposition[0m  ([1m7.3[0m) of FG for a finite semisimple
  algebra  of  a  strongly  monomial  group  (see  [BR]  for details). For non
  strongly   monomial   groups   the   algorithm  to  compute  the  Wedderburn
  decomposition just uses the Brauer characters.
  
